Bounds preserving temporal integration methods for hyperbolic conservation laws

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.

Automated summary

In this work, a modified explicit Runge-Kutta temporal integration scheme is proposed to guarantee the preservation of locally-defined quasiconvex set of bounds for the solution. The schemes use a bijective mapping to enforce bounds between the admissible set of solutions and the real domain. It is shown that these schemes can recover a wide range of methods, including positivity preserving, discrete maximum principle satisfying, entropy dissipative, and invariant domain preserving schemes. The additional computational cost is the evaluation of two nonlinear mappings which generally have closed-form solutions. The approach is demonstrated in numerical experiments using a pseudospectral spatial discretization without explicit shock capturing schemes for nonlinear hyperbolic problems with discontinuities.