A novel discrete variational derivative method using "average-difference methods"

TL;DR

This paper introduces a new average-difference method for structure-preserving numerical schemes that enhances stability and reduces oscillations in solutions of conservative systems.

Contribution

The paper proposes a novel average-difference method that improves the stability of structure-preserving schemes for conservative systems, addressing issues with standard difference operators.

Findings

The proposed method reduces spatial oscillations in numerical solutions.

Theoretical analysis confirms the method's superiority in linear cases.

Numerical experiments demonstrate improved stability and accuracy.

Abstract

We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of difference operators is essential to the discrete conservation law. Unfortunately, however, when we employ the standard central difference operator, the simplest one, the numerical solutions often suffer from undesirable spatial oscillations. In this letter, we propose a novel "average-difference method," which is tougher against such oscillations, and combine it with an existing conservative method. Theoretical and numerical analysis in the linear case show the superiority of the proposed method.