On energy consistency of intermediate states in HLL-type MHD Riemann solvers

TL;DR

The paper tackles the problem of positivity and energy-consistency in MHD Riemann solvers by introducing an energy-consistency condition for intermediate states within the Riemann fan. It develops two energy-consistent variants, HLLC-ec and HLLD-ec, that adjust intermediate energies to be consistent with split energy fluxes, improving robustness in low-$\beta$ regimes while maintaining solver simplicity. Through analysis and numerical tests, including 1D shock-tube, varying-$B_{\parallel}$ cases, and rotor problems, the authors demonstrate reduced spurious energy transfer and smaller $\nabla\cdot\mathbf{B}$ errors compared with conventional HLL-type schemes. The results suggest meaningful gains in reliability for MHD simulations under strong magnetic fields, with potential extensions to broader nonconservative or implicit formulations.

Abstract

Approximate Riemann solvers are widely used for solving hyperbolic conservation laws, including those of magnetohydrodynamics (MHD). However, due to the nonlinearity and complexity of MHD, obtaining accurate and robust numerical solutions to MHD equations is non-trivial, and it may be challenging for an approximate MHD Riemann solver to preserve the positivity of scalar variables, particularly when the plasma \b{eta} is low. As we have identified that the inconsistency between the numerically calculated magnetic field and magnetic energy may be at least partly responsible for the loss of positivity of scalar variables, we propose a consistency condition for calculating the intermediate energies within the Riemann fan and implement it in HLL-type MHD Riemann solvers, thereby alleviating erroneous magnetic field solutions that break scalar positivity. In addition, (I) for the HLLC-type scheme, we have designed a revised two-state approximation, specifically reducing numerical error in magnetic field solutions, although sacrificing the contact-resolving capability, and (II) for the HLLD-type scheme, we replace the constant total pressure assumption by a three-state assumption for the intermediate thermal energy, which is more consistent with our other assumptions. The proposed schemes perform better in numerical examples with low plasma \b{eta}. Moreover, we explained the energy error introduced during time integration.

Figures (11)

• Figure 0: Envelopes of the divergence $\nabla\cdot\mathbf{B}$ of a two-dimensional low-plasma $\beta$ test. The present HLLC/D-ec schemes produce less divergence error than their conventional counterparts. For details, see Section \ref{['sec:lowbetarotor']}.
• Figure 1: A $x-t$ spatio-temporal schematic of eigen-wave propagation in the MHD Riemann problem. Here, the subscripts $f$, $A$, and $s$ denote fast, Alfvén, and slow modes and the corresponding eigen-speeds.
• Figure 2: Density and velocity ($x$-component) distributions of HLLC-type schemes.
• Figure 3: Density and velocity ($x$-component) distributions of HLLD-type schemes.
• Figure 4: Results of 1D simulations with varying $B_x$.