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

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(τ 4 + τ 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.

On properties of an explicit in time fourth-order vector compact scheme for the multidimensional wave equation

“…Let us compare the above stability conditions with the corresponding conditions arising in the frequently used spectral method (it was used in [7] as well). In this method, one can consider the system of eigenvectors of the operator −L h in H h :…”

“…In the recent paper [7], a new third type of compact schemes has been suggested for the 2D wave equation in a square, with the non-homogeneous Dirichlet boundary conditions. The scheme is three-level explicit in time and conditionally stable; the uniform mesh in time and the square spatial mesh are taken.…”

“…In this paper, the scheme from [7] is generalized to the case of the n-dimensional wave equation, n 1, and the uniform rectangular mesh (recall that the square mesh cannot be constructed in any rectangular domain), and the scheme is classified as a vector one more systematically. The new explicit two-level vector equations for the approximate solution at the first time level are constructed too.…”

“…They are similar to the main equations and exploit only second order finite-differences of the initial functions. Notice that we do not use derivatives of the free term and initial functions in contrast to [7] that is essential for applications of the scheme in the non-smooth case, for example, see [8,10,22,24]. The scheme is new even in the simplest 1D case.…”

“…We need also to define v 1 at the first time level t = h t with the 4th order of accuracy. This can be done explicitly by using Taylor's formula and the wave equation, for example, see [2,5,7,18]. But these formulas involve higher-order derivatives (or differences) of the initial functions u 0 and u 1 that is inconvenient in the non-smooth case.…”