COMSOL Multiphysics for Efficient Solution of a Transient

Reaction-Diffusion System with Fast Reaction

Matthias K. Gobbert, Aaron Churchill, Guan Wang, and Thomas I. Seidman

Department of Mathematics and Statistics, University of Maryland, Baltimore County,1000 Hilltop Circle, Baltimore, MD 21250, gobbert@math.umbc.edu

Abstract: A reaction between chemical speciesis modeled by a particular reaction pathway, inwhich one reaction is very fast relative to the otherone. The diffusion controlled reactions of thesespecies together with a reaction intermediate aredescribed by a system of three transient reaction-diffusion equations over a two-dimensional spatialdomain. In the asymptotic limit of the reaction pa-rameter tending to infinity, an equivalent two com-ponent model can be used in numerical simulationsthat is significantly more computationally efficient.But this model has features such as an unusual cou-pling of the two components in a boundary condi-tion that bring out important advantages of COM-SOL Multiphysics. We discuss the use of COM-SOL and present representative simulation resultsstarting with an initial condition with interestingstructure throughout the domain.

Key words: Reaction-diffusion equation, Inter-nal layer, Transient layer, Initial transient.

1 Introduction

Our objective is to study the evolution of the chem-ical reactions involving the two reactive species Aand B, the reaction intermediate C, and a product(not tracked) modeled by the reaction pathway

A + B λ−→ C, A + Cµ−→ (∗)

subject to diffusion in the two-dimensional domainΩ := (0, 1) × (0, 1) ⊂ R2. In the reaction model,λ and µ are reaction coefficients scaled such thatλ µ = 1, which makes the first reaction a fastreaction relative to the second one. We denotethe concentrations of the three chemically reactivespecies A, B, C by u(x, y, t), v(x, y, t), w(x, y, t),

respectively. Standard chemical kinetics then givea system of three transient reaction-diffusion equa-tions for these concentrations as

ut = uxx + uyy − λuv − uwvt = vxx + vyy − λuvwt = wxx + wyy + λuv − uw

(1.1)

for (x, y) ∈ Ω and 0 < t ≤ tfin with boundaryconditions

u = αvx = 0wx = 0

at x = 0,ux = 0v = βwx = 0

at x = 1

(1.2)

on the left and the right side of Ω, respectively, and

uy = 0vy = 0wy = 0

at y = 0 and y = 1 (1.3)

on the top and the bottom of Ω, and the initialconditions

u = uini(x, y)v = vini(x, y)w = wini(x, y)

at t = 0. (1.4)

This setup represents the case of unlimited supplyof A to the left and unlimited supply of B to theright of the domain Ω.

Due to the increasingly steep gradients in themodel, the numerical simulation of (1.1)–(1.4) withlarge values of the fast reaction parameter λ arechallenging and costly. This is demonstrated in[1] for values λ = 103, 106, and particularly 109.However, studies for the model in one spatial di-mension in [3] demonstrate that the problem is es-sentially in the asymptotic limit for λ ≥ 106 and wecan therefore consider the appropriate limit prob-lem of (1.1)–(1.4) as λ→∞. Standard techniques

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of singular perturbation analysis would give onlythe reduced problem uv ≡ 0. But [5] recently pre-sented another approach to the problem (1.1)–(1.4)in asymptotic limit by deriving a two componentmodel that is equivalent to the three species model.The derivation from [5] is sketched in Section 2, ex-tended to the case of two spatial dimensions. Thistwo component model does not have any steep gra-dients any more and is thus significantly cheapercomputationally, as was confirmed in [1] and in [7]for the analogous problem in one spatial dimension.

However, the two component model has featuressuch as a boundary condition that couples the(derivatives of the) solution components. This andother properties of the desired simulations makethis an excellent example to bring out the bene-fits of several features of COMSOL Multiphysics,which are discussed in Section 3. Finally, Section 4presents representative simulation results for theproblem using the two component model as com-putational tool. We refer to [1] for additional plotsfrom these simulations and to [1, 2, 7] for detailedstudies that confirm the accuracy and compare theefficiency of studies based on the two componentmodel to ones for the three species model withlarge values of λ.

2 Two Component Model

Computationally, the greatest difficulty in han-dling the system (1.1) is the occurrence of the re-action term λuv in each of the equations as λ 1.Since the difference A− B and the sum B + C areeach conserved in the first reaction, this difficultreaction term cancels when one combines pairs ofequations in (1.1) to get

(u− v)t = (u− v)xx + (u− v)yy − uw(v + w)t = (v + w)xx + (v + w)yy − uw

(2.1)

such that the term λuv no longer appears. Thismotivates the transformation from u, v, w to thetwo components

u1 := u− v,u2 := v + w.

(2.2)

It is not immediately intuitive that it is possible torecover the original three variables u, v, w from thetwo variables u1 and u2. The key is to notice thatin the asymptotic limit λ→∞, we always have thecomplementarity condition uv ≡ 0 (meaning that

uv is negligible for large λ, even though, when thisis multiplied by λ 1, λuv can become large).Together with the facts u ≥ 0 and v ≥ 0 for thespecies concentrations u and v, we have that whenu1 = u − v > 0 we must have u 6= 0 so v = 0 andu1 = u, while u1 = u− v < 0 similarly gives u = 0whence u1 = −v. It then becomes possible to take

u = u+1 := maxu1, 0,

v = −u−1 := −minu1, 0,w = u2 + u−1 .

(2.3)

The formulas in (2.2) and (2.3) in effect consti-tute forward and backward transformations be-tween the three species and the two componentmodels and make the two models equivalent to eachother [5].

The equivalent two component model is then

u1,t = u1,xx + u1,yy − u+1 u2

u2,t = u2,xx + u2,yy − u+1 u2

(2.4)

for (x, y) ∈ Ω and 0 < t ≤ tfin with boundaryconditions

u1 = αu2,x = 0

at x = 0,

u1 = −βu2,x = −u1,x

at x = 1

(2.5)

on the left and the right side of Ω, respectively, and

u1,y = 0u2,y = 0

at y = 0 and y = 1, (2.6)

on the top and the bottom of Ω, and initial condi-tions

u1 = u1,ini(x, y)u2 = u2,ini(x, y)

at t = 0. (2.7)

Most transformations of the boundary conditionsand initial conditions follow directly from the def-initions in (2.2); to see the second boundary con-dition at x = 1, notice that u2,x = vx + wx = vx,since wx = 0 at x = 1. Using u1 = u+

1 + u−1 givesu1,x = u+

1,x + u−1,x = ux − vx. Since ux = 0 atx = 1, we thus have u1,x = −vx and the boundarycondition u2,x = vx = −u1,x at x = 1.

The complete two component system (2.4)–(2.7) is self-contained and computationally effi-cient, since it does not have any steep gradients[1]. The price is a rather unusual coupling in thesecond boundary condition at x = 1.

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3 Use of COMSOL

The computations use COMSOL Multiphysics 3.5aon the cluster hpc in the UMBC High PerformanceComputing Facility (www.umbc.edu/hpcf). COM-SOL is run on one node with two dual-core AMDOpteron processors (2.6 GHz, 1 MB cache per core)and 13 GB of memory. The cluster runs the LinuxRedHat EL5 operating system. Several features ofthe problem under consideration serve to highlightthe features of COMSOL:

• One of the crucial features of the reaction--diffusion equations under consideration aretheir non-linear reaction terms. Since thephysics of this problem relies on a subtle bal-ance between the diffusion effects and thesereactions, we wish to represent them as ac-curately as possible. Thus, we use the Gen-eral Form of the PDEs in COMSOL in or-der to enable COMSOL to compute symbolicderivatives of all terms in the PDEs by auto-matic differentiation for the highest accuracyin the evaluation of the Jacobian in the non-linear solver inside the implicit ODE method.By contrast, the Coefficient Form in COM-SOL approximates the derivatives of all termsnumerically, which is only suitable for mildlynon-linear PDEs.

• An interesting feature of the two componentproblem is the boundary condition u2,x =−u1,x at x = 1 that couples the (derivatives ofthe) solution components. Many PDE solvershave difficulty handling a boundary conditionof this type or do not permit specifying it atall, even for the analogous problem in only onespatial dimension. COMSOL has no troublewith this, as all terms specified are allowed toinclude also the dependent variables and theirderivatives.

• We couple COMSOL with Matlab using theLinux command comsol matlab. With this,a script is written so that each computationwould be using the same solver parametersinside COMSOL. This is crucial to ensure re-producibility of all results and useful to fa-cilitate parameter studies, by that with re-spect to problem parameters or with respectto numerical parameters. Scripting is alsovery useful in cases such as the two compo-nent model, where significant post-processing

such as back-transforming to the original vari-ables by (2.3) is necessary. But we use theflexibility of scripting even more for the prob-lems under consideration here. COMSOL eas-ily allows the user to specify functional expres-sions for boundary and initial conditions. Thisis suitable for the boundary conditions here,and changing one of them is possible in thescript. But the initial conditions desired herewill have a fairly complicated structure insidethe domain, and we moreover wish to makechanging the initial condition convenient. Us-ing functional expressions in the COMSOLscript becomes too cumbersome and poten-tially limiting. Therefore, we merely specifythe names of functions for the initial condi-tions in the script, which enables the user towrite these functions outside of COMSOL us-ing the full capabilities of high-level program-ming languages such as if-statements, for-loops, and vector operations on the vectors of(x, y) values.

• We use linear Lagrange finite elements on auniform quadrilateral mesh with N × N ele-ments. The studies below report results froma 128×128 mesh, but also studies with 64×64and 256×256 elements were performed [1]. Atthe coarser mesh resolutions, some of the geo-metric features are not as accurate as desired.The finer mesh did not add any appreciabledetail for the particular initial conditions con-sidered here. Clearly, computations increasesignificantly in complexity, hence we do notwish unnecessarily large values of N in pro-duction runs. Note that the defaults in COM-SOL are an unstructured mesh and quadraticLagrange finite elements. By using a struc-tured, uniform, quadrilateral mesh, we avoidany incidental biasing that might affect the so-lution as result of, e.g., from element bound-aries being at different random non-horizontalor non-vertical angles or similar properties in-herent to an unstructured mesh. Because wewill be interested in the exact location of in-terfaces later, it is important to avoid anysuch biasing for this problem. Because thesolutions have jump discontinuities at the ini-tial time and will have discontinuities in theirderivatives at the reaction interfaces [5], weuse the lowest order Lagrange finite elementsavailable in COMSOL.

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• COMSOL offers several ODE solvers, all ofwhich use implicit time stepping, which isa necessity for efficiently solving PDEs ofparabolic type such as reaction-diffusion equa-tions. Since we desire to compute to a largefinal time approaching steady state, sophisti-cated ODE solver features such as automatictime stepping and method order selection upto a high order are vital. This becomes par-ticularly clear if you consider that we have toexpect steep initial gradients from the cho-sen initial conditions, which requires smalltime steps and low ODE method orders, butthat we wish to use large time steps and highmethod orders, when the solution is smooth inits approach to steady state. The ODE solverused is BDF-DASPK. We use a relative tol-erance of 10−3 and an absolute tolerance of10−6 for the local error control in the ODEsolver, which is slightly tighter than COM-SOL defaults. We experimented with coarseras well as tighter tolerances. At coarser tol-erances, some features of the solution are notas clear. Tighter tolerances confirm the re-sults obtained for the tolerances used, thusthese are the most effective tolerances to use.The simulations reported below using the twocomponent model required 229 time steps andtook 95 seconds. This is markedly faster thansimulations for the three species model withlarge λ values, which can be, say, 1,743 timesteps and 1,802 seconds for λ = 109 [1, Ta-ble 1]. The increased numerical difficulty as-sociated with such large λ values materializesalso in other ways, e.g., that the ODE solverbroke down for the coarse 64 × 64 mesh atan intermediate time and we were forced touse a coarser ODE tolerance for this case. Wechoose the ODE solver BDF-DASPK becauseBDF-IDA, the default solver, has trouble con-verging at the initial conditions for the case oflarge λ values. The ODE solver GeneralizedAlpha was also experimented with, but thistoo had some difficulties.

• We use the linear solver PARDISO, since itis supposed to profit most from the multi-threading available on the dual-core proces-sors used; see [4] in the same proceedings asthis paper.

4 Results

In this section, we present representative numeri-cal results in the form of surface and contour plotsobtained by COMSOL Multiphysics. These simu-lations use the two component model (2.4)–(2.7) toefficiently compute u1 and u2, and then we back-transform to the original variables u, v, w of (1.1)–(1.4) using (2.3).

We are interested in initial condition functionsuini, vini, wini such as shown in Figure 1, whereeach connected piece of each initial concentrationhas some constant value. For u in Figure 1 (a),which has an unlimited supply with concentrationα = 1.6 at x = 0, the left hand portion of thedomain, all portions of the domain connected to it,as well as the small disjoint disk in the upper rightof the domain have values of uini = α, and uini = 0everywhere else. For v in Figure 1 (b), which hasan unlimited supply with concentration β = 0.8 atx = 1, the values are vini = 0 wherever uini > 0and wini = β wherever uini = 0. We note that bythis construction the product uv ≡ 0 at t = 0. Theconcentration of wini is 0 throughout the domain att = 0 and not shown in the figure. The fast reactionin the reaction model is restricted to areas of Ωwhere u and v co-exist. This is only the case alongthe interface through the domain where u and vcome in contact due to diffusion. The locations ofthese interface lines are shown in Figure 1 (c). Theinterface location is determined numerically as the0 level of a contour plot of the quantity u−v (withonly this one contour level shown in the plot). Toallow us a concrete reference to various parts ofthe domain, we use the following terminology: Theportion on the left side of the domain is called the‘body,’ and the piece protruding from the body iscalled the ‘head,’ which is connected to the bodyby the ‘neck.’ The inspiration for these terms ismost evident in the interface plot in Figure 1 (c).

Figures 1, 2, 3, and 4 each show the concentra-tions u and v and the interface where u and v comein contact due to diffusion, at times 0, 10−3, 10−2,and 20, respectively. After starting with sharp in-terfaces between areas with positive u or v and0 in Figure 1, there is a rapid initial transient intime during which the reaction-diffusion equationssmooth out the concentrations. This effect is vis-ible in Figures 2 and 3 for u and v. The mostintriguing feature of the model under study is thebehavior resulting for the location of the reaction

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interface, where u > 0 and v > 0 meet by diffusionand react rapidly. Recall that at the initial time,these two species do not coexist and mathemati-cally uv ≡ 0. It is only by diffusion that positivevalues of both get in contact with each other andthis defines the reaction interface. At the initialtime t = 0 in Figure 1 (c), we can still clearly seethe ‘head’ connected by the ‘neck’ to the ‘body’ onthe left portion of the domain and an disjoint ‘disk’in the upper right of the domain. By t = 10−3 inFigure 2 (c), we see the head separating from thebody in the left portion of the domain, before it re-attaches to the body by t = 10−2 in Figure 3 (c).Also by the time of Figure 3, the disjointed disk inthe upper right of the domain has vanished withits supply of the u species completely consumed.By t = 20 in Figure 4, the solution of the problemhas reached its steady state; this is confirmed bycomparing the solutions for u, v, and w as func-tions of 0 < x < 1 for each y to the solution of thestationary problem (with ut = vt = wt = 0) as-sociated with (1.1)–(1.2) in one spatial dimension[6]. Numerically, we can moreover compare the lo-cation of the interface in Figure 4 (c) to its knownsteady state location at x∗ ≈ 0.6; see [7, Table 1].

The chosen initial conditions give rise to an in-teresting behavior in that the head separates andthen re-attaches to the body in the left portion ofthe domain, while the disjoint disk disappears. Ad-ditional plots for this simulation based on the twocomponent model are shown in [1], along with com-parisons to studies with the three species model forseveral values of λ with respect to accuracy and ef-ficiency.

Acknowledgments

The work reported in this paper is based on con-sulting projects in the Center for InterdisciplinaryResearch and Consulting (www.umbc.edu/circ).The hardware used in the computational stud-ies is part of the UMBC High Performance Com-puting Facility (HPCF). The facility is supportedby the U.S. National Science Foundation throughthe MRI program (grant no. CNS–0821258) andthe SCREMS program (grant no. DMS–0821311),with additional substantial support from the Uni-versity of Maryland, Baltimore County (UMBC).See www.umbc.edu/hpcf for more information onHPCF and the projects using its resources.

References

[1] Aaron Churchill, Matthias K. Gobbert, andThomas I. Seidman. Efficient computation fora reaction-diffusion system with a fast reac-tion in two spatial dimensions using COMSOLMultiphysics. Technical Report HPCF–2009–7, UMBC High Performance Computing Facil-ity, University of Maryland, Baltimore County,2009.

[2] Aaron Churchill, Guan Wang, Matthias K.Gobbert, and Thomas I. Seidman. Efficientsimulation of a reaction-diffusion system with afast reaction in the asymptotic limit. In prepa-ration.

[3] Michael Muscedere and Matthias K. Gobbert.Parameter study of a reaction-diffusion systemnear the reactant coefficient asymptotic limit.Dynamics of Continuous, Discrete and Impul-sive Systems Series A Supplement, pages 29–36, 2009.

[4] Noemi Petra and Matthias K. Gobbert. Par-allel performance studies for COMSOL Multi-physics using scripting and batch processing. InYeswanth Rao, editor, Proceedings of the COM-SOL Conference 2009, Boston, MA, 2009.

[5] Thomas I. Seidman. Interface conditions fora singular reaction-diffusion system. Discreteand Cont. Dynamical Systems, to appear.

[6] Ana Maria Soane, Matthias K. Gobbert, andThomas I. Seidman. Numerical exploration ofa system of reaction-diffusion equations withinternal and transient layers. Nonlinear Anal.:Real World Appl., 6(5):914–934, 2005.

[7] Guan Wang, Aaron Churchill, Matthias K.Gobbert, and Thomas I. Seidman. Efficientcomputation for a reaction-diffusion systemwith a fast reaction with continuous and dis-continuous initial data using COMSOL Mul-tiphysics. Technical Report HPCF–2009–3,UMBC High Performance Computing Facil-ity, University of Maryland, Baltimore County,2009.

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(a) u(x, y) vs. (x, y)

(b) v(x, y) vs. (x, y)

(c) interface vs. (x, y)

Figure 1: u, v, and the interface at t = 0.

(a) u(x, y) vs. (x, y)

(b) v(x, y) vs. (x, y)

(c) interface vs. (x, y)

Figure 2: u, v, and the interface at t = 10−3.

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(a) u(x, y) vs. (x, y)

(b) v(x, y) vs. (x, y)

(c) interface vs. (x, y)

Figure 3: u, v, and the interface at t = 10−2.

(a) u(x, y) vs. (x, y)

(b) v(x, y) vs. (x, y)

(c) interface vs. (x, y)

Figure 4: u, v, and the interface at t = 20.

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