Numerical preservation of multiple local conservation laws

TL;DR

This paper extends a symbolic algebra-based method to construct finite difference schemes that preserve multiple local conservation laws for a broader class of PDEs, including systems, demonstrating improved robustness and accuracy.

Contribution

It adapts a recent conservation-preserving strategy to non-Kovalevskaya PDEs and systems, broadening its applicability and effectiveness.

Findings

Conservative schemes for the Benjamin-Bona-Mahony equation show high accuracy.

The method effectively preserves multiple local conservation laws.

Benchmarks indicate superior robustness compared to existing methods.

Abstract

There are several well-established approaches to constructing finite difference schemes that preserve global invariants of a given partial differential equation. However, few of these methods preserve more than one conservation law locally. A recently-introduced strategy uses symbolic algebra to construct finite difference schemes that preserve several local conservation laws of a given scalar PDE in Kovalevskaya form. In this paper, we adapt the new strategy to PDEs that are not in Kovalevskaya form and to systems of PDEs. The Benjamin-Bona-Mahony equation and a system equivalent to the nonlinear Schroedinger equation are used as benchmarks, showing that the strategy yields conservative schemes which are robust and highly accurate compared to others in the literature.