@Article{CiCP-9-2,
author = {},
title = {Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations},
journal = {Communications in Computational Physics},
year = {2011},
volume = {9},
number = {2},
pages = {324--362},
abstract = {<p style="text-align: justify;">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 semi-conservative
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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.171109.070510a},
url = {https://global-sci.com/article/80850/approximate-riemann-solvers-and-robust-high-order-finite-volume-schemes-for-multi-dimensional-ideal-mhd-equations}
}