An explicit fourth-order compact difference scheme for solving the 2D wave equation

Yongbin Ge,Yunzhi Jiang

doi:10.1186/s13662-020-02870-z

Abstract

In this paper, an explicit fourth-order compact (EFOC) difference scheme is proposed for solving the two-dimensional(2D) wave equation. The truncation error of the EFOC scheme is O({tau ^{4}} + {tau ^{2}}{h^{2}} + {h^{4}}), i.e., the scheme has an overall fourth-order accuracy in both time and space. Because the scheme is explicit, it does not need any iterative processes. Afterwards, the stability condition of the scheme is obtained by using the Fourier analysis method, which has a wider stability range than other explicit or alternation direction implicit (ADI) schemes. Finally, some numerical experiments are carried out to verify the accuracy and stability of the present scheme.

Highlights

In this paper, we consider the 2D wave equation as follows: ∂2u ∂t2 = a2∂2u ∂2u ∂x2 + ∂y2 + f (x, y, t),(x, y, t) ∈ Ω × [0, T], (1)with the initial conditions u(x, y, 0) = φ(x, y), (x, y) ∈ Ω, (2)∂u(x, y, 0) = ψ(x, y)
We assume that all the functions are sufficiently smooth for achieving the accuracy order of the difference scheme
It is easy to know that the truncation error of the scheme is O(τ 4 + τ 2h2 + h4), i.e., it has the fourth-order accuracy for both time and space

Summary

Introduction

We consider the 2D wave equation as follows:. where Ω = (x, y) : 0 ≤ x, y ≤ l and ∂Ω is the boundary of Ω. The most commonly used for solving this equation is the second-order central difference scheme Since it is lower accuracy and lower resolution, at least 20 grid points is necessary to be input into each propagating wavelength. Das et al [23] proposed some ADI schemes for solving the 2D wave equation, which are fourth-order accuracy in both time and space. They are conditionally stable and suffer from the stability conditions which only allow Courant– Friedrichs–Lewy (CFL) numbers from 0.7321 to 0.8186. Liao and Sun [19] extended this method to the 3D wave equation It is fourth-order accuracy in both time and space and the stability condition allows CFL number to be about 0.6079.

EFOC difference scheme

Total time marching scheme

Findings

Conclusion