Multiderivative Runge-Kutta Flux Reconstruction for Hyperbolic Conservation Laws

Abstract

We extend the fourth-order, two-stage multiderivative Runge-Kutta (MDRK) scheme to the flux reconstruction (FR) framework by writing both stages in terms of a time-averaged flux and then using the approximate Lax-Wendroff (LW) procedure to compute the time-averaged flux. Numerical flux is carefully constructed to enhance Fourier CFL stability and accuracy. A subcell-based blending limiter is developed for the MDRK scheme which ensures that the limited scheme is provably admissibility preserving. Along with being admissibility preserving, the blending scheme is constructed to minimize dissipation errors using Gauss-Legendre (GL) solution points and performing the MUSCL-Hancock (MH) reconstruction on subcells. The accuracy enhancement of the blending scheme is numerically verified on compressible Euler equations, with test cases involving shocks and small-scale structures.

Keywords

Conservation laws / Hyperbolic PDE / Multiderivative Runge-Kutta (MDRK) / Flux reconstruction (FR) / Admissibility preservation / Shock capturing / 65M22 / 65M60 / 65M70

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Funding

Tata Institute of Fundamental Research (TIFR)

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