Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Abstract

Abstract. We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three-and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semiconservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO-and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first-and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

“…The system is threaded by a uniform magnetic field along the x-axis and the problem is defined on the 2D Cartesian domain ðx; yÞ 2 ½À0:5; 0: The adiabatic index c ¼ 1:4. For this case, when the ECUSP scheme with the third order WENO or the fifth order WENO reconstruction is used, we found the same problem as that in [63], that is: the reconstructed densities and pressures may become negative. Hence, in this paper, the first order upwind reconstruction is used to replace the reconstruction of those negative points.…”

Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations

“…Recent papers [17,19] have demonstrated that the added source term in (1.14) needs to be discretized in a very careful manner for numerical stability. Another difficulty with this approach lies in the non-conservative form of (1.14) which may result in wrong shock speeds [44].…”

Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws

“…This source term is proportional to the divergence and allows divergence errors to be swept out of the domain. Numerical stability can only be ensured by a careful upwinding of the source term, see [16]. 3.…”

“…Divergence constraint. The divergence constraint in the MHD equations [16] is handled in ALSVID by adding the Godunov-Powell source term to the MHD equations. This source term is proportional to the divergence and allows divergence errors to be swept out of the domain.…”

“…Approximate Riemann solver: The numerical fluxes in the Finite Volume Scheme (1.5) used in ALSVID are based on approximate Riemann solvers of the HLL type for the Euler and MHD equations [16]. For shallowwater equations with bottom topography, the energy stable well-balanced schemes of [15] have been implemented in ALSVID-UQ.…”