Dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations

TL;DR

This paper develops high-order local discontinuous Galerkin methods that are either energy dissipative or conservative for Fornberg-Whitham equations, with proven properties and demonstrated effectiveness in numerical experiments.

Contribution

It introduces novel high-order LDG schemes with proven dissipation and conservation properties specifically tailored for Fornberg-Whitham equations.

Findings

Dissipative schemes effectively handle shock solutions.

Conservative schemes reduce shape error over long-term simulations.

Error estimates are rigorously established for the proposed methods.

Abstract

In this paper, we construct high order energy dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. We give the proofs for the dissipation and conservation for related conservative quantities. The corresponding error estimates are proved for the proposed schemes. The capability of our schemes for different types of solutions is shown via several numerical experiments. The dissipative schemes have good behavior for shock solutions, while for a long time approximation, the conservative schemes can reduce the shape error and the decay of amplitude significantly