Bounds Preserving Temporal Integration Methods for Hyperbolic Conservation Laws

TL;DR

This paper introduces bounds-preserving modifications to explicit Runge-Kutta schemes, ensuring solutions stay within desired bounds for hyperbolic conservation laws without sacrificing accuracy, demonstrated through numerical experiments.

Contribution

It presents a novel framework for modifying Runge-Kutta methods to guarantee bounds preservation across various schemes, independent of spatial discretization.

Findings

Preserves bounds such as positivity and maximum principles.

Maintains the order of accuracy of the original Runge-Kutta methods.

Effective in numerical experiments with discontinuities.

Abstract

In this work, we present a modification of explicit Runge-Kutta temporal integration schemes that guarantees the preservation of any locally-defined quasiconvex set of bounds for the solution. These schemes operate on the basis of a bijective mapping between an admissible set of solutions and the real domain to strictly enforce bounds. Within this framework, we show that it is possible to recover a wide range of methods independently of the spatial discretization, including positivity preserving, discrete maximum principle satisfying, entropy dissipative, and invariant domain preserving schemes. Furthermore, these schemes are proven to recover the order of accuracy of the underlying Runge-Kutta method upon which they are built. The additional computational cost is the evaluation of two nonlinear mappings which generally have closed-form solutions. We show the utility of this approach in numerical experiments using a pseudospectral spatial discretization without any explicit shock capturing schemes for nonlinear hyperbolic problems with discontinuities.