Maximum norm error estimates of a linear CADI method for the Klein–Gordon–Schrödinger equations

Abstract

In this paper, an efficient linear compact alternating direction implicit (CADI) scheme with second order temporal accuracy and fourth order spatial accuracy is proposed for approximating solution to the initial–boundary problem for the two-dimensional Klein–Gordon–Schrödinger equations. Furthermore, maximum norm error estimates of numerical solutions are obtained by using the energy method and the mathematical induction method. Finally, some numerical experiments are presented to support the theoretical results.

Introduction

In this paper, we consider the following two-dimensional nonlinear Klein–Gordon–Schrödinger (KGS) equations

$$i{\varphi }_t+\frac{1}{2}\Delta \varphi +u\varphi =f(x,y,t),(x,y)\in \Omega ,t\in (0,T],$$$$u_{tt}-\Delta u+u-{\left|\varphi \right|}^2=g(x,y,t),(x,y)\in \Omega ,t\in (0,T],$$

subject to the initial and boundary conditions

$$(\varphi ,u,u_t)(x,y,0)=({\varphi }_0,u_0,u_1)(x,y),(x,y)\in \bar{\Omega }=\Omega \cup \partial \Omega ,$$$$\varphi (x,y,t)=u(x,y,t)=0,(x,y)\in \partial \Omega ,t\in (0,T],$$

where the complex unit $i=\sqrt{-1}$, $\Delta$ is the Laplacian operator, $\Omega =(a,b)\times (c,d)$, $\partial \Omega$ is the boundary of $\Omega$, $f,{\varphi }_0$ are given regular complex functions, and $g,u_0,u_1$ are given regular real-valued functions.

The KGS equations are classical models which describe the interaction between complex neutron field and neutral meson Yukawa in quantum field theory [1], [2], where $\varphi (x,t)$ and $u(x,t)$ represent a complex scalar nucleon field and a real scalar meson field, respectively. A number of research studies have been conducted for the KGS equations. We refer to [1], [2], [3], [4], [5], [6], [7] for various mathematical properties of the system, for instance, the existence, uniqueness, attractors, asymptotic behavior and stability of solutions. Along the numerical front, different numerical methods such as the spectral method [8], [9], Chebyshev pseudo-spectral multidomain method [10] and multi-symplectic method [11], [12], have been investigated for the KGS equations.

Thanks to the simplicity of implementation on machines, finite difference method is a widely used method in solving the KGS equations. There are many accurate and effective difference schemes for one-dimensional homogeneous KGS equations, see e.g. [13], [14], [15], [16], [17]. Recently, Wang et al. [18] developed two second-order linear difference schemes for three-dimensional homogeneous KGS Equations. They also proved that their schemes were stable and convergent without any restrictions on the mesh size. Nevertheless, in practical computation, one scheme was coupled and then cannot be solved simultaneously by parallel computing. Although the other scheme is decoupled, it led to a block tri-diagonal system of linear equations and may require considerable computational effort.

In the past few years, there has been growing interest in developing, analyzing and implementing ADI or compact ADI (CADI) methods for solving multidimensional partial differential equations [19], [20], [21], [22], [23], [24], [25], [26]. For instance, two second-order ADI difference schemes were discussed in [23], and a fourth-order CADI method was suggested for the linear Schrödinger equations with periodic boundary conditions [20]. Obviously, in terms of computational efficiency, ADI or CADI methods are preferable because they can simplify the solution of a multidimensional problem into a set of independent one-dimensional problems. However, to our best knowledge, few works on ADI or CADI schemes have so far been published for the coupled KGS equations.

When numerical errors are measured in practice, it is more convenient to perform the error estimate in the grid-independent maximum norm than in the $L_2$ norm. The purpose of this paper is therefore to develop a high-order CADI scheme for solving the KGS equations, and derive the maximum norm error estimates of the difference solution. However, in general, due to the difficulties caused by small perturbations and compact operators in CADI schemes, it is usually hard to obtain the maximum norm error estimates of numerical solutions for the multi-dimensional nonlinear Schrödinger-type equations. In this paper, we will use the energy method and the mathematical induction method to derive the desired error estimates.

The rest of this paper is organized as follows: a decoupled and linear CADI scheme is proposed in Section 2, and some important auxiliary lemmas are given in Section 3. What is more, Section 4 is devoted to achieving the maximum norm error estimates for the numerical solution. Finally, in Section 5, we verify our theoretical analysis by numerical experiments.