Multidimensional, Self-similar, strongly-Interacting, Consistent (MuSIC) Riemann Solvers – Applications to Divergence-Free MHD and ALE Schemes
By Dinshaw S. Balsara ( Этот адрес электронной почты защищен от спам-ботов. У вас должен быть включен JavaScript для просмотра. ) University of Notre Dame, USA
Please also see : http://www.nd.edu/~dbalsara/Numerical-PDE-Course


    Large-scale, multidimensional flow simulations are now commonplace and there is a considerable interest in very accurate algorithms for such simulations. Introducing true multidimensionality in such algorithms is very valuable for a complete representation of the physics of the problem. Reconstruction strategies for hyperbolic PDEs (WENO, DG, PNPM, MOOD) are already fully multidimensional as are methods for their temporal update (RK, ADER). The majority of Riemann solvers are still one-dimensional. The present talk describes the design of multidimensional Riemann solvers and their applicability to higher order schemes.
    Such multidimensional Riemann solvers act at the vertices of the mesh, where the multidimensional flow structure becomes visible to the Riemann solver. Instead of two input states, the input states consist of states from all the zones that meet at that vertex. At any zone interface that separates two states, a one dimensional Riemann problem emanates, as always. However, at any vertex, all the adjacent one-dimensional Riemann problems interact to form a strongly interacting state. The strongly interacting state evolves self-similarly in spacetime. By evolving the structure of the strongly interacting state in a set of self-similar variables we show that the structure of the strongly interacting state can be elucidated. Self-similarity is crucially important in the development of multidimensional Riemann solvers (Balsara (2010, 2012, 2014, 2015), Balsara, Dumbser & Abgrall (2014), Balsara & Dumbser (2015), Balsara et al. (2015)). This has prompted the name of MuSIC Riemann solvers, where MuSIC stands for “Multidimensional, Self-similar, strongly-Interacting, Consistent”. For a video introduction to multidimensional Riemann solvers see: http://www.nd.edu/~dbalsara/Numerical-PDE-Course.
    Numerical MHD has come into its own in the last several years. MHD forms an involution-constrained system where the magnetic field, once divergence-free, remains so forever. As a result, one has to find a strategy to represent the magnetic field in divergence-free fashion. Keeping the magnetic field divergence-free requires solving the problem on a Yee-type mesh. This necessarily requires identifying the multidimensionally upwinded electric field at the edges of a computational mesh. I proceed to show that recent advances in designing multidimensional Riemann solvers give a unique, multidimensionally-upwinded representation of the electric field.
    The benefits of the multidimensional Riemann solver go beyond numerical MHD and apply to any hyperbolic system. I show that the multidimensional Riemann solver gives more isotropic flow on resolution-starved meshes. The permitted CFL number is also increased. Even more importantly, the multidimensional Riemann solver gives us a physically-motivated node solver for any ALE application involving any manner of hyperbolic system. The talk ends with presentation of hydrodynamical and magnetohydrodynamical ALE results at all orders.