Linear high-resolution schemes for hyperbolic conservation laws: TVB numerical evidence

TL;DR

This paper introduces linear high-resolution schemes based on the Osher-Chakrabarthy family for hyperbolic conservation laws, demonstrating their effectiveness and TVB behavior across various complex fluid dynamics problems.

Contribution

It provides improved lower bounds on compression factors and combines these schemes with the LLF flux for efficient finite-difference algorithms, tested on diverse equations.

Findings

Schemes exhibit total-variation-bounded behavior in multiple tests.

Unlimited version of schemes is viable due to improved bounds.

Schemes struggle with compound shocks from non-convex fluxes.

Abstract

The Osher-Chakrabarthy family of linear flux-modification schemes is considered. Improved lower bounds on the compression factors are provided, which suggest the viability of using the unlimited version. The LLF flux formula is combined with these schemes in order to obtain efficient finite-difference algorithms. The resulting schemes are applied to a battery of numerical tests, going from advection and Burgers equations to Euler and MHD equations, including the double Mach reflection and the Orszag-Tang 2D vortex problem. Total-variation-bounded behavior is evident in all cases, even with time-independent upper bounds. The proposed schemes, however, do not deal properly with compound shocks, arising from non-convex fluxes, as shown by Buckley-Leverett test simulations.