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Multidimensional Riemann Problem with Self-Similar Internal Structure – Part III– A Multidimensional Analogue of the HLLI Riemann Solver for Conservative Hyperbolic Systems

Dinshaw S Balsara 1 Boniface Nkonga 2, 3 
3 CASTOR - Control, Analysis and Simulations for TOkamak Research
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné
Abstract : Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come together at the vertex of a mesh. The interaction of the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The fastest way of endowing such sub-structure consists of making a multidimensional extension of the HLLI Riemann solver for hyperbolic conservation laws. Presenting such a multidimensional analogue of the HLLI Riemann solver with linear sub-structure for use on structured meshes is the goal of this work. The multidimensional MuSIC Riemann solver documented here is universal in the sense that it can be applied to any hyperbolic conservation law. The multidimensional Riemann solver is made to be consistent with constraints that emerge naturally from the Galerkin projection of the self-similar states within the wave model. When the full eigenstructure in both directions is used in the present Riemann solver, it becomes a complete Riemann solver in a multidimensional sense. I.e., all the intermediate waves are represented in the multidimensional wave model. The work also presents, for the very first time, an important analysis of the dissipation characteristics of multidimensional Riemann solvers. The present Riemann solver results in the most efficient implementation of a multidimensional Riemann solver with sub-structure. Because it preserves stationary linearly degenerate waves, it might also help with well-balancing. Implementation-related details are presented in pointwise fashion for the one-dimensional HLLI Riemann solver as well as the multidimensional MuSIC Riemann solver. Several stringent test problems drawn from hydrodynamics, MHD and relativistic MHD are presented to show that the method works very well on structured meshes. Our results demonstrate the versatility of our method. The reader is also invited to watch a video
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Preprints, Working Papers, ...
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Submitted on : Wednesday, January 4, 2017 - 7:56:01 PM
Last modification on : Friday, November 18, 2022 - 9:24:02 AM
Long-term archiving on: : Wednesday, April 5, 2017 - 3:19:47 PM


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  • HAL Id : hal-01426759, version 1



Dinshaw S Balsara, Boniface Nkonga. Multidimensional Riemann Problem with Self-Similar Internal Structure – Part III– A Multidimensional Analogue of the HLLI Riemann Solver for Conservative Hyperbolic Systems. 2017. ⟨hal-01426759⟩



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