A fourth-order exponential wave integrator Fourier pseudo-spectral method for the Klein–Gordon equation

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Abstract

This paper is concerned with a fourth-order exponential wave integrator Fourier pseudo-spectral method for solving the Klein–Gordon equation. We suggest a new numerical integration formula to approximate the time integral term in the phase space. The error estimate shows the suggested numerical method is fourth-order accurate in time and spectral accurate in space. The theoretical findings are confirmed by numerical experiments.

Introduction

In this work, we develop a new exponential wave integrator Fourier pseudo-spectral (EWI-FP) method for the Klein–Gordon equation [1], uttΔu+u+f(u)=0,xΩ,0<tT,u,ut(x,0)=u0,u1(x),xΩ̄, where the nonlinear function f(u)=u3, the spatial variables x=(x,y)T and domain Ω=(0,L)2R2 with its closure Ω̄ for brevity. Note that the boundary conditions are set to be periodic so as not to complicate the analysis with unwanted details.

The EWI-type numerical strategies have significant interest in designing numerical algorithms for solving time dependent PDEs [2], [3], [4], [5], [6], [7]. Bao et al. [2] developed and analyzed a Gautschi-type EWI Fourier pseudo-spectral method for the Klein–Gordon equation in the nonrelativistic limit regime. The resulting numerical scheme is explicit, time symmetric and implemented effectively via the fast discrete Fourier transform. Also, the authors pointed out that the trapezoidal rule can be used to approximate the time integral term in the phase space. Dong [3], [4] considered the trigonometric integrator pseudo-spectral methods with or without filters for the N-coupled nonlinear Klein–Gordon equations. Numerical analysis shows that the proposed numerical schemes are second-order accurate in time and spectral-order accurate in space. Zhao et al. [5], [6] designed and analyzed the EWI sine pseudo-spectral methods for the Klein–Gordon–Zakharov system. In addition, Zhao [7] investigated an EWI-FP method for the symmetric regularized long wave equation. As can be seen, the EWI-type numerical algorithms introduced above have limited second-order time accurate, which comes from the trapezoidal or Gautschi-type numerical rules. In [8], Wang and Zhao introduced a new numerical strategy to develop arbitrarily high-order EWI-type numerical algorithms, i.e., they approximated the unknown integrals by using Taylor’s expansion of the integrand up to any even order and then carried out the rest integrations exactly. The resulting scheme is fully explicit and efficient; while it always requires to evaluate higher derivatives of nonlinearity.

In this paper, to improve the time accuracy and to avoid using the potential information of derivatives of nonlinear terms, we suggest a fourth-order time accurate EWI-FP method for the Klein–Gordon equation (1.1)–(1.2). In the next section, a second ODE equation (2.2) is first obtained by using the Fourier spectral method, and a corrected trapezoidal rule (2.5) is then introduced to approximate the time integral term in the phase space. Although the formula (2.5) involves the values of first-order derivative of integral function, the special structure of formula (2.4) eliminates the unwanted values automatically. As a result, the numerical scheme (2.6) is almost as straightforward as the earlier ones [2], [5]; while the error estimate reported in Section 3 shows that the new numerical approach can achieve fourth-order accurate in time, which may be more sufficient than the ones having the second-order time accuracy. Also, numerical results in Section 4 confirm the theoretical findings. Note that, throughout this paper, any subscripted c, such as c1 and cu, denotes a generic positive constant that only depends on the solution and given data but is always independent of the time and space mesh sizes.

Section snippets

Numerical scheme

For positive integer M, let the time levels tn=nτ,0nM with the time-step τTM. For the domain Ω=(0,L)2, let the grid lengths hx=hy=hLN with an even positive integer N for simplicity. Define the discrete domains Ωh{xh=(ih,jh)1i,jN}. For a periodic function v(x)L2(Ω), let PN:L2(Ω)N be the standard L2-projection operator onto the space N, consisting of all trigonometric polynomials of degree up to N2, and IN:L2(Ω)N be the trigonometric interpolation operator [9], [10], that is, PNv(

Numerical analysis

Let Cper(Ω) be a set of infinitely differentiable periodic functions defined on Ω for spatial variables. Hperr(Ω) is the closure of Cper(Ω) in Hr(Ω) that endowed with the semi-norm ||Hperr and the norm Hperr. For simplicity, denote ||Hr||Hperr, HrHperr, and L2.

To facilitate the convergence analysis of EWI-FP scheme (2.6), we assume the solution u of problem (1.1)–(1.2) satisfies uC([0,T];Hperr)C1([0,T];Hper2)C2([0,T];Hper1)C3([0,T];L2)C4([0,T];L2)for some r3

Numerical example

We run the EWI-FP scheme (2.6) for solving the problem (1.1)–(1.2) numerically. The Taylor expansion formula is used to obtain the first-level solution, and a simple iteration is employed to solve the nonlinear system at each time level. Always, we use the solution at previous level as the initial guess for each iteration and the numerical tolerance is taken as 10−12.

Example 4.1

Consider the Klein–Gordon equation (1.1)–(1.2) with the initial data [12], u0(x)=exp((x+2)2y2)+exp((x2)2y2),u1(x)=exp(x2y2)

CRediT authorship contribution statement

Bingquan Ji: Investigation, Methodology, Writing - original draft. Luming Zhang: Supervision, Writing - review & editing.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which are very helpful for improving the quality of the article.

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Luming Zhang is supported by the NSFC, PR China grant No. 11571181.

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