Engineering Analysis with Boundary Elements 37 (2013) 1052–1058

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Engineering Analysis with Boundary Elements

0955-79http://d

☆Resen CorrE-m

1011018

journal homepage: www.elsevier.com/locate/enganabound

Conservative multiquadric quasi-interpolation method for Hamiltonianwave equations$

Zongmin Wu, Shengliang Zhang n

Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, PR China

a r t i c l e i n f o

Article history:Received 6 February 2013Accepted 23 April 2013Available online 21 May 2013

Keywords:Meshless methodQuasi-interpolationEnergy conservationHamiltonian wave equationsSymplectic integrator

97/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.enganabound.2013.04.011

arch partially supported by Grant 12DZ22728esponding author. Tel.: +86 13681731799.ail addresses: zmwu@fudan.edu.cn (Z. Wu),0035@fudan.edu.cn (S. Zhang).

a b s t r a c t

Hamiltonian PDEs have some invariant quantities such as energy and momentum, etc., which should bewell conserved with the numerical integration. In this paper we concentrate on the nonlinear waveequation. We study how a space discretization by using multiquadric quasi-interpolation method makesthe space discretized system also possess some invariants which are good approximation of thecontinuous energy. Then, appropriate symplectic scheme is employed for the integration of the semi-discretized system. Theoretical results show that the proposed method has not only high order accuracybut also good properties of long-time tracking capability. Some numerical examples are presented todemonstrate the effectiveness of the proposed method.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Hamiltonian wave equations can be treated as Hamiltoniansystems (in infinite dimensions) [20]. Hamiltonian formalism hasthe important property of being area-preserving (symplectic). Tosolve these Hamiltonian PDEs numerically, one hopes that thenumerical solution will hold this property too. A standard methodto obtain symplectic scheme for an infinite-dimensional Hamilto-nian PDEs is that, first discretize the Hamiltonian PDEs in space toobtain a finite-dimensional Hamiltonian system, and then evolvethe semi-discrete system by symplectic integrators [2]. In thisnumerical procedure, the key for success is to ensure that theobtained semi-discrete system is a finite-dimensional HamiltonianODEs system, for which finite difference method (FDM) [6,11],finite element method (FEM) [27], Fourier pseudospectral method[13] can be utilized. However, most of those methods depend on asuitable generation of meshes, which is difficult for problems withvery complicated and irregular geometries. To develop a meshlesssymplectic integrator or meshless energy-conserving numericalscheme on scattered nodes motivates the current work.

It is well known that the multiquadric is one of the most oftenapplied kernels in meshless methods. Multiquadric kernels wereproposed by Hardy [9]. Franke designed lots of numerical experi-ments, among which multiquadrics performed best [8]. Thereforemultiquadric quasi-interpolation method has caught the

ll rights reserved.

00.

attentions of many researchers. For the meshless collocation (orinterpolation) method for PDEs by using multiquadric functions,one is required to solve a large scaled linear system of equations;moreover, the coefficients matrix is usually very ill-conditionedand the results are sensitive to the shape parameter c [16]. Themost important advantage of quasi-interpolation is that one canevaluate the approximant directly without needs to solve anylinear system of equations. Beatson and Powell first proposedsome quasi-interpolation scheme by using multiquadric [1]. Beat-son even used the multiquadric quasi-interpolation as a computeraided design tool in the film “The Lord of the Rings III”. Ref. [23]improved these schemes and discussed their approximation orderand the shape preserving property. Lately [12] proved that multi-quadric quasi-interpolation can approximate not only the functionitself but also its high order derivatives. Ref. [24] used the multi-quadric quasi-interpolation to solve free boundary diffusion pro-blem. The multiquadric kernel method is one of the radial basisfunctions (RBFs) methods. RBFs method for solving PDEs hasbecome one part of the new numerical meshless methods. Moredetails about RBFs meshfree approximation methods for PDEs canbe found in [5].

To be more precise, define the multiquadric functionϕðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ c2

pand ϕjðxÞ ¼ ϕðx−xjÞ, where c is a shape parameter.

Multiquadric quasi-interpolation of a function f : R↦R on thescattered knots

⋯ox−1ox0ox1o⋯oxNo⋯; h≔maxj

ðxj−xj−1Þ;

takes the form

ðLf Þ ¼∑f ðxjÞψ jðxÞ; ð1:1Þ

Z. Wu, S. Zhang / Engineering Analysis with Boundary Elements 37 (2013) 1052–1058 1053

where ψ jðxÞ are the following linear combinations of the multi-quadrics, that

ψ jðxÞ ¼ ψ ðx−xjÞ ¼ϕjþ1ðxÞ−ϕjðxÞ2ðxjþ1−xjÞ

−ϕjðxÞ−ϕj−1ðxÞ2ðxj−xj−1Þ

:

The purpose of this paper is to present a meshless energy-conserving numerical method by using multiquadric quasi-interpolation as an approximation scheme for numerical solutionof Hamiltonian wave equation

utt−uxx þ F′ðuÞ ¼ 0; ð1:2Þwhere F : R↦R is a smooth function.

An outline of the paper is as follows. In Section 2, preliminariesabout Hamiltonian wave equation and the properties of multi-quadric function are recalled. In Section 3 we will discuss how aspace discretization by using multiquadric quasi-interpolationmethod makes the space discretized system that also has someinvariants which well approximate the continuous energy. InSection 4, the conservative multiquadrics quasi-interpolationmethod is introduced by using staggered Störmer–Verlet scheme

Vnþ1=2 ¼ Vn−1=2 þ τΦ2MUn−τF′ðUnÞ

Unþ1 ¼Un þ τVnþ1=2:

This scheme conserve a quantity ~HΔ satisfying

HΔ ¼ ~HΔ þOðτ2Þ:Both the truncation error and globe error are also studied. InSection 5, numerical examples are tested to verify the effect of themethod. Finally, concluding remarks show that by further studythe proposed method can be applied to construct not onlyconservative moving knots but also high-order schemes.

2. Preliminaries

2.1. Nonlinear wave equation

We consider the nonlinear wave equation (1.2). This equation isused to model nonlinear phenomena such as the propagation ofdislocation in crystal and the behavior of elementary particles. It isalso used in soliton theory. The equation is a classical example ofHamiltonian PDEs (infinite-dimensional Hamiltonian system) [20].By defining a new variable v¼ ut , the Hamiltonian formulationgoes as

ut ¼ þ δHδv

¼ v

vt ¼−δHδu

¼ uxx−F′ðuÞ;

8>><>>: ð2:1Þ

where

Hðu; vÞ ¼ 12

Z½v2 þ u2

x þ 2FðuÞ� dx ð2:2Þ

is invariant with respect to time under an appropriate initialboundary-value condition. Here δH=δu, δH=δv are the variational,or Gateaux derivatives defined by

ddϵ

H½uþ ϵϕ�Þϵ ¼ 0

≡Z

δHδu

ϕ dx;ddϵ

H½vþ ϵϕ�Þϵ ¼ 0

≡Z

δHδv

ϕ dx:��

The symplectic form of this system

Ω¼Z

du∧dv dx ð2:3Þ

is also invariant with respect to time. More details can be found in[13,15].

2.2. Discretization method for the NLW equation

Classical methods to solve (2.1) numerically are those, where astandard procedure starts with the discretization of the equationin space and then in time. After the discretization in space, thefollowing semi-discretized problem arises:

ddtUh ¼ Vh

ddtVh ¼ AhUh−F′ðUhÞ

8>><>>: ð2:4Þ

where UhðtÞ ¼ ð…;uðjh; tÞ;…ÞT . To preserve the symplectic form of(2.1), an appropriate numerical discretization scheme needs to bedeveloped in the sense that the above resulting semi-discretesystem (continuous in time) can be written as a finite-dimensionalHamiltonian system. For this purpose, the numerical scheme isrequired to be able to preserve the symmetric property of second-order differential operator embedded in (2.1). That means asuitable Ah must be symmetric [3]. Several methods can be chosensuch as the finite difference method [2], finite element method[27] and Fourier pseudospectral method [13] on a uniform grid.

As for nonuniform knots, consider the finite divided differenceapproximation of uxx at point xi

ðuxxÞi ¼uiþ1−ui

xiþ1−xi−ui−ui−1

xi−xi−1

� �xiþ1−xi−1

2;

.

by defining Δxi ¼ ðxiþ1−xi−1Þ=2, αi ¼ 1=ðxi−xi−1Þ, βi ¼ 1=xiþ1−xi, and

M≔

⋱ αiαi −ðαi þ βiÞ βi

βi ⋱

0B@

1CA;

taking ~ui ¼ffiffiffiffiffiffiffiffiΔxi

pui, we can get

⋮ð ~uxxÞi⋮

0B@

1CA≈

⋱1ffiffiffiffiffiffiΔxi

p⋱

0B@

1CAM

⋱1ffiffiffiffiffiffiΔxi

p⋱

0B@

1CA

⋮~ui

⋮

0B@

1CA:

Sometimes the sampling data points (knots) should be movedaccording to the equation, e.g. [10,21]; it is hard to solve thenonlinear partial differential propagations equation with movingknots by using the traditional methods which depend on a suitablegeneration of meshes. Multiquadric quasi-interpolation method isa true meshless method, it can be used for constructing movingknots schemes [21] and can be generalized to high-dimensionalspace [22]. In this paper, a multiquadric quasi-interpolationmethod will be used for the spatial discretization of the waveequation.

2.3. The Multiquadric function

Multiquadric function satisfiesRϕ″ðxÞ=2 dx¼ 1, some impor-

tant properties are given in the following lemmas. Based on theapproach of Cheney [4,12] Lemmas 2.1 and 2.2 were proved.

Lemma 2.1. If f∈C2ðRÞ, then the following inequality��� Z þ1

−1f ðtÞϕ″ðx−yÞ

2dt−f ðxÞ

���≤Oðc2Þ ð2:5Þ

holds.Lemma 2.2. If f ðxÞ∈C2ðRÞ and f, f ′ and f ″ are bounded by apolynomial of degree2, 1 and 0, respectively, then the followinginequality��� Z ϕ″ðx−tÞ

2f ðtÞ dt−∑

jf ðxjÞ

ϕ″ðx−xjÞ2

Δj

���oOðh=cÞ ð2:6Þ

holds, where Δj ¼ ðxjþ1−xj−1Þ=2.

Z. Wu, S. Zhang / Engineering Analysis with Boundary Elements 37 (2013) 1052–10581054

Theorem 2.1. For a large enough real number M, the scattered knots

−1o−M¼ x−Nox−Nþ1o⋯ox0ox1o⋯xN ¼Mo þ1;

h≔maxjðxj−xj−1Þ. If vðxÞ∈C2ðRÞ ,R jvðxÞjdxo1 and

Rv2ðxÞ dxo1,

denote V ¼ ð…; vðxjÞ;…ÞT , then the following inequality��� Z 1

−1v2 dx−VTΦ−1

2 V���≤Oðh2=3Þ

holds, provided that c¼Oðh1=3Þ, where Φ2 ¼ ðϕ″ðxi−xjÞ=2ÞNi;j ¼ −N .

Proof. According to Lemmas 2.1 and 2.2, we have the followinginequality���∑jvðxjÞ

ϕ″ðx−xjÞ2

Δj−vðxÞ���≤Oðh=cÞ þOðc2Þ;

the optimum estimate can be obtained by setting h=c¼ c2 i.e.c¼ h1=3, then���∑jvðxjÞ

ϕ″ðx−xjÞ2

Δj−vðxÞ���≤Oðh2=3Þ

uniformly, independent of x.Define X≔diagðΔjÞ, then

Φ2XV ¼ V þOðh2=3Þ:Therefore, the following quadratic form

VTΦ−12 V ¼ ðVTXΦ2 þOðh2=3ÞÞΦ−1

2 ðΦ2XV þOðh2=3ÞÞ¼ VTXΦ2XV þ

ZvðxÞ dxOðh2=3Þ þ 1TΦ−1

2 1Oðh2=3Þ

¼ VTXΦ2XV þOðh2=3Þ¼ZZ

dx dyvðxÞϕ″ðx−yÞ2

vðyÞ þOðh2=3Þ

¼Z

v2ðxÞ dxþOðh2=3Þ:

This completes the proof. □

3. Multiquadric quasi-interpolation method for spacediscretization

In this section multiquadric quasi-interpolation method is usedfor space discretization. We approximate uðx; tÞ by multiquadricquasi-interpolation operator ðLuÞðx; tÞ (1.1), by rearranging thescheme

unðx; tÞ ¼ ðLuÞðx; tÞ ¼∑j

ujþ1−uj

xjþ1−xj−uj−uj−1

xj−xj−1

� �ϕjðxÞ2

:

Then one can obtain

ðuxxÞi≈un

xxðxi; tÞ ¼∑j

ujþ1−uj

xjþ1−xj−uj−uj−1

xj−xj−1

� �ϕj″ðxiÞ

2

with an error of Oðh2=3Þ, provided that c¼Oðh1=3Þ [12], thenUxx ¼Φ2MU þOðh2=3Þ;where Uxx ¼ ð…;uxxðxiÞ…Þ⊤ and U ¼ ð…;uðxiÞ…Þ⊤, the definition ofthe matrix M as before in Section 2.2. By using the secondderivative un

xx to approximate uxx, Eq. (2.1) can be written as

ddtU ¼ V

ddtV ¼Φ2MU−F′ðUÞ;

8>><>>: ð3:1Þ

since Φ2M is not a symmetric matrix, we need to introduce newvariables ~U and ~V which can make the semi-discretized system(3.1) to be a Hamiltonian system. Because ϕ″ðxÞ ¼ c2=ðc2 þ x2Þ3=2 ispositive definite, then the matrix Φ2 is positive definite. Thereexist a positive definite matrix Q satisfying Φ2 ¼ Q2. By employing

transformations

~U ¼ Q−1U; ~V ¼Q−1V ;

the system (3.1) will be (For simplicity, in what follows we omitthe nonlinear term F ′ðuÞ).

ddt

~U ¼ ~V

ddt

~V ¼QMQ ~U :

8>><>>: ð3:2Þ

Because QMQ is symmetric, the semi-discrete system (3.2) is afinite-dimensional Hamiltonian system with respect to ~U and ~V .Let Z ¼ ð ~U ; ~V Þ⊤ and

J ¼ 0−IN

−IN0

� �;

Eq. (3.2) can be rewritten as

Zt ¼ J∇ZHðZÞ;with the Hamiltonian function

Hð ~U ; ~V Þ ¼ 12 ⟨

~V ; ~V ⟩−12⟨

~U ;QMQ ~U⟩:

Actually, by means of the multiquadric quasi-interpolationmethod, not only the Hamiltonian function (2.2) but also thesymplectic structure (2.3) can be approximated with the sameorder of accuracy. To discuss the symplecticity of (3.2), we start bypreparing the follow lemma, which can be found in [18].

Lemma 3.1. If f and g are vector-valued functions and S a matrixthen

dðSf Þ∧dg¼ df∧dðS⊤gÞ:

Theorem 3.1. Hamiltonian function and symplectic structure of thefinite-dimensional Hamiltonian system (3.2) are approximations tothe infinite-dimensional Hamiltonian (2.1)'s, those are

12

Zv2 þ u2

x dx¼12⟨ ~V ; ~V ⟩−

12⟨ ~U ;QMQ ~U⟩þOðh2=3Þ

andZdu∧dv dx¼ d ~U∧d ~V þOðh2=3Þ:

Proof. According to the Theorem 2.1,

Hð ~U ; ~V Þ ¼ 12⟨ ~V ; ~V ⟩−

12⟨ ~U ;QMQ ~U⟩

¼ 12⟨Q−1V ;Q−1V⟩−

12⟨Q−1U;QMQQ−1U⟩¼ 1

2⟨V ;Φ−1

2 V⟩−12⟨U;MU⟩

¼ 12VTΦ−1

2 V þ 14∑j

ui−ui−1

xi−xi−1

� �2

ðxi−xi−1Þ þ14∑j

uiþ1−ui

xiþ1−xi

� �2

ðxiþ1−xiÞ

¼ 12

Zv2 þ u2

x dxþOðh2=3Þ;

and by Lemma 3.1,

d ~U∧d ~V ¼ dQ−1U∧dQ−1V

¼ dU∧dΦ−12 V

¼Z

du∧dvþOðh2=3Þ:

This completes the proof. □

According to the Theorem 3.1, we obtain the followingtheorem.

Z. Wu, S. Zhang / Engineering Analysis with Boundary Elements 37 (2013) 1052–1058 1055

Theorem 3.2. For the semi-discretized system (3.1), define a discreteenergy function

HΔ≔12⟨V ;Φ−1

2 V⟩−12⟨U;MU⟩¼ 1

2ðUT ;VT ÞS U

V

� �;

where

S≔−M0

0Φ−1

2

� �;

then

dHΔ

dt¼ 0 and HΔ ¼

12

Zv2 þ u2

x dxþOðh2=3Þ:

Proof. The system (3.1) can be written in a matrix form

∂∂t

U

V

� �¼D

U

V

� �

where

D≔0

Φ2MI0

� �:

By taking derivatives of HΔ with respect to time, one can check

ddt

HΔ ¼12ðUT ;VT ÞðDTS þ SDÞ U

V

� �¼ 0:

That means the discretized energy function HΔ is an invariant of(3.1) and is also an approximation of the continuous energy. □

Theorem 3.2 shows an important idea for constructing energy-conserving numerical methods. The continuous energy functionHðu; vÞ ¼ 1

2

R ½v2 þ u2x � dx can be approximated in many sensible

ways by the quadratic

ðUT ;VT ÞS U

V

� �;

conservativeness in the sense that

U;V : ðUT ;VT ÞS U

V

� �¼ const

� :

As discussed above, the differentiation matrix D that appearsshould be skewed with respect to the same S that appears in thequadrature formula (i.e. DTS þ SD¼ 0).

4. Symplectic methods for time discretization

4.1. Symplectic methods for time discretization

The finite-dimensional Hamiltonian system (3.2) can be dis-cretized in time by lots of symplectic methods. Feng and hiscollaborators developed implicit symplectic schemes based on thegenerating function [6,7]. Sanz-Serma and Lasagni proposed sym-plectic Runge–Kutta methods, who found a condition for implicitRunge–Kutta methods to be symplectic [18]. Ruth developed anidea of explicit symplectic schemes for separable Hamiltoniansystem. Along the line of Ruth [17], Neri [14] and Yoshidapresented higher order integrators based on the Baker–Camp-bell–Hausdorff formula [26]. The Hamiltonian function whichconsists of a sum of kinetic and potential energies

Hðu; vÞ ¼ T ðvÞ þ VðuÞis often discretized by the staggered Störmer–Verlet scheme

unþ1 ¼ un þ τ∇vT ðvnþ1=2Þvnþ1=2 ¼ vn−1=2−τ∇uVðunÞ; ð4:1Þwhere ðun; vnþ1=2Þ ¼ ðuðtnÞ; vðtnþ1=2ÞÞ and tn ¼ t0 þ nτðτ¼ΔtÞ. Thisscheme is second-order symplectic with respect to a staggered

symplectic form (e.g. [18]), that is

ω¼ dun∧dvnþ1=2;

where dun and dvnþ1=2 are solution of the discrete variationalequation associated with (4.1), then

ωnþ1−ωn

τ¼ 0 or ωnþ1 ¼ ωn:

According to the discussion above, the following theorem can beobtained

Theorem 4.1. Integrating the Hamiltonian system (3.2) in time bythe staggered Störmer–Verlet scheme, we can obtain a symplecticscheme

Unþ1 ¼Un þ τVnþ1=2

Vnþ1=2 ¼ Vn−1=2 þ τΦ2MUn; ð4:2Þwith a conserved quantity ~HΔ satisfying HΔ ¼ ~HΔ þOðτ2Þ, where

HΔ ¼12ðU;VÞ

−M 00 Φ−1

2

!U

V

� �:

In order to estimate the error of the scheme (4.2), the followingequation is prepared which can be obtained by eliminating thevalue Vnþ1=2:

Unþ1−2Un þ Un−1

τ2¼Φ2MUn: ð4:3Þ

4.2. Convergence analysis

In this paper, our analysis is restricted to the conservativemultiquadric quasi-interpolation scheme by using the staggeredStörmer–Verlet scheme in time. We shall estimate the error of theproposed numerical methods and prove that the scheme is stableand convergent. First the truncation error is considered. Denote∥ � ∥ the L2 norm.

Theorem 4.2. Suppose uðx; tÞ∈HsðRÞ, s≥3, for any t∈½0; T �,uðx; tÞ∈C4ðRÞ, ∀x. Then the truncation error Rn of the scheme (4.2)satisfies

∥Rn∥≤Oðτ2 þ h2=3Þ;provided that c¼Oðh1=3Þ.

Proof. According to Eq. (4.3),

Unþ1−2Un þ Un−1

τ2¼Φ2MUn

¼Unxx þOðh2=3Þ:

Based on Taylor expansion, the following equation can beobtained:

Unþ1−2Un þ Un−1 ¼ τ2Untt þOðτ4Þ:

Hence, the truncation error goes as

Rn ¼ Unþ1−2Un þ Un−1

τ2−Φ2MUn−ðUn

tt−UnxxÞ

≤Oðτ2 þ h2=3Þ: □

Then, the following error estimate of the scheme (4.2) isobtained.

Theorem 4.3. Suppose uðx; tÞ satisfies the same condition as inTheorem4.2. Let uðx; tnÞ and uapprðx; tnÞ be the solution of (1.2) and

Table 1A posteriori estimate of the scheme (4.2) with ðτ¼ 0:002Þ.

N Error ðL∞Þ Error (L2) Order (L2)

128 9.2e−3 3.8259e−3256 5.9e−3 1.9726e−3 0.96512 9.85e−4 8.5758e−4 1.10

Z. Wu, S. Zhang / Engineering Analysis with Boundary Elements 37 (2013) 1052–10581056

(4.2), respectively. Denote Un ¼ ð…;uðxi; tnÞ;…Þ⊤, Unappr ¼

ð…;uapprðxi; tnÞ;…Þ⊤ and en ¼Un−Unappr . Then the error estimate of

the scheme (4.2) at time T satisfies

∥eðTÞ∥¼ ∥eL∥≤Oðτ2 þ h2=3Þ; L¼ Tτ:

Proof. Based on Taylor expansion, the following equation can beobtained:

Unþ1 þ 2Un þ Un−1 ¼ 4Un þ τ2Untt þOðτ4Þ;

Define B≔Φ2M, one can get

~Rn ¼ enþ1−2en þ en−1

τ2−B

enþ1 þ 2en þ en−1

4

� �≤Oðτ2 þ h2=3Þ: ð4:4Þ

Define

δtenþ1=2 ¼ enþ1−en

τ; enþ1=2 ¼ enþ1 þ en

2

and take inner product to the both sides of (4.4) withδtenþ1=2 þ δten−1=2, it follows that

1τð∥δtenþ1=2∥2−∥δten−1=2∥2Þ

þ 1τð⟨−Benþ1=2; enþ1=2⟩−⟨−Ben−1=2; en−1=2⟩Þ

¼ ⟨ ~Rn; δtenþ1=2 þ δten−1=2⟩:

Next, define Wn as

Wn ¼ ∥δtenþ1=2∥2 þ ∥en∥2 þ ∥enþ1∥2 þ ⟨−Benþ1=2; enþ1=2⟩;

in order to estimate Wn, several inequations are employed here.First

⟨−Benþ1=2; enþ1=2⟩¼ ⟨−BðUnþ1=2−Unþ1=2appr Þ;Unþ1=2−Unþ1=2

appr ⟩

≈−⟨ðU″Þnþ1=2−ðU″apprÞnþ1=2;Unþ1=2−Unþ1=2appr ⟩

¼ ⟨ðU′Þnþ1=2−ðU′apprÞnþ1=2; ðU′Þnþ1=2−ðU′apprÞnþ1=2⟩

≥0:

In addition

∥enþ1∥2−∥en−1∥2

τ¼ ⟨enþ1 þ en−1; δtenþ1=2 þ δten−1=2⟩

≤∥en∥2 þ ∥enþ1∥2 þ ∥en−1∥2 þ ∥δtenþ1=2∥2 þ ∥δten−1=2∥2:

Therefore

Wn−Wn−1

τ¼ ⟨ ~R

n; δtenþ1=2 þ δten−1=2⟩þ

∥enþ1∥2−∥en−1∥2

τ

≤∥ ~Rn∥2 þ ∥δtenþ1=2∥2 þ ∥δten−1=2∥2 þ ∥en∥2 þ ∥enþ1∥2

þ∥en−1∥2 þ ∥δtenþ1=2∥2 þ ∥δten−1=2∥2

≤An þ 2ðWn þWn−1Þ:

The estimate of WL can be obtained by using Gronwall inequality[28]:

WL ≤ W0 þ τ ∑L

n ¼ 1An

� �e8T ;

Because

∥e0∥2 ¼ ∥R0∥2 ¼ 0; ∥e1∥2 ¼Oðτ2 þ h2=3Þ2;one can get

W0 ¼Oðτ2 þ h2=3Þ2;from Theorem 4.2

An ¼ ∥ ~Rn∥2 ¼ ∥Rn∥2 ¼Oðτ2 þ h2=3Þ2;

based on the Gronwall inequality

WL ¼ ∥δteLþ1=2∥2 þ ∥eL∥2 þ ∥eLþ1∥2

þ ⟨−BeLþ1=2; eLþ1=2⟩≤Oðτ2 þ h2=3Þ2:

Finally, the error estimate

∥eL∥≤Oðτ2 þ h2=3Þ

is obtained. □

5. Numerical examples

5.1. Test for one-dimensional Hamiltonian wave equation

In this subsection, to test the accuracy and the efficiency of thesymplectic multiquadric quasi-interpolation method (4.2), we givetwo examples, one is linear wave equation for posteriori estimateand the other is nonlinear wave equation for long-time simulation.There are several ways to handle the boundary, here LE whichappears in [12] is employed.

Example 5.1. Consider the linear wave equation

utt ¼ uxx ð5:1Þ

with initial conditions

uðx;0Þ ¼ gðxÞ ¼ expð−4x2Þ; vðx;0Þ ¼ 0: ð5:2Þ

The real solution is

uðx; tÞ ¼ 12½gðxþ tÞ þ gðx−tÞ�: ð5:3Þ

The problem is considered in the spatial interval ½−1;1� and thetime to T¼1. In our computations, we choose Δx¼ 2=N where N isthe space grid number. The L2 and L1 errors are shown in Table 1with c¼ 0:1

ffiffiffiffiffiffiΔx3

pand Δt ¼ 0:002. The root mean square error L2ðtnÞ

and the max error L1ðtnÞ are, respectively, defined as

L2ðtnÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N þ 1∑N

j ¼ 0ðuðxj; tnÞ−uapprðxj; tnÞÞ2

s

and

L1ðtnÞ ¼max1 ≤ j ≤N juðxj; tnÞ−uapprðxj; tnÞj

As expected, the scheme has nearly first order accuracy inspace, which performs better than Theorem 4.3.

Example 5.2. Consider Sine–Gordon equation

utt−uxx þ sin ðuÞ ¼ 0 ð5:4Þ

with initial conditions

uðx;0Þ ¼ 4 arctan expx

1−μ2

� �; utðx;0Þ ¼−

2μffiffiffiffiffiffiffiffiffiffiffi1−μ2

p sechx

1−μ2

� �

ð5:5Þ

Z. Wu, S. Zhang / Engineering Analysis with Boundary Elements 37 (2013) 1052–1058 1057

where μ is a constant. The soliton solution to this equation is

uðx; tÞ ¼ 4 arctan expx−μtffiffiffiffiffiffiffiffiffiffiffi1−μ2

p !

: ð5:6Þ

Eq. (5.4) has been used for numerical methods of differentialequation by multiquadric quasi-interpolation in many literature.However, usually those method are nonsymplectic so that they cannot capture the long-time dynamic of the soliton. Here we use thesymplectic Eq. (4.2) to simulate the traveling of the soliton wavewith time step τ¼ 0:0004, spatial step Δx¼ 0:1 and c¼ 0:19

ffiffiffiffiffiffiΔx3

p.

In Fig. 1, we give three snap shots of the soliton wave at timet¼0.0, t¼2.5 and t¼12.5. The numerical results still give theprefect soliton wave even when it is computed up to t¼12.5, after31 250 discrete time steps.

5.2. Test for two-dimensional Hamiltonian wave equation

Table 2A posteriori estimate of the scheme (5.9) ðτ¼ 0:002Þ.

M ðh¼ 1=MÞ L2 L∞ Order (L2)

20�20 9.1�10−3 11.5�10−3

40�40 5.8�10−3 10.1�10−3 0.9560�60 3.8�10−3 9.1�10−3 1.0480�80 2.5�10−5 8.0�10−3 1.03

Example 5.3. Consider the linear wave equation

∂2u∂t2

¼ ∂2u∂x2

þ ∂2u∂y2

; ðx; y; tÞ∈Ω� ð0; TÞ; ð5:7Þ

with initial conditions

uðx; y;0Þ ¼ 0; vðx; y;0Þ ¼ffiffiffi2

pπ sin ðπxÞ sin ðπyÞ:

and boundary condition

uðx; y; tÞ ¼ 0; ðx; yÞ∈∂Ωwhere Ω¼ fðx; yÞ : −1≤x; y≤1g. The real solution is

uðx; tÞ ¼ sin ðffiffiffi2

pπtÞ sin ðπxÞ sin ðπyÞ: ð5:8Þ

The numerical algorithm is

Unþ1 ¼Un þ τVnþ1=2

Vnþ1=2 ¼ Vn−1=2 þ τLn; ð5:9Þwhere Un ¼ ðUðxi; yj; tnÞÞij, Vnþ1=2 ¼ ðVðxi; yj; tnþ1=2ÞÞij andLn ¼Φ2xMxU

nMyΦy þΦxMxUnMyΦ2y:

Here Φx ¼ ðϕðxi−xjÞ=2Þij, Φ2x ¼ ðϕ″ðxi−xjÞ=2Þij,

Mx ¼⋱ αiαi −ðαi þ βiÞ βi

βi ⋱

0B@

1CA;

(αi ¼ 1=ðxi−xi−1Þ, βi ¼ 1=ðxiþ1−xiÞ), and the definitions of Φy, Φ2y andMy are similar like Φx, Φ2x and Mx. The problem is considered till

−10 −5 0 50

5

10

−10 −5 0 50

5

10

−10 −5 0 50

5

10

Fig. 1. The numerical solution to Example 5.2. Solid line represents the

time T¼1. In our computations, we choose Δx¼Δy¼ 2=N, whereN2 is the space grid numbers. The L2 and L1 errors, as well as thenumerical order of accuracy are contained in Table 2.

Remark 5.1. For multi-dimensional Hamiltonian wave equations,one can construct conservative numerical schemes by using thetensor product algorithm of one-dimensional quasi-interpolation.One also can use high dimensional quasi-interpolation for multi-dimensional cases, more details about high dimensional quasi-interpolation can be found in [22].

Remark 5.2. For multi-dimensional Hamiltonian wave equations,one can construct conservative numerical schemes by using radialbasis interpolation with high-order accuracy, more detail can befound in the coming paper [25].

The above numerical tests show that the proposed conservativemultiquadric quasi-interpolation scheme has not only high orderof accuracy but also good behavior in long time simulation.

6. Conclusions

Remark 6.1. The meshless feature of the multiquadric quasi-interpolation method is the most important advantage of thisscheme over the traditional mesh-dependent techniques. At thesame time, it still possesses high accuracy. Similarly, highlyaccurate quasi-interpolation operators can be employed if onewant to construct high-order conservative meshless numericalschemes, e.g. construction techniques for highly accurate quasi-interpolation can be found in [19].

Remark 6.2. The proposed method can be used for constructingconservative moving knots schemes by using the techniques formoving knots scheme, e.g. the methods of construction techniquesfor moving knots scheme can be found in [10,21].

Remark 6.3. The proposed method can be generalized to solveany Hamiltonian systems with second-order spatial derivatives

10 15 20

10 15 20

10 15 20

analytic solution, while the circle represents numerical solution.

Z. Wu, S. Zhang / Engineering Analysis with Boundary Elements 37 (2013) 1052–10581058

embedded in, such as the nonlinear Schrödinger equation

iut−Δuþ f ðuÞ ¼ 0:

How to apply the proposed quasi-interpolation method to solvemulti-symplectic PDEs is our next research subject.

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