Convergence of summation-by-parts finite difference methods for the wave equation

TL;DR

This paper investigates the convergence properties of summation-by-parts finite difference methods for the wave equation, revealing conditions under which accuracy gains are achieved or limited, supported by theoretical analysis and numerical validation.

Contribution

It provides a detailed analysis of convergence rates for SBP finite difference schemes for the wave equation, especially when the determinant condition is not satisfied.

Findings

Normal mode analysis explains convergence behavior near boundaries.

Large truncation errors do not always lead to two-order gains.

Numerical experiments confirm theoretical error estimates.

Abstract

In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid interface conditions are imposed by the simultaneous-approximation-term technique. Typically, the truncation error is larger at the grid points near a boundary or grid interface than that in the interior. Normal mode analysis can be used to analyze how the large truncation error affects the convergence rate of the underlying stable numerical scheme. If the semi-discretized equation satisfies a determinant condition, two orders are gained from the large truncation error. However, many interesting second order equations do not satisfy the determinant condition. We then carefully analyze the solution of the boundary system to derive a sharp estimate for the error in the solution and acquire the gain in convergence rate. The result shows that stability does not automatically yield a gain of two orders in convergence rate. The accuracy analysis is verified by numerical experiments.