Asymptotic-preserving and energy-conserving methods for a hyperbolic approximation of the BBM equation

TL;DR

This paper introduces asymptotic-preserving, energy-conserving numerical methods for a hyperbolic approximation of the BBM equation, ensuring invariant preservation and demonstrating effectiveness through numerical experiments.

Contribution

It develops novel implicit-explicit Runge-Kutta schemes that are both asymptotic-preserving and energy-preserving for the hyperbolic BBM approximation.

Findings

Discretizations conserve invariants converging to BBM invariants.

Numerical experiments confirm the effectiveness of the proposed methods.

Abstract

We study the hyperbolic approximation of the Benjamin-Bona-Mahony (BBM) equation proposed recently by Gavrilyuk and Shyue (2022). We develop asymptotic-preserving numerical methods using implicit-explicit (additive) Runge-Kutta methods that are implicit in the stiff linear part. The new discretization of the hyperbolization conserves important invariants converging to invariants of the BBM equation. We use the entropy relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.