Semi-implicit relaxed finite volume schemes for hyperbolic multi-scale systems of conservation laws

TL;DR

The paper tackles the stiffness challenges of multi-scale hyperbolic conservation laws by introducing a semi-implicit scheme based on Jin-Xin relaxation and a flux splitting that separates slow from fast dynamics. By projecting relaxation variables onto equilibrium and reformulating the system as linear wave type equations, the method achieves efficient linearly implicit time stepping with explicit treatment of slow processes. It delivers first-order two-split and higher order schemes, extends to three-split configurations for complex multi-physics like ideal MHD, and demonstrates asymptotic preserving and contact preserving properties through Euler and MHD benchmark tests. The results show accurate, robust performance across a wide range of Mach and Alfvén numbers, making the approach practical for simulating multi-scale aero- and magnetohydrodynamic flows.

Abstract

In this paper a new semi-implicit relaxation scheme for the simulation of multi-scale hyperbolic conservation laws based on a Jin-Xin relaxation approach is presented. It is based on the splitting of the flux function into two or more subsystems separating the different scales of the considered model whose stiff components are relaxed thus yielding a linear structure of the resulting relaxation model on the relaxation variables. This allows the construction of a linearly implicit numerical scheme, where convective processes are discretized explicitly. Thanks to this linearity, the discrete scheme can be reformulated in linear decoupled wave-type equations resulting in the same number of evolved variables as in the original system. To obtain a scale independent numerical diffusion, centred fluxes are applied on the implicitly treated terms, whereas classical upwind schemes are applied on the explicit parts. The numerical scheme is validated by applying it on the Toro & Vázquez-Cendón (2012) splitting of the Euler equations and the Fambri (2021) splitting of the ideal MHD equations where the flux is split in two, respectively three sub-systems. The performance of the numerical scheme is assessed running benchmark test-cases from the literature in one and two spatial dimensions.

Figures (6)

• Figure 1: Euler Riemann Problems RP1 and RP2 from Table \ref{['tab:RPEuler']}: Sod and slow moving contact test. Comparison of the new second order semi-implicit SIFV-PC scheme against the fully implicit IMFV-PC scheme from Thomann2023 and the exact solution on 1000 cells.
• Figure 2: Euler Riemann Problem RP3 from Table \ref{['tab:RPEuler']}: Travelling contact wave at $t_f= 0.5$ computed with the first order SIFV-PC scheme on two grids with $N=500$ and $N=1000$ cells.
• Figure 3: Kelvin-Helmholtz instability: Density $\rho$ obtained with the second order SIFV-PC scheme computed with $N\times N$ cells in the Mach number regime $M\approx 0.25$ at $t=2$ displayed with 31 equidistant contour lines. Left: $N=128$. Center: $N=256$. Right: $N=512$.
• Figure 4: MHD Riemann Problems RP1-4 from Table \ref{['tab:RPMHD']}. Comparison with the second-order semi-implicit SIFV-EB scheme from Boscheri2024 on 1000 cells.
• Figure 5: Field loop advection test at $t_f = 1$: Magnitude of the magnetic field with $A_0 = 1$ in the top row and $A_0 = 10$ in the bottom row computed on $N\times N$ cells displayed with 31 equidistant contour lines. Left collumn: $N=128$, Right column: $N=256$.

Theorems & Definitions (7)

• Lemma 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 4.1: Contact preserving property
• proof
• Lemma 4.2
• proof