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

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 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. Moreover, we explained the energy error introduced during time integration.