List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I A. Nissen and G. Kreiss. An optimized perfectly matched layer for theSchrödinger equation, Commun. Comput. Phys. 9:147–179, 2011. 1

The ideas were developed in close collaboration between the authors. Theauthor of this thesis performed part of the analysis and all the computations.The manuscript was written in close cooperation between the authors.

II K. Kormann and A. Nissen. Error control for simulations of a dissocia-tive quantum system, In Numerical mathematics and advanced appli-cations: 2009, pp 523-531, Springer-Verlag, Berlin, 2010. 2

The author of this thesis was mainly responsible for the parts regarding thePML. Both authors performed the computations, wrote the manuscript anddrew conclusions in close collaboration.

III A. Nissen, H. O. Karlsson and G. Kreiss. A perfectly matched layerapplied to a reactive scattering problem, J. Chem. Phys. 133:054306,2010. 3

The author of this thesis implemented the methods, performed all thecomputations and had the main responsibility for preparing the manuscript.The ideas were developed in collaboration between the authors.

1With permission from Global Science Press.2With kind permission from Springer Science and Business Media.3With permission from American Institute of Physics.

IV A. Nissen, G. Kreiss and M. Gerritsen. High order stable finite dif-ference methods for the Schrödinger equation, Technical report 2011-014, Department of Information Technology, Uppsala University, 2011.(Submitted)

V A. Nissen, G. Kreiss and M. Gerritsen. Stability at nonconforming gridinterfaces for a high order discretization of the Schrödinger equation,Technical report 2011-017, Department of Information Technology,Uppsala University, 2011. (Submitted)

The ideas were developed in close collaboration between the authors.The author of this thesis had the main responsibility for the theoreticaldevelopment of the stability analysis, performed most of the detailed analysisand all the computations. The manuscript was written in close cooperationbetween the authors.

VI M. Gustafsson, A. Nissen and K. Kormann. Stable difference methodsfor block-structured adaptive grids, Technical report 2011-022, Depart-ment of Information Technology, Uppsala University, 2011.

The ideas were developed in close collaboration between the authors inthis paper. The author of this thesis contributed mainly in the development ofthe numerical method and wrote a part of the manuscript. The computationswere performed in close collaboration between the authors.

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 The time-dependent Schrödinger

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 The Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . 102.2 Bound states and non-bound states . . . . . . . . . . . . . . . . . . . . . . 11

3 Numerical methods and computational challenges . . . . . . . . . . . . . 134 Boundary treatment for open systems . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Absorbing layers in quantum dynamics . . . . . . . . . . . . . . . . . . 184.2 Absorbing layers and absorbing boundary

conditions in the numerical analysis community . . . . . . . . . . . . 204.2.1 Perfectly matched layers . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Spatial adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.1 Adaptive mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Summation-by-parts operators . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.6 Paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1. Introduction

The investigation of the dynamics of chemical reactions, both from the theo-retical and experimental side, has drawn an increasing interest since AhmedH. Zewail was awarded the 1999 Nobel Prize in Chemistry for his work onfemtochemistry. On the experimental side, new techniques such as femtosec-ond lasers and attosecond lasers enable laser control of chemical reactions.Numerical simulations serve as a valuable complement to experimental tech-niques, not only for validation of experimental results, but also for simulationof processes that cannot be investigated through experiments. With increas-ing computer capacity, more and more physical phenomena fall within therange of what is possible to simulate. Also, the development of new, efficientnumerical methods further increases the possibilities.

The focus of this thesis is twofold; numerical methods for open quantummechanical systems and methods that can handle problems with large varia-tions in spatial scales. Firstly, we consider numerical methods for open quan-tum mechanical systems, in particular chemical reactions involving dissocia-tive states. An open quantum mechanical system is a system in interactionwith its surroundings. It could be a nano-electric device, into which electronsenter and leave, where the positions of the electrons can only be describedas a time-dependent probability distribution. Dissociative chemical reactionsare reactions where molecules break up into smaller components. The disso-ciation can occur spontaneously, e.g. by radioactive decay, or be induced byadding energy to the system, e.g. in terms of a laser field. Quantities of interestcan for instance be the reaction probabilities of possible chemical reactions.

Secondly, we consider methods that can deal with coexistence of spatialregions with very different physical properties. One application of interest islong-range molecules, where the atoms are affected by chemical attractiveforces that lead to fast movement in the region where they are close to eachother and exhibits a relatively slow motion of the atoms in the long-rangeregion. The ability of the method to adapt to different scales is important inthe study of more complex chemical systems.

The key equation in the mathematical modeling of quantum mechanics isthe Schrödinger equation, which is defined on an infinite spatial domain. Mostnumerical models involve the introduction of a finite computational domainand boundary conditions. In the context of dissociative problems we focusspecifically on the boundary treatment. In order to avoid the introduction ofadditional errors that are due to an ad hoc imposition of boundary conditions,

7

we use an analytic approach to impose the boundary conditions. Papers I, IIand III in this thesis are based on the perfectly matched layer (PML) tech-nique, which stands on a well-founded mathematical ground, while it is alsovery similar to methods employed in the field of chemical physics. We inves-tigate the limitations and quantify the errors introduced by the PML.

In order to deal with different spatial scales, the computational domaincan be decomposed into smaller subdomains where the computational workis distributed according to the work needed to resolve the physical featuresin each subdomain. Our approach for the domain decomposition is based onthe summation-by-parts-simultaneous approximation term (SBP-SAT) frame-work, a finite difference methodology for which robust and accurate approxi-mations often can be achieved. In papers IV and V we consider the numericaltreatment of the boundaries surrounding the computational domain and of theartificial boundaries between subdomains. The accuracy and robustness of thenumerical method is investigated. In paper VI the SBP-SAT methodology isextended to multiblock-domains.

The outline of the thesis is as follows. The underlying equations and prob-lem formulation are described in section 2. Numerical techniques and chal-lenges are reviewed in section 3. In section 4, we describe different ways ofimposing absorbing boundary conditions for the Schrödinger equation. Sec-tion 5 deals with adaptive discretizations in space and the numerical treatmentacross interior grid boundaries. A summary of the papers included in the thesisis given in section 6, and section 7 concludes with a discussion and an outlineof possible future work.

8

2. The time-dependent Schrödingerequation

Fundamental processes at the atomic level cannot be modelled by Newtonianmechanics, but require a quantum mechanical description. Theoretical under-standing of such processes, like the dynamics of chemical reactions, can beobtained by solving the time-dependent Schrödinger equation (TDSE),

ih∂ψ(x, t)

∂ t= Hψ(x, t), (2.1)

where the Hamiltonian of the system, H = − h2

2m∆ +V , consists of a kineticand a potential energy operator. h is the reduced Planck’s constant and m isthe mass of the system. The TDSE (2.1) was introduced in 1926 in a seriesof papers by Erwin Schrödinger [72, 74, 71, 73] and gives a comprehensivedescription of a physical system through the space- and time-dependent wave-function ψ(x, t). Although the physical interpretation of ψ(x, t) is not trivial,all obtainable information is accessible through it. However, the informationis limited due to the uncertainty principle. Instead of considering the complex-valued ψ(x, t) directly, the probability density |ψ(x, t)|2 gives the likelihoodof finding the system in a particular state for the given x at time t. In a one-particle system, x denotes the position of the particle in space. For a largersystem, e.g. a polyatomic molecule, x describes the spatial relation betweenthe particles, usually in terms of distances and angles between the particles.

Analytic solutions are in most cases not obtainable for the TDSE, and there-fore we need to turn to numerical simulations in order to obtain approximatesolutions. However, to directly solve the TDSE numerically is unfeasible fornearly all real systems. A seemingly small carbon dioxide molecule (CO2),consists of 25 particles (3 nuclei and 22 electrons) and gives rise to a prob-lem with 75 degrees of freedom. In addition, the time-scales for the relativelylight electrons and the heavy nuclei differ with several orders of magnitude.Due to the high requirements of computational resources in order to solve theTDSE for both nuclei and electrons, the interaction induced by the electronsis in practical computations reduced to potential energy surfaces through theBorn-Oppenheimer approximation, which will be described in the followingsection.

9

2.1 The Born-Oppenheimer approximationThe Born-Oppenheimer approximation was derived by Born andOppenheimer [14] in 1927, and is still a crucial tool in quantum chemistryand chemical physics. By separating the nuclear and the electronic partsof the wavefunction and solving the Schrödinger equation for each partseparately, the dimensionality of the problem can be reduced significantly.The Born-Oppenheimer approximation is derived starting from thetime-independent Schrödinger equation

H(x,X)Ψ(x,X) = EΨ(x,X), (2.2)

where the full molecular Hamiltonian can be written as

H(x,X) = ∑i−

h2∇2

e,i

2m+ ∑

j>i

e2

|xi− x j|+∑

i−

h2∇2

N,i

2Mi+

∑j>i

ZiZ je2

|Xi−X j|−∑

i j

Z je2

|xi−X j|≡ Te +Ve +TN +VN +VeN ,

(cf. [81]). x and X refer to the electron and nuclear coordinates, respectively,e to the elementary charge and Zi to the charge on nucleus i. m is the electronmass, Mi the mass of nucleus i, and ∇e and ∇N refer to the electron and nuclearmomenta, respectively. By assuming that the full wavefunction Ψ(x,X) can beseparated into an electronic part at a fixed nuclear position X, ϕ(x;X), and anuclear part, χ(X), we get

Ψ(x,X) = ϕ(x;X)χ(X), (2.3)

where Ψ(x,X) is an energy eigenfunction in the full coordinate space and asolution to (2.2). In the first step, the electronic eigenvalue problem,

Heϕ(x;X) = E(X)ϕ(x;X), He = Te +Ve +VeN ,

is solved for various values of X, yielding E(X) as a function of the nu-clear coordinates X. Substituting (2.3) into (2.2) leads to the time-independentSchrödinger equation for the nuclear wavefunction,

H(x,X)χ(X)≈ [TN +E(X)+VN(X)]χ(X),

by neglecting the two terms with derivatives of the electronic wavefunctionwith respect to the nuclear coordinates. The omission of these terms is knownas the Born-Oppenheimer approximation. Once the effective potential un-der which the nuclei move, V (X) = E(X)+VN(X), is determined, the Born-Oppenheimer approximation allows us to solve (2.1) separately for the nuclearcoordinates. In terms of the CO2 example in the previous section, the influenceof the 22 electrons has been modeled into the potential V , leaving 9 degrees of

10

freedom corresponding to the 3 nucei remaining. Moreover, 5 of these involvetranslation and rotation of the linear CO2 molecule, and thus only 4 degreesof freedom are sufficient to describe the internal motion of the atoms.

The task of computing the effective potentials, or potential energy surfaces,is cumbersome and an important branch of quantum chemistry [24]. However,here we are interested in solving the nuclear TDSE and we will assume thatthe potential energy surfaces are given.

2.2 Bound states and non-bound statesThe TDSE (1) is defined on an infinite spatial domain, which needs to be trun-cated and supplied with boundary conditions and initial data for the purposeof numerical simulation. When a system involves particles that do not separateunless sufficient energy is added there are so-called bound states, i.e. localizedeigenfunctions with corresponding discrete eigenvalues. At low energy onlybound states are involved in the dynamics of the system, and the wavefunc-tion will also be localized and naturally confined to a smaller domain. Forexample, the internal vibration of a diatomic molecule can be modeled by aharmonic oscillator. Then the range of the most probable distance betweenthe two particles corresponds to the distance between the turning points ofthe harmonic oscillator. Hence, the domain can be truncated closely beyondthe turning points, where the localized probability distribution is vanishinglysmall, with e.g. periodic boundary conditions.

In many cases the wave function is localized in space and exhibits fast os-cillations compared to the length scale of the domain of interest. For suchproblems it is important to include a sufficiently large region to capture inter-esting phenomena, and yet to resolve the high spatial frequencies. For compu-tational efficiency it is convenient with a high grid density in areas with highfrequency modes and a lower grid density where the frequency componentsare slower. Different ways of adapting the grid to the physical properties areconsidered in this thesis.

Many interesting problems involve not only bound states, but also the con-tinuous spectrum of the Hamiltonian. Investigating chemical reactions involv-ing dissociation, where compounds break up into smaller subsystems e.g. byinteraction with a laser field, is one example of interest. Another is calculatingthe energies of resonance states. These correspond to eigenfunctions whichare not square integrable, unlike the eigenfunctions corresponding to the dis-crete spectrum of the bound states. In the time-dependent setting, solutionswith energies in the continuous spectrum lead to waves that travel out fromthe designated computational domain, as the distance between the scatteredparticles increases. As a result of the outgoing waves, reflections are gener-ated unless adequate boundary conditions are used. In this thesis we discussdifferent boundary models employed to absorb outgoing waves for the time-

11

dependent Schrödinger equation, and how they fit in with commonly usedspatial and temporal discretizations. Although our focus is on time-dependentdynamics, we stress that similar techniques are of importance and can be usedalso for time-independent problems. In that case they can e.g. provide a meansto find the energies of the resonance states.

12

3. Numerical methods andcomputational challenges

Applying the Born-Oppenheimer approximation to the full description of amolecule, where both the spatial coordinates of the electrons and the nucleiare accounted for, considerably reduces the dimensionality of the problem.

However, the dimensionality of the nuclear TDSE increases with the num-ber of atoms in the molecule, and a straightforward discretization will result ina very high number of degrees of freedom. This is due to the so-called curse ofdimensionality, caused by the exponential increase in grid-points as a functionof the number of independent variables. As of today, only problems up to a fewspatial dimensions are within reach of direct solution of the TDSE. Thus, in or-der to undertake large-scale problems, further approximations have been nec-essary. Approaches used to tackle high-dimensional problems include MultiConfiguration Time Dependent Hartree (MCTDH) [57] and Hagedorn wavepackets [29]. However, these methods will not be addressed further here. Inthis thesis we focus on how to expand the range of simulations based directlyon the TDSE. Efficient numerical methods are essential. Moreover, they mustbe incorporated into a powerful parallel implementation framework [37, 15].

Discretization of (2.1) usually follows the method of lines, i.e. the spa-tial discretization is first introduced and the resulting system of ODEs canbe solved by some suitable time-stepping method. The standard spatial dis-cretizations in chemical physics are based on pseudospectral methods [31],e.g. the Fourier method and sinc-DVR [22]. Pseudospectral methods cap-ture the dispersion relation correctly and accurate solutions can often be ob-tained with relatively few grid points. However, pseudospectral methods areglobal approximations, which in one spatial dimension leads to a full spa-tial discretization matrix. In the multi-dimensional case, the matrix is not full,but block-structured according to the independent variables. The popularityof pseudospectral methods is due to that efficient implementations can beachieved in combination with the fast Fourier transform (FFT).

An alternative approach to pseudospectral methods is finite differencemethods. The locality of finite differences is advantageous for parallelimplementations, since the communication between processors is reduced,compared to the pseudospectral case. However, unlike for pseudospectralmethods, the dispersion relation is not accurately captured for finitedifference methods. Gray and Goldfield [34] propose dispersion-fitted finitedifference methods, especially for problems where low accuracy is tolerable.

13

Moreover, finite differences are flexible in terms of boundary conditions andspatial adaptivity, whereas pseudospectral methods are not. Large variationsin the solution to the Schrödinger equation are often concentrated to smallparts of the full computational domain, and spatial adaptivity can thus serveas a way of enhancing the computational efficiency. The global nature ofthe basis functions makes it difficult to combine a pseudospectral approachwith an adaptive spatial discretization, whereas the local support of thefinite difference stencils makes them well suited to combine with local meshrefinement. We will elaborate on this topic in chapter 5.

Finite element methods have also been used for the Schrödinger equation,although to a lesser extent than finite difference methods and pseudospec-tral methods. Efficient implementations are more difficult to achieve for finiteelement methods. On the other hand, the stability properties can be advanta-geous. For example, Antoine and Besse employ a finite element discretizationin space in order to preserve the stability for their numerical scheme, see [4].For an adaptive finite element discretization of the TDSE in one spatial di-mension, Dörfler derived a posteriori error estimates [27].

As already mentioned in the context of open systems, a computational chal-lenge is to truncate the unbounded domain of (2.1), without destroying theaccuracy of the model. In the next section we will present boundary treatmentprocedures that address this issue. As a first step we will briefly describe somemethods for temporal discretization and how restrictions can be inflicted uponthem by the boundary treatment.

Time-propagation methods developed and used for the TDSE have beentailored to share some properties with the physical system, to be detailed be-low. If there is no explicit time-dependence in the Hamiltonian operator H,the solution to (2.1) with initial condition ψ(x, t0) is given exactly at time t by

ψ(x, t) = U(t− t0)ψ(x, t0),

where

U(t− t0) = e−iH(t−t0)/h (3.1)

is the evolution operator. Primarily, the numerical approximation of the evo-lution operator should be unitary. Unitarity implies norm-preservation of thewavefunction, i.e. the integral∫

∞

−∞

ψ(x, t)∗ψ(x, t)dx = 1, (3.2)

holds for all times. Here, ∗ denotes the complex conjugate. Suppose that thewavefunction describes the state of a single particle, then (3.2) corresponds tothe probability of finding the particle anywhere in space being equal to one.

Secondly, the TDSE is time-reversible and that should be true also for thenumerical time-propagation. By retaining the exponential form of (3.1) for

14

the temporal discretization, both unitarity and time-reversibility can be pre-served. Besides methods based on an exponential form, partitioned Runge-Kutta methods are commonly used to propagate the Schrödinger equation intime. Partitioned Runge-Kutta methods own the advantage of being symplec-tic, a property which is described in [69]. A discussion of time-propagationmethods is found in [51]. In [46], different propagators for the TDSE with anexplicitly time-dependent Hamiltonian are compared.

Kormann et al. [47] developed an h, p-adaptive Magnus-Lanczos algorithmwith global error control for systems with explicitly time-dependent Hamilto-nians. They use the truncated Magnus expansion to average the Hamiltonianoperator. Instead of direct application of the matrix exponential of the dis-cretized Hamiltonian, a subspace spanned by a few dominating eigenvectorsis generated by the Lanczos algorithm [25], and used as a substitute for thediscretized Hamiltonian in (3.1). This method is used in both Paper II andIII along with a high-order finite difference spatial discretization and a per-fectly matched layer (PML) for boundary truncation. A three-state systemwith explicit time-dependence in terms of a laser pulse is treated with thefull propagator in Paper II. In Paper III the time-independent version is ap-plied to a dissociative problem for which the Hamiltonian is only spatiallydependent. However, the Hamiltonian with PML boundary treatment is nolonger Hermitian. This is generally the case when including boundary mod-eling for dissociative states. Hence, the use of numerical methods developedfor bound states are not always directly applicable to dissociative problems.For example, explicit partitioned Runge-Kutta methods are not available forHamiltonians with complex entries [45]. In Paper II and III, the Lanczos al-gorithm is substituted by the Arnoldi algorithm for general matrices to enablethe application to dissociative problems.

The SBP-SAT spatial discretizations considered in Papers IV-VI do not giverise to a Hermitian discretized Hamiltonian in the regular l2 norm. However, inmost cases it is Hermitian in specific norms compatible with the SBP operatorsand Lanczos iteration can be used if the scalar product in the algorithm isbased on the SBP norms. In Paper IV-VI the Lanczos algorithm is used forthe temporal propagation in the numerical experiments for which we have thesymmetry described above.

15

4. Boundary treatment for opensystems

Consider the one-dimensional Schrödinger equation with compactlysupported initial conditions and a decay condition at infinity,

ih∂ψ(x,t)∂ t =− h2

2m∂ 2ψ(x,t)

∂x2 +V (x, t)ψ(x, t) = Hψ(x, t),lim|x|→∞ ψ(x, t) = 0,

ψ(x,0) = ψ0(x).

(4.1)

In order to perform numerical simulations, the infinite domain needs to betruncated to a finite computational domain and closed with boundary condi-tions. We are looking for boundary conditions for which the solution on thecomputational domain coincides with the solution on the infinite domain. Thisis obtained by truncating the domain beyond the compact support of the ini-tial data at a point where the potential V is constant, and imposing boundaryconditions that absorb waves that leave the computational domain. At large,there are two types of boundary treatments that can be used for this purpose,absorbing boundary conditions (ABCs) and absorbing layers. An ABC is im-posed exactly at the numerical boundary, while an absorbing layer is a bufferzone where the equation is modified so that waves are damped immediatelyoutside the domain of interest. The absorbing layer is then terminated with astable boundary condition, at some point outside the interior domain where theoutgoing waves are sufficiently damped. A two-dimensional computationaldomain surrounded by an absorbing layer is illustrated in figure 4.1.

17

Figure 4.1: Computational domain (dark gray) extended with an absorbing layer (lightgray).

Most methods for absorbing boundary treatment in the field of chemicalphysics are based on absorbing layers. In the field of numerical analysis, on theother hand, the work on absorbing layers for the TDSE is rather limited com-pared to the work on ABCs [87]. Also in the cases where similar approacheshave been used, there are surprisingly few references between the two commu-nities, as Hein et al. [39] point out. Increased communication between the twocommunities is of potential benefit for the development of efficient boundarytreatment models for problem settings of interest in the quantum chemistrycommunity.

In this section we review the techniques of absorbing boundary treatmentthat have been used for the Schrödinger equation, from the quantum chemist’sand the numerical analyst’s point of view, respectively.

4.1 Absorbing layers in quantum dynamicsThe vast majority of absorbing boundary methods in quantum dynamics arebased on absorbing layers. The most common approach is the complex ab-sorbing potential (CAP), where a complex potential −iW (x) is added to theoperator on the right hand side of (4.1),

ih∂ψ(x, t)

∂ t=− h2

2m∂ 2ψ(x, t)

∂x2 +V (x, t)ψ(x, t)− iW (x)ψ(x, t).

18

W (x) is typically zero everywhere, except near the numerical boundarywhere energy is absorbed by letting the real part of W (x) be positive.The CAP method has been used extensively in both time-dependent andtime-independent scattering problems [61, 66, 75], which to a large part isdue to the simplicity of the method and its compatibility with pseudospectralmethods. Much effort has been devoted to finding effective and optimizedCAPs for different applications, by minimizing the reflection and thetransmission coefficients [83, 67, 63]. The reflection and transmissioncoefficients describe the relation between the amplitudes of an incidentwave and the waves that are reflected on and transmitted through the CAP,respectively. By minimizing the sum of the two the absorption is maximized.However, the reflection of the CAP increases with increasing frequency,while the transmission is reduced with increasing frequency. Thus, a CAPcan only be optimized for certain frequencies. Also, numerical reflectionsarise from the region where the CAP is active, and pollute the solutionin the interior. A transmission-free CAP was derived by Manolopoulos[52, 32], who used a pole at the end of the absorbing region to eliminateall transmission. Since only minimization of the reflection coefficient isrequired, the transmission-free CAP is not as frequency-dependent as aregular CAP.

An elegant and mathematically rigorous approach for dealing with the con-tinuum in the eigenvalue spectrum is complex scaling of the spatial coordinate[1, 6, 76]. Complex scaling involves an analytical continuation into the com-plex plane,

x→ xeiθ , (4.2)

where θ is a positive and real constant. Simon [76] showed that the result-ing complex scaled Hamiltonian, Hθ (x) = H(xeiθ ), has complex eigenvaluesthat are identified as resonances. The method of complex scaling providesa means of dealing with both bound states and resonance states through thesame formalism [59], since the eigenfunctions of Hθ are square-integrable. Al-though most frequently used in time-independent settings [16, 58], Bengtssonet al. [9] recently used complex scaling for the time-dependent Schrödingerequation. The scaling of the Hamiltonian through (4.2) implies scaling of apotential energy surface. This might be difficult, for instance if the potentialis given as a set of ab initio points instead of an analytic expression. Exteriorcomplex scaling (ECS) circumvents this issue to a large extent, by introducingthe rotation into the complex plane beyond some x = x0, where the potentialis close to constant. For instance, if the potential is given as a set of points inthe interaction region, but analytically continued in the asymptotic region, it ispossible to perform the coordinate transformation for the potential for x≥ x0.McCurdy et al. [56] recommend ECS for practical use of complex scalingin time-dependent problems. A similar approach to ECS is smooth exterior

19

scaling (SES). The difference between ECS and SES consists of the transitionto the complex plane being simply rotated or rotated through a smooth tran-sition function. In the continuous setting, the SES interface is non-reflecting,whereas the ECS interface is not. From a numerical point of view, the smooth-ness of the transition function is of importance in order to avoid numerical re-flections from the absorbing layer. This is discussed in Paper III. The relationbetween CAP and SES is investigated both in [68] and [42].

4.2 Absorbing layers and absorbing boundaryconditions in the numerical analysis communityIn the numerical analysis community, most of the effort on absorbing bound-ary techniques for the TDSE has been directed towards absorbing boundaryconditions. A recent and extensive review on absorbing boundary techniquesfor the TDSE, mainly treating ABCs, was written by Antoine et al [3]. Ashorter comparison is given by Yevick et al [86]. Also, the dissertations ofJiang [41], and Schädle [70], deal with absorbing boundary conditions for theSchrödinger equation.

An exact transparent boundary condition for (4.1), is a boundary conditionfor which the solution on the computational domain coincides exactly with theinfinite space solution on the computational domain. It is in general difficult tofind an exact transparent boundary condition in the case of a space- and time-dependent potential. However, an exact condition can be derived by assumingthat V (x, t) = V is constant for |x| ≥ x0. This condition can for instance beexpressed as the Dirichlet-to-Neumann (DtN) map

∂nψ(x, t) =−e−π4 i√

πe−iVt d

dt

∫ t

0

ψ(x,τ)eiV τ

√t− τ

dτ, (4.3)

see [3]. The imposition of (4.3) at x = ±x0 solves the problem of reducingthe infinite domain of (4.1) to [−x0,x0] completely. However, (4.3) is a globalcondition in time for the one-dimensional problem. In the multi-dimensionalcase, the corresponding relation is global both in time and space. In order toavoid the global coupling, (4.3) is often approximated to a local condition intime, which naturally introduces errors. Also, numerical errors are generatedas an effect of the spatial discretization.

The Crank-Nicolson scheme is frequently used to discretize (4.1)on the truncated domain [−x0,x0] with absorbing boundary conditions.Crank-Nicolson is unconditionally stable, unitary and time-reversible,which makes it suitable for solving the Schrödinger equation. However,Crank-Nicolson is rather expensive due to it being an implicit method. InPaper II, we investigate the performance of the Crank-Nicolson schemecompared to that of the Magnus-Arnoldi propagator [47], and conclude

20

that the latter is much more efficient, at least in the case of an explicitlytime-dependent Hamiltonian. Even though the Crank-Nicolson discretizationis unconditionally stable in the interior, caution needs to be taken in theprocess of imposing the boundary conditions. An unfortunate discretizationof the boundary conditions may lead to the full discretization being unstable,which was discovered by Mayfield [55] and Baskakov and Popov [7].

An alternative approach is to first discretize (4.1) and thereafter find anexact transparent boundary condition for the semi-discrete or fully discreteformulation of (4.1). Antoine and Besse [4] used a conform finite elementsubspace for the spatial discretization in order to preserve the stability oftheir fully discrete numerical scheme including absorbing boundary condition,which was constructed for the semi-discrete equation in time. The approachof Ehrhardt and Arnold [28] was to first discretize both in space and time,using the Crank-Nicolson scheme. An exact transparent boundary conditionwas then derived for the fully discretized scheme, retaining the stability of theinterior scheme. The trade-off for their non-reflecting boundary condition isthe lack of flexibility. The discretization needs to be uniform, and extension tomore than one dimension is a difficult matter.

4.2.1 Perfectly matched layersThe approach we employ for boundary treatment throughout this thesis is theperfectly matched layer (PML). The PML is an absorbing layer, for whicha modified set of equations are solved in the layer. The significant featureof the PML in contrast to a general absorbing layer is the perfect matching,which guarantees that the continuous PML is non-reflecting at the interface tothe interior domain for all frequencies, and all incoming angles in the multi-dimensional case. However, discretization introduces numerical reflections atthe interface.

The PML method was developed for Maxwell’s equations by Berenger in1994 [10]. It has since then become the standard method in electromagnetics,and is widely used for hyperbolic problems in general [5, 8, 77]. The PMLapproach has been used also for the Schrödinger equation [2, 23], but theextent of the investigations is far from the extent of the analysis of PMLs forhyperbolic equations. A common approach for the TDSE is the modal ansatzPML [38], where the starting point of the PML derivation is modal solutions inthe frequency domain. An alternative way is to introduce a coordinate change[20], as in the case of ECS and SES.

For most hyperbolic problems, the modified equations in the frequency do-main can not be transformed back into the time domain without the introduc-tion of auxiliary variables, which leads to additional equations in the system.However, no auxiliary variables need to be introduced for the Schrödingerequation. Thus, when solving the TDSE with PML no additional equationneeds to be solved. Extra computational effort may be required, compared

21

to the solution with ABC, due to the additional grid points in the layer. Onthe other hand, compared to ABC the PML formulation is easier to extend tomulti-dimensional problems.

Application of PMLs have also been made to the nonlinear TDSE by Zhengin [87] and by Dohnal [26], who constructed a PML for a system of two-dimensional coupled nonlinear Schrödinger equations with mixed derivatives.Due to the mixed derivatives, waves with opposite phase and group velocitiesare supported. For many hyperbolic problems, this would lead to unstable lay-ers, in the sense of exponentially growing solutions for the continuous prob-lem, see [8]. Here, it only leads to a stability condition on the layer parameters.However, stability analysis for Schrödinger PMLs is an area of research whichneeds more consideration.

In order to derive a PML for (4.1), we need to make the assumption that thepotential is independent of x, i.e. V (x, t) = V (t). For the multi-dimensionalcase, the requirement is that the potential in the layer is independent of thenormal direction in which the PML is imposed, but it may depend on otherspatial variables. By performing the coordinate transformation

x→ x+ eiγ∫ x

±x0

σ(ω)dω, |x| ≥ x0 (4.4)

in (4.1), we arrive at the PML equationih∂ψ(x,t)

∂ t =− h2

2m1

1+eiγ σ(x)∂

∂x

(1

1+eiγ σ(x)∂ψ(x,t)

∂x

)+V (t)ψ(x, t),

ψ(−x0−d, t) = ψ(x0 +d, t) = 0,

ψ(x,0) = ψ0(x),

(4.5)

on the domain [−x0−d,x0 +d], after domain truncation and imposing Dirich-let boundary conditions. The absorption function σ(x) is a nonnegative, realfunction in [−x0−d,−x0] ∪ [x0,x0 +d], and zero in [−x0,x0], and γ should bein the range 0≤ γ ≤ π

2 for well-posedness, as discussed in Paper I.Perfect matching means that there are no reflections at the interface. This is

achieved if σ(±x0) = 0, since it implies continuity of ψ and ψx at the inter-face. In the continuous setting, the damping introduced by the PML dependsonly on the value of the integral in (4.4), and on the parameter γ . Hence, alarge value of the integral results in a large damping, and allows the use of athin layer. However, the case is different for the discretized problem and σ(x)needs to be chosen carefully in order to avoid numerical reflections. Thus,a balance between a sufficiently steep slope of σ(x), such that the width ofthe PML can be kept small, and a sufficiently smooth function in order toavoid numerical reflections is advisable. Such an optimization process wasperformed by Sjögreen and Petersson for Maxwell’s equation [77]. We em-ploy a similar approach in Paper I for optimization of the Schrödinger PML.

22

Equation (4.5) can be written as a complex symmetric expression, see [42].This form, often used for SES, is easily implemented with high-order finitedifferences in space, which is exploited in Paper I-III. In contrast to the SESapproach the potential is not analytically continued into the complex plane.Except for this difference the PML and the SES approaches are equivalent, asdiscussed in Paper I.

23

5. Spatial adaptivity

Many problems in chemical physics demonstrate large local variations of spa-tial scales where a uniform discretization either leads to unneccessary refine-ment in parts of the computational domain or to unresolved physical featuresin the fine scale regions. For applications that exhibit such variety in the spa-tial scales, local grid refinement can lead to considerable savings of computermemory and computational time. Also, the limit of feasible problems can bepushed further, especially for problems of high dimensionality.

The development of techniques for spatial adaptivity is an important branchof scientific computing and a considerable amount of work has been done forinstance in computational fluid dynamics. Spatial adaptivity for the TDSE hasbeen investigated to a much lesser extent. One reason for this may be thecommonly used pseudospectral methods for the spatial discretization, whichdue to the global nature of the basis functions are difficult to combine withlocal mesh refinement. The majority of the work done on adaptive grid baseddiscretizations for the TDSE is either based on a coordinate transformation forthe continuous equation, or on a combination of semi-classical and quantummechanical methods.

Fattal et al. [30] developed a method where a mapping of the wave functionoptimizes the phase space of the new wave function to fit with the Fouriermethod, and applied it in the context of time-independent problems includinga Coulomb potential. Their approach is very similar to the complex scaling co-ordinate transformation described in the previous chapter, with the distinctionthat the coordinate transformation in the mapped Fourier method is along thereal line instead of along a complex contour. Kokoouline et al. [44] extendedthe method to a broader class of long-range potentials. The mapped Fouriermethod was later extended to time-dependent problems by Kleinekathöfer andTannor [43] and applied in numerical simulations up to three dimensions.They claim that a three-dimensional calculation would be out of reach forthe original Fourier method, but also remark on the limitations of the methodwhen it comes to dealing with dissociative problems, as well as problemsfor which the spectral components of the wave packet change in time. In thetime-independent setting similar methods have been developed that also in-clude the treatment of resonances [85, 21], and could possibly be extended totime-dependent dissociative problems.

Pettey and Wyatt [62] use a combination of a semi-classical and a quan-tum mechanical approach. In their hybrid method a uniform, fixed grid fol-

25

lows the wave packet by moving the boundary points surrounding the fixedmesh. The boundary conditions are determined using a semi-classical trajec-tory method and the uniform grid is discretized using a fourth order finitedifference method.

In order to deal with more general situations and distribute the grid pointsaccording to the temporal evolution of the wave packet, a more flexible tech-nique would be advantageous. Local approximations such as finite differencemethods or finite element methods could suit this purpose well.

In the following section we describe adaptive mesh refinement techniquesthat have been successful for example for problems in computational fluiddynamics. We continue by describing the methodology that we suggest asa building block in an adaptive mesh refinement framework for the TDSE.The work on spatial adaptivity for the TDSE in this thesis has primarily beenfocused on the numerical aspects, with an efficient parallel implementation ofa dynamically evolving adaptive discretization in mind.

5.1 Adaptive mesh refinementAdaptive mesh refinement (AMR) techniques use computational resources ef-ficiently by adapting the computational grid dynamically in time. Grid pointsare automatically added in regions where the solution changes rapidly, andremoved from parts of the computational domain that no longer need a highergrid resolution.

When discretizing complicated domains flexible approaches that can adaptto complex geometries have been successful. One way is to use unstructuredmeshes to represent the geometry of the domain [84]. An alternative techniqueis overlapping grids [40], where the domain is discretized using structuredcomposite grids that overlap. Typically, boundary-fitted curvilinear grids areused to discretize regions close to boundaries. If the domains are simple, as isthe case for the TDSE, a structured approach is beneficial. Structured meshescan be very efficient since the relations between neighboring grid points canbe inferred from a less complicated data structure than in the case of unstruc-tured meshes [64]. Additionally, unstructured meshes can often lead to loss inaccuracy in comparison to structured meshes [60].

Berger and Oliger [13] developed an approach for structured adaptive meshrefinement (SAMR) based on the idea of multiple component grids for finitedifference methods. Each grid point for which the numerical solution does notmeet a user-defined local error tolerance is marked and regions that cover themarked points are refined. In the original method the alignment of the refinedregions could vary with respect to the underlying grid patches. Berger andColella [12] proposed a modified method where the boundaries of the refinedregions are required to be parallel to the underlying grid. To allow for simplerdata structures and reduce overhead for grid-fitting, block-structured versions

26

of SAMR have been developed [78]. In the block-structured approach, refine-ment is carried out concurrently for all points in a, typically rectangular, gridpatch if the local error tolerance is not met for any point belonging to that gridpatch. The block decomposition in Paper VI is based on a block-structured ap-proach, where each grid block contains the same number of grid points. Sucha division of grid blocks facilitates the load balancing.

From a computer science perspective, block structured AMR has many ad-vantages that can lead to an efficient implementation if the communicationbetween processors can be limited. From a numerical analysis point of viewa high order accurate discretization that leads to a stable numerical approx-imation is desirable. Berger [11] used normal mode analysis to investigatestability at a two-dimensional grid interface for the advection-diffusion equa-tion for different finite difference approximations of low order. For high or-der discretizations this type of analysis often becomes very techniqual and itmay be unfeasible to derive analytic stability conditions. Alternatively, a nu-merical procedure can be used. Thuné [82] presented an algorithm for auto-matic stability investigation using normal mode analysis for hyperbolic initial-boundary value problems.

Summation-by-parts operators are finite difference operators with specialboundary closures that in combination with the simultaneous approximationterm often lead to stable approximations via the energy method [17]. Also,stability-, accuracy-, and conservation-properties shown for a single block canoften be extended to a multi-block configuration [53]. SBP-SAT discretiza-tions for grid interfaces have been considered by Carpenter et al. [18, 19] andLindström and Nordström [50] to mention a few. An advantage of the SBP-SAT method is that the stability results are independent of the order of thespatial discretization as long as the operators possess certain so-called SBPproperties, described in section 5.2. Also, the SBP operators contain one-sided approximations close to the interface, which limit the communicationbetween grid blocks to the SAT terms that couple adjacent grid blocks witheach other. A drawback is that the operators are of lower order close to bound-aries and interfaces. In our case the order of accuracy close to boundaries isonly half the order of accuracy in the interior. A straightforward application ofthe energy method leads to error estimates that imply the lower convergenceorder. However, earlier results have shown that the accuracy of the numericalsolution is often one or two orders higher than the order the boundary approx-imation would suggest [36, 35]. Svärd and Nordström [80] proved that twoorders of accuracy are recovered for numerical approximations of equationswith second derivatives for which uniform stability can be shown. In Paper IVand Paper V we investigate the accuracy for SBP-SAT discretizations of theTDSE using normal mode analysis. In this case uniform stability estimatescannot be derived. However, we show that two orders of accuracy are gainedboth for the boundary case with homogeneous Dirichlet boundary conditionsand for the situation with interfaces.

27

To prove stability of a grid interface coupling using normal mode analy-sis may become very technical, or even unfeasible, especially for high orderapproximations. However, if the grid coupling is proven stable by e.g. the en-ergy method, normal mode analysis can be used to consider the accuracy ofthe grid interface coupling. For accuracy it suffices to restrict the analysis toa limited region of Laplace space near the origin, as opposed to the stabilityinvestigation where the whole right half-plane needs to be considered.

The use of SBP operators in AMR has been very limited. Until recently ithas not been known how to couple adjoining nonconforming grid blocks, i.e.grid blocks for which collocation points across grid boundaries do not match,in a stable way using SBP operators. Recently, Mattsson and Carpenter [53]derived interpolation and projection operators that together with commonlyused SBP operators preserve stability between nonconforming grid blocks.

In order to show stability for a multiblock SBP-SAT configuration, cornerswith different refinement levels, illustrated by A and B in figure 5.1, need tobe treated properly. Kramer et al. [48] considered stability and accuracy forgrids in two dimensions with corners illustrated by A, for possible extensionto AMR. They derived conditions for stability at corners and showed that theorder of accuracy at the corner points is considerably lower compared to theaccuracy order at grid points away from corners.

By imposing the SAT interfaces denoted by 2, 3, 4 and 5, a stable discretiza-tion can be shown for the corner point B, using the interpolation operatorsfrom [53] for 4 and 5. However, in a multiblock configuration with severallayers with different refinement levels this solution is not optimal with respectto load balancing. Continuing the interfaces that start in the corner (corre-sponding to 2 and 3) may lead to grid blocks with only a few grid points in theouter layer of the multiblock configuration grid. In order to avoid this situationit is important to be able to treat the junction, illustrated by J in figure 5.1, in astable way. In paper VI we consider the imposition of interface 1 for a stablecoupling at the junction J.

5.2 Summation-by-parts operatorsSummation-by-parts operators are finite difference approximations that satisfya summation by parts formula. First developed by Kreiss and Scherer [49], theidea is that the summation by parts rule mimics the corresponding integrationby parts formula for the continuous operators. The operators in [49] approx-imate first derivatives. Since then, SBP operators have been constructed forinstance by Strand [79] for first derivatives and by Mattsson and Nordström[54] for second derivatives. If boundary conditions are imposed using the si-multaneous approximaion term method, it is often possible to show stabilityof the numerical scheme. We will demonstrate the SBP-SAT technique by asimple example below.

28

AJ

B

1

2

3

4

5r r r

Figure 5.1: Discretization with a junction (J) and corners (A, B). The red lines denoteSAT interfaces.

Consider the normalized TDSE in one dimension on the domainΩ = [−1, 1], with h = 1, 2m = 1, and homogeneous Dirichlet boundaryconditions,

i∂ψ

∂ t=−∂ 2ψ

∂x2 +V ψ,

ψ(−1, t) = ψ(1, t) = 0, (5.1)ψ(x,0) = f (x),

where ψ is complex valued and V is real valued. We define the scalar productand L2 norm as

(ψ,φ) =∫ 1

−1ψ∗φ dx, ‖ψ‖2 =

∫ 1

−1ψ∗ψ dx. (5.2)

We introduce the semi-discrete complex valued grid functions u,v ∈ CN+1,defined on an equidistant grid with spacing h. The discrete scalar product andnorm are defined as

(u,v)P = uHPv, ‖u‖2P = uHPu, (5.3)

where H denotes the Hermitian transpose and P is a positive definite operator.The elements of the corresponding matrix are proportional to the grid spacingh. With P = hI we get the standard discrete norm, defined by

‖u‖2h = huHu. (5.4)

29

We use the SBP operators approximating second derivatives derived by Matts-son and Nordström [54]. They fulfill the properties

D = P−1(−A+BS), A = AT , A≥ 0, (5.5)

where P is the same operator as in (5.3) and B = diag(−1,0, · · · ,0,1). In thematrix that represents the operator S, the first and last rows are approximationsof a first derivative. We consider operators for which P is diagonal.

The semi-discrete approximation of (5.1) is given by

idudt

=−Du+diag(V (xi))u, (5.6)

u(0) = f ,

where boundary conditions have been omitted temporarily. By multiplyingthe continuous equation (5.1) with ψ∗, adding the conjugate transpose, andintegrating by parts, we get

ddt‖ψ‖2 =

∫ 1

−1

∂ψ∗

∂ tψ +ψ

∗ ∂ψ

∂ tdx =

[−i

∂ψ∗

∂xψ + iψ∗

∂ψ

∂x

]1

−1

=−i∂ψ(1, t)∗

∂xψ(1, t)+ i

∂ψ(−1, t)∗

∂xψ(−1, t)

+iψ(1, t)∗∂ψ(1, t)

∂x− iψ∗(−1, t)

∂ψ(−1, t)∂x

. (5.7)

The corresponding procedure is done for the semi-discrete formulation, andfor (5.6) we get

ddt‖u‖2

P =duH

dtPu+uHP

dudt

=−i(Su)HN uN + i(Su)H

0 u0 + iuHN (Su)N− iuH

0 (Su)0,

(5.8)

where indices 0 and N indicate the first and last vector components, respec-tively. Note that the right hand side of (5.8), which is the semi-discrete coun-terpart to (5.7), is indefinite. If we impose the boundary conditions weaklywith the SAT method, and add the term iσP−1ST (uN−0)− iσP−1ST (u0−0)to equation (5.6), where u0 = (u0,0, · · · ,0)T , uN = (0, · · · ,0,uN)T , the esti-mate becomes

ddt‖u‖2

P =duH

dtPu+uHP

dudt

= (−i+σ)(Su)HN uN +(i−σ)(Su)H

0 u0

+(i+σ∗)uH

N (Su)N +(−i−σ∗)uH

0 (Su)0. (5.9)

30

By choosing the penalty parameter σ = i, the right hand side vanishes andintegration in time gives

‖u(t)‖2P = ‖ f‖2

P, (5.10)

i.e. we obtain a stable semi-discrete approximation. Since the P norm is equiv-alent to the standard discrete norm, we have by 5.10

c1‖ f‖P ≤ ‖u(t)‖h ≤ c2‖ f‖P, (5.11)

for some positive numbers c1, c2 independent of the grid spacing h.The SBP-SAT technique can also be used to show stability at grid block

interfaces. Reula [65] derived interface conditions for the TDSE using an in-teraction term only at the grid point at the interface. In Paper V we insteadderive interface conditions between adjacent grid blocks by weakly imposingcontinuity in the solution and the normal first derivative across the interface.The stability estimates are extended to multiple dimensions including non-conforming grid interfaces by using the interpolation and projection operatorsconstructed in [53]. Unlike for the boundary case, there is some freedom inchoosing the interface parameters. We choose the penalty parameters to pre-serve the P norm and to yield a symmetric discretization in the P norm.

31

6. Summary of papers

The content of the papers in the thesis will briefly be reviewed in thissection. Papers I-III deal with the application of the perfectly matched layertechnique to the time-dependent Schrödinger equation. Papers IV-VI dealwith an adaptive discretization of the Schrödinger equation by the SBP-SATmethodology.

6.1 Paper IPaper I considers the properties of a modal ansatz PML for the TDSE and dis-cusses the errors that arise due to the non-reflecting boundary treatment in thecontext of a Crank-Nicolson finite difference discretization. Well-posednessis shown for the continuous equation. The estimates are used to derive errorestimates for the continuous error equation with truncation error as forcing,as a model for the semi-discrete error equation. The estimates indicate thatthe damping function in the PML should have 2m + 1 bounded derivatives,where 2m is the order of the spatial discretization. However, numerical exper-iments show that the estimates are not sharp. Approximate error formulas arederived for the PML errors, establishing how various parameters in the PMLconstruction should be chosen in order to effectively balance error with cost.The optimized PML is applied to the free-particle Schrödinger equation in oneand two dimensions, demonstrating that it is possible to construct the PML sothat numerical results fall within a given accuracy, from one preliminary com-putation.

6.2 Paper IIIn Paper II we investigate a dissociative three-state system describing a one-dimensional IBr molecule subjected to a laser field. A molecule in its groundstate is excited to a higher, unstable state, from which parts of the moleculedissociates. High-order finite differences with PML boundary treatment areused in space and the h, p-adaptive Magnus-Arnoldi algorithm devised in [47]is used for the time-propagation. The global errors from the spatial and tem-

33

poral discretizations are controlled using a posteriori error estimation theory,and the error from the boundary treatment is controlled using the error formu-las derived in Paper I. The optimization of the PML requires some beforehandknowledge of the dominant frequencies in the outgoing wavepacket, and un-like in Paper I and Paper III, these can in the present case not be determinedthrough initial conditions. In Paper II we show that the error formulas fromPaper I can be applied to more general data, after retrieving the energy spec-tra of the excited wavepacket from the time-independent Schrödinger equa-tion. Paper II also provides a comparison between the implicit Crank-Nicolsonscheme and the Magnus-Arnoldi propagator, demonstrating the latter to be su-perior already in one dimension.

6.3 Paper IIIThe objective of Paper III is to illustrate the potential of the PML techniquefor dissociative problems in chemical physics. The process of dissociative ad-sorbtion and associative desorption of a H2 molecule on a solid surface isexplored through a two-dimensional model. The dissociation channels aretruncated using a PML at the end of each channel, and the spatial and tem-poral discretizations are carried out using high-order finite differences andthe Arnoldi method, respectively. The performance of the PML is comparedto that of other absorbing boundary treatments for the Schrödinger equation,commonly used in quantum dynamics simulations, in terms of numerical re-flections from the boundary and outgoing flux. The exterior complex scalingmethod is shown to produce large numerical reflections, despite a wide ab-sorption region. Manolopoulos transmission-free absorbing potential provesto be competitive for low accuracy levels, where a thin absorbing layer canbe utilized. The PML, on the other hand, is more effective for higher accu-racy levels. The possibility to tune the accuracy of the PML, due to the errorformulas in Paper I, is also demonstrated.

6.4 Paper IVThe purpose of Paper IV is to demonstrate that the SBP-SAT methodologyis applicable to the TDSE. We derive stability estimates for the TDSE withhomogeneous Dirichlet boundary conditions using the energy method. Theaccuracy of the boundary closures is investigated using normal mode analysisfor operators of interior order 2m, m = 1,2,3. We show that boundary closuresof order m lead to global accuracy of order m + 1 for m = 1 and m + 2 form = 2,3. Thus, full accuracy is obtained for discretizations of second andfourth order, and a sixth order discretization leads to global accuracy of fifthorder. The theroretical accuracy results are verified in numerical simulations.

34

6.5 Paper VIn Paper V we extend the SBP-SAT method presented in Paper IV to gridinterfaces and derive stability estimates for grid interfaces between noncon-forming grid blocks. Theoretical investigations in one spatial dimension showthat one and two orders of accuracy are recovered with respect to the lower or-der approximations close to the interface for second and fourth order methods,respectively. The accuracy and stability of the numerical method is examinedthrough several numerical tests, where we investigate convergence rates, longtime stability and dispersion effects. The gain in using an adaptive mesh isconsidered.

6.6 Paper VIIn Paper VI the SBP-SAT methodology is extended to multiblock grids, im-plemented using a block-structured partitioning of the mesh where differentblocks can have different refinement levels. We conduct theoretical investiga-tions for stability between grid blocks with a combination of SBP-SAT dis-cretization and central finite difference schemes, with the objective to treatmultiblock structures in a stable way. Grid convergence studies for smallertest problems demonstrate the stability and accuracy of the numerical method.Furthermore, the performance of the methodology for multiblock configura-ration grids is considered.

35

7. Discussion and outlook

The unbounded domain on which the Schrödinger equation is defined needsto be truncated to a finite computational domain for numerical simulations.Large-scale chemical physics computations involving dissociative states aredependent on an efficient and accurate absorbing boundary technique. Also,the numerical method should be capable of handling large variations in spatialscales. This thesis discusses a boundary treatment model based on the per-fectly matched layer (PML) approach, and a means to include spatial adap-tivity using the summation-by-parts-simulteneous approximation term (SBP-SAT) methodology.

A next step would be to extend the SBP-SAT framework to include the ab-sorbing boundary treatment. This would be done by coupling the interior do-main with an absorbing layer using SBP operators and a numerical interface.Since the stencils close to the interface are one-sided, this approach could leadto less strict continuity requirements on the absorption function in the PML.

In order to handle multiblock grids, corners that appear where patches ofdifferent refinement levels meet need to be treated in a stable manner. Sucha treatment would be valuable also in other application areas, for instancein computational fluid dynamics, where AMR techniques are important. Thecombination of high order finite difference methods with provable stable nu-merical approximations make the SBP-SAT framework attractive. A long termgoal concerning the adaptive discretization is to develop a robust AMR frame-work based on the SBP-SAT methodology.

The work in this thesis is a part of a larger project developing andanalyzing efficient numerical techniques for simulation of the time-dependentSchrödinger equation. Other building blocks are efficient time-marchingmethods, electronic structure calculations and a high-performance parallelimplementation. The theoretical and experimental findings of this work areintended to be included in a powerful parallel implementation for solving thetime-dependent Schrödinger equation. For further information of the project,visit http://www.it.uu.se/research/project/qd.

37

8. Summary in Swedish

Intresset för kemisk reaktionsdynamik har tilltagit sedan Ahmed H. Zewail1999 blev tilldelad Nobelpriset i kemi för sitt bidrag till utvecklingen av fem-tokemin. Genom femtokemin har det blivit möjligt att följa kemiska reaktionerpå den tidsskala reaktionerna äger rum, nämligen femtosekunder. Med hjälpav nya forskningsområden inom experimentell kemi såsom femtokemi ochattokemi ökar möjligheterna att kunna kontrollera kemiska reaktioner medhjälp av laserpulser. Den kemiska processen där en molekyl upplöses i mindrebeståndsdelar genom att molekylbindningarna bryts kallas dissociation. Dis-sociativa processer är en av två utmaningar inom kemisk fysik som ligger ifokus i den här avhandlingen. Det andra området är kemiska system som in-nehåller komponenter med stor lokal variation i rumsskalorna, där intressantafenomen äger rum i ett utsträckt område, samtidigt som det är viktigt att kunnalösa upp högfrekventa komponenter.

Ett värdefullt komplement till experimentell kemi är att användamatematiska modeller som underlag för att utföra datorberäkningar somsimulerar kemiska reaktioner. De numeriska beräkningarna kan användasbåde för validering av experimentella resultat och för att simulera kemiskaprocesser som inte kan undersökas genom experiment. I takt med att datorerblir mer kraftfulla ökar omfattningen av de fenomen som kan simuleras meddatorberäkningar. Utvecklingen av nya, effektiva numeriska metoder utökarmöjligheterna ytterligare.

För att kunna förstå fundamentala processer på atomär nivå behövsen kvantmekanisk beskrivning. Den matematiska modell som teoretisktbeskriver detta är den tidsberoende Schrödingerekvationen. Schrödingerek-vationen är formulerad på en oändlig definitionsmängd och för att kunnautföra datorsimuleringar behöver ett ändligt beräkningsområde definieras.Dessutom behöver randvillkor preciseras på ränderna som omsluterberäkningssområdet. Genom att diskretisera ekvationen införs en beskrivningför ett begränsat antal beräkningspunkter och med hjälp av datorsimuleringarkan en numerisk lösning till det diskretiserade problemet beräknas imotsvarande punkter. Punkterna som upptar beräkningsomtrådet kallasför beräkningsnät. Den här avhandlingen behandlar numeriska metoderför Schödingerekvationen baserade på finita differensmetoder, en klass avmetoder där derivator approximeras med differenser mellan lösningsvärden inärliggande punkter i beräkningsnätet.

39

För kemiska system med dissociation är lösningen till Schrödingerekvatio-nen inte begränsad till ett mindre beräkningsområde, och artificiella randvil-lkor för vilka den numeriska lösningen efterliknar lösningen på en oändligdefinitionsmängd är önskvärd. Denna egenskap gör att lösningskomponen-ter som rör sig ut ur beräkningsdomänen till synes absorberas av randen,därav namnet absorberande randvillkor. Artikel I–III behandlar en typ av ab-sorberande randvillkor, så kallat perfekt matchat lager (PML), vilket utgörsav ett absorberande skikt av några beräkningspunkter som beräkningsområdetutvidgas med. PML-metoden bygger på en modifiering av den kontinuerligaekvationen i skiktet utanför den önskade beräkningsdomänen där lösningendämpas ut. Artikel I undersöker metodens effektivitet och kvantifierar fel förmodellproblem. I Artikel II-III tillämpas metoden på mer realistiska problemi kemikalisk fysik.

För tillämpningsproblem med variation i rumsskalorna är högfrekventakomponenter ofta begränsade till en mindre del av beräkningsområdet. Föratt maximera effektiviteten i de numeriska simuleringarna är det önskvärtatt anpassa upplösningen hos nätet efter egenskaper hos lösningen. Ett sättatt hantera detta på är att dela upp beräkningsdomänen i delområden ochfördela nätpunkterna adaptivt, så att varje delområde innehåller lika månganätpunkter vardera, men upptar olika stor del av den fysiska domänen. Mellandelområden med olika nätpunktsdensiteter kommer artificiella inre ränder attuppstå. I Artikel IV–VI undersöker vi hur dessa inre ränder kan hanterasför att ge en robust numerisk metod. Vi använder finita differensoperatorersom uppfyller en partiell summationsregel, så kallade SBP-operatorer, somgenom att kombineras med en speciell randbehandling ofta kan bevisas gestabila numeriska approximationer. I Artikel IV behandlas stabilitet ochnoggrannhet för de yttre ränderna av beräkningsdomänen, och i ArtikelV utvidgas analysen till de artificiella ränderna mellan två delområden,eller nätblock, i beräkningsdomänen. I Artikel VI utvidgas metoden till enmulti-nätblockskonfiguration.

40

9. Acknowledgements

First of all I would like to express my gratitude to my advisor Prof. GunillaKreiss for sharing expertise and for providing guidance throughout my timeas a PhD student. Gunilla, I truly appreciate all your help, support and encour-agement.

I want to thank Katharina Kormann and Magnus Gustafsson for the collab-orations in papers II and VI. I have enjoyed and appreciated our collaborationvery much. Many thanks to Prof. Hans Karlsson for the collaboration in paperIII and for sharing expertise on the chemical aspects. I appreciate the collab-oration with Prof. Margot Gerritsen in papers IV and V very much and amgrateful for the opportunity to spend three months at Stanford University.

I wish to thank my co-advisor Prof. Sverker Holmgren for all the workhe has done to create a group in computational chemistry at the Division ofScientific Computing and for introducing me to this interesting subject.I acknowledge the numerical quantum dynamics group for being an importantplatform for sharing ideas and knowledge.

Thanks also to all friends and collegues at the Division of Scientific Com-puting. A special thanks to Jens Berg, Sofia Eriksson and Emil Kieri for read-ing this thesis and providing useful comments and suggestions.

I would like to thank my family for the warm athmosphere you provide andfor helping putting priorities in perspective. Per, thanks for many discussions,great companionship, and for reading this thesis.

Financial support from the Swedish Research Council grant no. VR2009-5852, Anna Maria Lundins stipendiefond, Liljewalchs stipendiefond and Wal-lenbergsstiftelsen is gratefully acknowledged.

41

Bibliography

[1] J. Aguilar and J. M. Combes. A class of analytic perturbations for one-bodySchrödinger Hamiltonians. Commun. Math. Phys., 22:269–279, 1971.

[2] A. Ahland, D. Schulz, and E. Voges. Accurate mesh truncation for Schrödingerequations by a perfectly matched layer absorber: Application to the calculationof optical spectra. Phys. Rev. B, 60:5109–5112, 1999.

[3] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle. A review oftransparent and artificial boundary conditions techniques for linear and nonlin-ear Schrödinger equations. Commun. Comput. Phys., 4:729–796, 2008.

[4] X. Antoine and C. Besse. Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation. J.Comput. Phys., 188:157–175, 2003.

[5] D. Appelö, T. Hagstrom, and G. Kreiss. Perfectly matched layers for hyper-bolic systems: general formulation, well-posedness, and stability. SIAM J. Appl.Math., 67:1–23, 2006.

[6] E. Balslev and J. M. Combes. Spectral properties of many-body Schrödingeroperators with dilation-analytic interactions. Commun. Math. Phys., 22:280–294, 1971.

[7] V. A. Baskakov and A. V. Popov. Implementation of transparent boundariesfor numerical solution of the schrödinger equation. Wave motion, 14:123–128,1991.

[8] E. Becache, S. Fauqueux, and P. Joly. Stability of perfectly matched layers,group velocities and anisotropic waves. J. Comput. Phys., 188:399–433, 2003.

[9] J Bengtsson, E. Lindroth, and S. Selstø. Solution of the time-dependentSchrödinger equation using uniform complex scaling. Phys. Rev. A, 78:032502,2008.

[10] J. P. Berenger. A perfectly matched layer for the absorption of electromagneticwaves. J. Comput. Phys., 114:185–200, 1994.

[11] M. J. Berger. Stability of interfaces with mesh refinement. Math. Comput.,45:301–318, 1985.

[12] M. J. Berger and P. Colella. Local adaptive mesh refinement for shock hydrody-namics. J. Comput. Phys., 82:64–84, 1989.

43

[13] M. J. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partialdifferential equations. J. Comput. Phys., 53:484–512, 1984.

[14] M. Born and R Oppenheimer. Zur Quantentheorie der Molekeln. Ann. Phys.,84:457–484, 1927.

[15] S. Borowski, S. Thiel, T. Klüner, H.-J. Freund, R. Tisma, and H. Lederer. High-dimensional quantum dynamics of molecules on surfaces: a massively parallelimplementation. Comput. Phys. Commun., 143:162–173, 2002.

[16] E. Brändas and P. Froelich. Continuum orbitals, complex scaling problem, andthe extended virial theorem. Phys. Rev. A., 16:2207–2210, 1977.

[17] M. H. Carpenter, D. Gottlieb, and S Abarbanel. Time-stable boundary condi-tions for finite-difference schemes solving hyperbolic systems: M ethodologyand application to high-order compact schemes. J. Comput. Phys., 111:220–236, 1994.

[18] M. H. Carpenter, J. Nordström, and D. Gottlieb. A stable and conservative in-terface treatment of arbitrary spatial accuracy. J. Comput. Phys., 148:341–365,1999.

[19] M. H. Carpenter, J. Nordström, and D. Gottlieb. Revisiting and extending inter-face penalties for multi-domain summation-by-parts operators. J. Sci. Comput.,45:118–150, 2010.

[20] W. C. Chew and W. H. Weedon. A 3D perfectly matched medium from modi-fied Maxwell’s equations with stretched coordinates. Microw. Opt. Techn. Let.,7:599–604, 1994.

[21] X. Chu and S.-I. Chu. Complex-scaling generalized pseudospectral method forquasienergy resonance states in two-center systems: Application to the floquetstudy of charge resonance enhanced multiphoton ionization of molecular ions inintense low-frequency laser fields. Phys. Rev. A, 63:013414, 2000.

[22] D. T. Colbert and W. H. Miller. A novel discrete variable representation forquantum mechanical reactive scattering via the S-matrix Kohn method. J. Chem.Phys., 96:1982–1991, 1992.

[23] F. Collino. Perfectly matched absorbing layers for the paraxial equations. J.Comput. Phys., 131:164–180, 1997.

[24] M. A. Collins. Molecular potential-energy surfaces for chemical reaction dy-namics. Theor. Chem. Acc., 108:313–342, 2002.

[25] J. K. Cullum and R. A. Willoughby. Lanczos algorithms for large symmetriceigenvalue computations., volume 1. Birkhäuser, 1985.

[26] T. Dohnal. Perfectly matched layers for coupled nonlinear Schrödinger equa-tions with mixed derivatives. J. Comput. Phys., 228:8752–8765, 2009.

44

[27] W. Dörfler. A time- and spaceadaptive algorithm for the linear time-dependentSchrödinger equation. Numer. Math., 73:419–448, 1996.

[28] M. Ehrhardt and A. Arnold. Discrete transparent boundary conditions for theSchrödinger equation. Riv. Mat. Univ. Parma, 6:57–108, 2001.

[29] E. Faou, V. Gradinaru, and C Lubich. Computing semi-classical quantum dy-namics with Hagedorn wavepackets. SIAM J. Sci. Comput., 31:3027–3041,2009.

[30] E. Fattal, R. Baer, and R. Kosloff. Phase space approach for optimizing grid rep-resentations: The mapped Fourier method. Phys. Rev. E, 53:1217–1227, 1996.

[31] B. Fornberg and D. M. Sloan. A review of pseudospectral methods for solvingpartial differential equations. Acta Numerica, 3:203–267, 1994.

[32] T. Gonzalez-Lezana, E. J. Rackham, and D. E. Manolopoulos. Quantum re-active scattering with a transmission-free absorbing potential. J. Chem. Phys.,120:2247–2254, 2003.

[33] V. Gradinaru. Strang splitting for the time-dependent Schrödinger equation onsparse grids. SIAM J. Numer. Anal., 46:103–123, 2007.

[34] S. K. Gray and E. M. Goldfield. Dispersion fitted finite difference method withapplication to molecular quantum dynamics. J. Chem. Phys., 115:8331–8344,2001.

[35] B. Gustafsson. The convergence rate for difference approximations to mixedinitial boundary value problems. Math. Comput., 29:396–406, 1975.

[36] B. Gustafsson. The convergence rate for difference approximations to gen-eral mixed initial boundary value problems. SIAM J. Numer.Anal., 18:179–190,1981.

[37] M. Gustafsson. A PDE solver framework optimized for clusters of multicoreprocessors. Master’s thesis, Uppsala University, 2009.

[38] T. Hagstrom. New results on absorbing layers and radiation boundary condi-tions. In Topics in computational wave propagation, volume 31, pages 1–42.2003.

[39] S. Hein, T. Hohage, and W. Koch. On resonances in open systems. J. FluidMech., 506:255–284, 2004.

[40] W. D. Henshaw and D. W. Schwendeman. Moving overlapping grids with adap-tive mesh refinement for high-speed reactive and non-reactive flow. J. Comput.Phys., 216:744–779, 2006.

[41] S. Jiang. Fast evolution of the nonreflecting boundary conditions for theSchrödinger equation. PhD thesis, Courant Institute of Mathematical Sciences,New York University, 2001.

45

[42] H. O. Karlsson. Accurate resonances and absorption of flux using smooth exte-rior scaling. J. Chem. Phys., 109:9366–9371, 1998.

[43] U. Kleinekathöfer and D. J. Tannor. Extension of the mapped Fourier method totime-dependent problems. Phys. Rev. E, 60:4926–4933, 1999.

[44] V. Kokoouline, O. Dulieu, R. Kosloff, and F. Masnou-Seeuws. Mapped Fouriermethods for long-range molecules: Application to perturbations in the Rb2(0+

u )photoassociation spectrum. J. Chem. Phys., 110:9865–9876, 1999.

[45] K. Kormann. Numerical methods for quantum molecular dynamics. UppsalaUniversity, Licentiate thesis, 2009.

[46] K. Kormann, S. Holmgren, and H. O. Karlsson. Accurate time-propagationfor the Schrödinger equation with an explicitly time-dependent hamiltonian. J.Chem. Phys., 128:18410, 2008.

[47] K. Kormann, S. Holmgren, and H. O. Karlsson. Global error control of the time-propagation for the Schrödinger equation with a time-dependent hamiltonian. J.Comput. Sci., 2:178–187, 2011.

[48] R. M. J. Kramer, C. Pantano, and D. I. Pullin. Nondissipative and energy-stablehigh-order finite-difference interface schemes for 2-D patch-refined grids. J.Comput. Phys., 228:5280–5297, 2009.

[49] H.-O. Kreiss and G. Scherer. Finite element and finite difference methods forhyperbolic partial differential equations. In Mathematical aspects of finite ele-ments in partial differential equations, volume 128, pages 459–466. AcademicPress inc., 1974.

[50] J. Lindström and J Nordström. A stable and high-order accurate conjugate heattransfer problem. J. Comput. Phys., 229:5440–5456, 2010.

[51] C. Lubich. Integrators in quantum dynamics: a numerical analysist’s brief re-view. In Quantum simulation of many-body systems: from theory to algorithms,volume 128, pages 459–466. 2002.

[52] D. E. Manolopoulos. Derivation and reflection properties of a transmission-freeabsorbing potential. J. Chem. Phys., 117:9552–9559, 2002.

[53] K. Mattsson and M. H. Carpenter. Stable an accurate interpolation operatorsfor high-order multiblock finite difference methods. SIAM J. Sci. Comput.,32:2298–2320, 2010.

[54] K. Mattsson and J. Nordström. Summation by parts operators for finite dif-ference approximations of second derivatives. J. Comput. Phys., 199:503–540,2004.

[55] B. Mayfield. Non-local boundary conditions for the Schrödinger equation. PhDthesis, University of Rhode Island, 1989.

46

[56] C. W. McCurdy, C. K. Stroud, and M. K. Wisinski. Solving the time-dependent Schrödinger equation using complex-coordinate contours. Phys. Rev.A, 43:5980–5990, 1991.

[57] H.-D. Meyer, U. Munthe, and L. S. Cederbaum. The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett., 165:73–78, 1990.

[58] N. Moiseyev. Quantum theory of resonances: calculating energies, widths andcross-sections by complex scaling. Physics Reports, 302:211–293, 1998.

[59] N. Moiseyev, P. R. Certain, and F. Weinhold. Resonance properties of complex-rotated hamiltonians. Molecular Physics, 36:1613–1630, 1978.

[60] P. Monk and K. Parrott. Phase-accuracy comparisons and improved far-fieldestimates for 3-D edge elements on tetrahedral meshes. J. Comput. Phys.,170:614–641, 2001.

[61] D. Neuhauser and M. Baer. The time-dependent Schrödinger equation: Appli-cation of absorbing boundary conditions. J. Chem. Phys., 90:4351–4355, 1988.

[62] L. R. Pettey and R. E. Wyatt. Wave packet dynamics with adaptive grids: Themoving boundary truncation method. Chem. Phys. Lett., 424:443–448, 2006.

[63] B. Poirier and T Carrington. Semiclassically optimized complex absorbing po-tentials of polynomial form. II. Complex case. J. Chem. Phys., 119:77–89, 2003.

[64] Rantakokko, J. and Thuné, M. Parallel structured adaptive mesh refinement. InParallel Computing, pages 147–173. Springer Verlag, 2009.

[65] O. Reula. Numerical treatment of interfaces in quantum mechanics. March2011.

[66] U. V. Riss and H.-D. Meyer. Calculation of resonance energies and widths usingthe complex absorbing potential method. J. Phys. B, 26:4503–4535, 1993.

[67] U. V. Riss and H.-D. Meyer. Investigation on the reflection and transmissionproperties of complex absorbing potentials. J. Chem. Phys., 105:1409–1419,1996.

[68] U. V. Riss and H.-D. Meyer. The transformative complex absorbing potentialmethod: a bridge between complex absorbing potentials and smooth exteriorscaling. J. Phys. B, 31:2279–2304, 1998.

[69] J. M. Sanz-Serna. Symplectic integrators for Hamiltonian problems: anoverview. Acta Numerica, 1:243–286, 1992.

[70] A. Schädle. Ein schneller Faltungsalgorithmus für nichtreflektierende Rand-bedingungen. PhD thesis, Universität Tübingen, 2002.

[71] E Schrödinger. Quantisierung als Eigenwertproblem (Dritte Mitteilung). Ann.Phys., 80:437–490, 1926.

47

[72] E. Schrödinger. Quantisierung als Eigenwertproblem (Erste Mitteilung). Ann.Phys., 79:361–376, 1926.

[73] E. Schrödinger. Quantisierung als Eigenwertproblem (Vierte Mitteilung). Ann.Phys., 81:109–139, 1926.

[74] E. Schrödinger. Quantisierung als Eigenwertproblem (Zweite Mitteilung). Ann.Phys., 79:489–527, 1926.

[75] T. Seideman and W. H. Miller. Quantum mechanical reaction probabilities via adiscrete variable representation-absorbing boundary condition Green’s function.J. Chem. Phys., 97:2499–2514, 1992.

[76] B. Simon. Resonances in n-body quantum systems with dilation analytic po-tentials and the foundations of time-dependent perturbation theory. Ann. Math.,97:247–274, 1973.

[77] B. Sjögreen and N. A. Petersson. Perfectly matched layers for Maxwell’s equa-tions in second order formulation. J. Comput. Phys., 209:19–46, 2005.

[78] Q. Stout, D. De Zeeuw, T. Gombosi, C. Groth, H. Marshall, and K. Powell.Adaptive blocks: A high-performance data structure. In Supercomputing ’97,1997.

[79] B. Strand. Summation by parts for finite difference approximations for d/dx. J.Comput. Phys., 110:47–67, 1994.

[80] M. Svärd and J. Nordström. On the order of accuracy for difference approx-imations of initial-boundary value problems. J. Comput. Phys., 218:333–352,2006.

[81] D. J. Tannor. Introduction to Quantum Mechanics: A Time-Dependent Perspec-tive. University Science Book, 2007.

[82] M. Thuné. Automatic GKS stability analysis. Siam. J. Sci. Stat. Comput., 7:959–977, 1986.

[83] A. Vibok and B. Balint-Kurti. Parametrization of complex absorbing potentialsfor time-dependent quantum dynamics. J. Phys. Chem., 96:8712–8719, 1992.

[84] L. White, E. Deleersnijder, and V. Legat. A three-dimensional unstructuredmesh finite element shallow-water model, with application to the flows aroundan island and in a wind-driven, elongated basin. Ocean Model., 22:26–47, 1992.

[85] G. Yao and S.-I.. Chu. Generalized pseudospectral methods with mappings forbound and resonance state problems. Chem. Phys. Lett., 204:381–388, 1993.

[86] D. Yevick, T. Friese, and F. Schmidt. A comparison of transparent boundaryconditions for the Fresnel equation. J. Comput. Phys., 168:433–444, 2001.

[87] C. Zheng. A perfectly matched layer approach to the nonlinear Schrödingerwave equations. J. Comput. Phys., 227:537–556, 2007.

48