Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics
Journal of Computational Physics
On Godunov-type methods for gas dynamics
SIAM Journal on Numerical Analysis
Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics
On Godunov-type methods near low densities
Journal of Computational Physics
A multidimensional flux function with applications to the Euler and Navier-Stokes equations
Journal of Computational Physics
Numerical solution of the Riemann problem for two-dimensional gas dynamics
SIAM Journal on Scientific Computing
On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation
Journal of Computational Physics
An unsplit 3D upwind method for hyperbolic conservation laws
Journal of Computational Physics
Notes on the eigensystem of magnetohydrodynamics
SIAM Journal on Applied Mathematics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
On WAF-type schemes for multidimensional hyperbolic conservation laws
Journal of Computational Physics
Wave propagation algorithms for multidimensional hyperbolic systems
Journal of Computational Physics
Journal of Computational Physics
On the Choice of Wavespeeds for the HLLC Riemann Solver
SIAM Journal on Scientific Computing
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Multidimensional upwinding. Part I. The method of transport for solving the Euler equations
Journal of Computational Physics
Multidimensional upwinding. Part II. Decomposition of the Euler equations into advection equations
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Two-dimensional Riemann solver for Euler equations of gas dynamics
Journal of Computational Physics
Divergence-free adaptive mesh refinement for Magnetohydrodynamics
Journal of Computational Physics
Hyperbolic divergence cleaning for the MHD equations
Journal of Computational Physics
ADER: Arbitrary High Order Godunov Approach
Journal of Scientific Computing
An HLLC Riemann solver for magneto-hydrodynamics
Journal of Computational Physics
ADER schemes for three-dimensional non-linear hyperbolic systems
Journal of Computational Physics
A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics
Journal of Computational Physics
Journal of Computational Physics
A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems
SIAM Journal on Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics
Journal of Computational Physics
A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems
Journal of Scientific Computing
Journal of Computational Physics
Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics
Journal of Computational Physics
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The goal of this paper is to formulate genuinely multidimensional HLL and HLLC Riemann solvers for unstructured meshes by extending our prior papers on the same topic for logically rectangular meshes Balsara (2010, 2012) [4,5]. Such Riemann solvers operate at each vertex of a mesh and accept as an input the set of states that come together at that vertex. The mesh geometry around that vertex is also one of the inputs of the Riemann solver. The outputs are the resolved state and multidimensionally upwinded fluxes in both directions. A formulation which respects the detailed geometry of the unstructured mesh is presented. Closed-form expressions are provided for all the integrals, making it particularly easy to implement the present multidimensional Riemann solvers in existing numerical codes. While it is visually demonstrated for three states coming together at a vertex, our formulation is general enough to treat multiple states (or zones with arbitrary geometry) coming together at a vertex. The present formulation is very useful for two-dimensional and three-dimensional unstructured mesh calculations of conservation laws. It has been demonstrated to work with second to fourth order finite volume schemes on two-dimensional unstructured meshes. On general triangular grids an arbitrary number of states might come together at a vertex of the primal mesh, while for calculations on the dual mesh usually three states come together at a grid vertex. We apply the multidimensional Riemann solvers to hydrodynamics and magnetohydrodynamics (MHD) on unstructured meshes. The Riemann solver is shown to operate well for traditional second order accurate total variation diminishing (TVD) schemes as well as for weighted essentially non-oscillatory (WENO) schemes with ADER (Arbitrary DERivatives in space and time) time-stepping. Several stringent applications for compressible gasdynamics and magnetohydrodynamics are presented, showing that the method performs very well and reaches high order of accuracy in both space and time. The present multidimensional Riemann solver is cost-competitive with traditional, one-dimensional Riemann solvers. It offers the twin advantages of isotropic propagation of flow features and a larger CFL number. Please see http://www.nd.edu/~dbalsara/Numerical-PDE-Course for a video introduction to multidimensional Riemann solvers.