A Provably Positive-Preserving HLLD Riemann Solver for Ideal Magnetohydrodynamics. Part I: The One-Dimensional Case (Communications in Computational Physics, 2025)
Combining robustness and high accuracy is one of the primary challengesin the magnetohydrodynamics (MHD) field of numerical methods. This paper investigates two critical physical constraints: wave order and positivity-preserving (PP) properties of the high-resolution HLLD Riemann solver, which ensures the positivity ofdensity, pressure, and internal energy. This method’s distinctiveness lies in its abilityto ensure that the wave characteristic speeds of the HLLD Riemann solver are strictlyordered. A provably PP HLLD Riemann solver based on the Lagrangian setting is established, which can be viewed as an extension of the PP Lagrangian method in hydrodynamics but with more and stronger constraint condition. In addition, the above twoproperties are ensured on moving grid method by employing the Lagrange-to-Eulertransform. Meanwhile, a novel multi-moment constrained finite volume method isintroduced to acquire third order accuracy, and practical limiters are applied to avoidnumerical oscillations. Selected numerical benchmarks demonstrate the robustnessand accuracy of our methods.
Keywords: Approximate Riemann solver; positivity-preserving method; multi-moment constrained finite volume method; arbitrary-Lagrangian-Eulerian (ALE) framework.