TVD-MOOD schemes based on implicit-explicit time integration

TL;DR

This paper develops TVD IMEX Runge-Kutta schemes that maintain stability and TVD properties for hyperbolic multi-scale equations, improving resolution while avoiding CFL restrictions related to fast waves.

Contribution

It introduces a convex combination approach to enhance first-order TVD IMEX schemes, ensuring stability and TVD properties for multi-scale hyperbolic equations.

Findings

The new schemes are unconditionally stable with scale-independent CFL conditions.

Numerical tests show improved resolution over standard schemes.

Applied successfully to the isentropic Euler equations.

Abstract

The context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its $L^\infty$ stability and TVD properties. In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme. We derive and analyse the TVD property for a scalar multi-scale equation and numerically assess the performance of our TVD schemes compared to standard $L$-stable and SSP IMEX RK schemes from the literature. Finally, the resulting TVD-MOOD schemes are applied to the isentropic Euler equations.