Simple bespoke preservation of two conservation laws

TL;DR

This paper introduces simplified symbolic-numeric methods to construct finite difference schemes that preserve two conservation laws of PDEs, demonstrated on the KdV and nonlinear heat equations, resulting in robust and accurate numerical schemes.

Contribution

It presents key simplifications to the symbolic approach for preserving multiple conservation laws in finite difference schemes, making the method more practical.

Findings

Schemes preserve two conservation laws for KdV and nonlinear heat equations.

Numerical tests show high accuracy and robustness of the schemes.

Simplified approach enhances feasibility of symbolic-numeric conservation law preservation.

Abstract

Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic-numeric approach feasible. To illustrate the simplified approach, we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg-de Vries (KdV) equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared to others in the literature.