ON COMPACT 4TH ORDER FINITE-DIFFERENCE SCHEMES FOR THE WAVE EQUATION

Alexander Zlotnik,Olga Kireeva

doi:10.3846/mma.2021.13770

Abstract

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional nonhomogeneous wave equation, n≥ 1. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for n≥ 2. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schro¨dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for n≥ 2. For n = 1 and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.

Highlights

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional wave equation with constant coefficients, n ≥ 1. Their construction on uniform meshes is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation related to the polylinear finite element method (FEM), together with further necessary improvements of the arising scheme for n ≥ 2
This alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schrodinger equations (TDSE)
We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation

Summary

Introduction

Compact higher-order finite-difference schemes for PDEs is a popular subject and a vast literature is devoted to them. In virtue of the last formula we have (sN − sN )Λtu = O(|h|4) and (sN − sN )(δtu)0 = O(|h|4) for functions u sufficiently smooth in QT , the approximation errors of the both equations of scheme (3.24)–(3.25) still have the order O(|h|4) as for the previous scheme (3.22)–(3.23). For the sufficiently smooth in QT solution u to the IBVP (3.1)– (3.2), v0 = u0 on ωh and under the hypotheses of Theorem 2 excluding g = 0, for all the schemes listed in it, the following 4th order error bound in the strong energy norm holds max ε20 δt(u − v)m st(u − v)m. Concerning exact schemes, see [7]

The case of non-uniform rectangular meshes

Numerical experiments