On compact 4th order finite-difference schemes for the wave equation

TL;DR

This paper develops and analyzes compact 4th order finite-difference schemes for the wave equation, including stability, error bounds, and numerical experiments demonstrating advantages over lower-order schemes, applicable to various PDEs and mesh types.

Contribution

It introduces novel 4th order compact schemes for the wave equation using classical and alternative averaging techniques, with stability analysis and generalization to non-uniform meshes.

Findings

Schemes are stable under certain energy norms.

Error bounds are established for smooth solutions.

Numerical experiments show higher accuracy over second-order schemes.

Abstract

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. 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 $n\geq 2$. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schr\"{o}dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for $n\geq 2$. 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.