March 30th – Mohammad Zakerzadeh: Analysis of space-time discontinuous Galerkin scheme for hyperbolic and viscous conservation laws

Mohammad Zakerzadeh: Thursday 30 March at 10 am, A415 Inria Paris.
The well-posedness of the entropy weak solutions for scalar conservation laws is a classical result. However, for multidimensional hyperbolic systems, some theoretical and numerical evidence cast doubt on that entropy solutions constitute the appropriate solution paradigm, and it has been conjectured that the more general EMV solutions ought to be considered the appropriate notion of solution. In the numerical framework and building on previous results, we prove that bounded solutions of a certain class of space-time discontinuous Galerkin (DG) schemes converge to an EMV solution. The novelty in our work is that no streamline-diffusion (SD) terms are used for stabilization. While SD stabilization is often included in the analysis of DG schemes, it is not commonly found in practical implementations. We show that a properly chosen nonlinear shock-capturing operator success to provide the necessary stability and entropy consistency estimates. In case of scalar equations this result can be strengthened and the reduction to the entropy weak solution is obtained. We prove the boundedness of the solutions as well as the consistency with all entropy inequalities, and consequently the convergence to the entropy weak solution is obtained. For viscous conservation laws, we extend our framework to general convection-diffusion systems, with both nonlinear convection and nonlinear diffusion, such that the entropy stability of the scheme is preserved. It is well-known that this property is not guaranteed, even if the convective discretization is entropy stable with respect to the purely hyperbolic problem with a naive formulation of the viscous fluxes. We use a mixed formulation, and handle the difficulties arising from the nonlinearity of the viscous flux by an additional Galerkin projection operator. We prove the entropy stability of the method for different treatments of the viscous flux, thus unifying and extending some results already existing in the literature.

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