Compact Runge-Kutta flux reconstruction methods for non-conservative hyperbolic equations

TL;DR

This paper develops a compact Runge-Kutta flux reconstruction method that efficiently handles hyperbolic equations with stiff source terms and non-conservative products, ensuring robustness and physical admissibility.

Contribution

It introduces an IMEX scheme combined with a blending approach for non-conservative terms, preserving positivity and stability in complex hyperbolic problems.

Findings

Successfully applied to Burgers' and reactive Euler equations.

Demonstrated robustness for non-smooth solutions.

Preserves physical admissibility like positivity of density and pressure.

Abstract

Compact Runge-Kutta (cRK) Flux Reconstruction (FR) methods are a variant of RKFR methods for hyperbolic conservation laws with a compact stencil including only immediate neighboring finite elements. We extend cRKFR methods to handle hyperbolic equations with stiff source terms and non-conservative products. To handle stiff source terms, we use IMplicit EXplicit (IMEX) time integration schemes such that the implicitness is local to each solution point, and thus does not increase inter-element communication. Although non-conservative products do not correspond to a physical flux, we formulate the scheme using numerical fluxes at element interfaces. We use similar numerical fluxes for a lower order finite volume scheme on subcells of each element, which is then blended with the high order cRKFR scheme to obtain a robust scheme for problems with non-smooth solutions. Combined with a flux limiter at the element interfaces, the subcell based blending scheme preserves the physical admissibility of the solution, e.g., positivity of density and pressure for compressible Euler equations. The procedure thus leads to an admissibility preserving IMEX cRKFR scheme for hyperbolic equations with stiff source terms and non-conservative products. The capability of the scheme to handle stiff terms is shown through numerical tests involving Burgers' equations, reactive Euler's equations, and the ten moment problem. The non-conservative treatment is tested using variable advection equations, shear shallow water equations, the GLM-MHD, and the multi-ion MHD equations.