Stable and convergent discontinuous galerkin methods for hyperbolic and viscous systems of conservation laws
Zakerzadeh, Mohammad; May, Georg (Thesis advisor); Noelle, Sebastian (Thesis advisor); Tadmor, Eitan (Thesis advisor)
Aachen (2017, 2018)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2017
Despite the classical well-posedness theorem for entropy weak solutions of scalar conservation laws, some theoretical and numerical evidence cast doubt on the appropriateness of this solution paradigm for multidimensional hyperbolic systems. It has been conjectured that the more general entropy measure-valued (EMV) solutions ought to be considered as the adequate notion of the solution. 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 terms are used for stabilization, in contrary to the main role of such stabilizations in the existing analysis of DG schemes. Our approach conforms to the way DG schemes were originally proposed, and are most often used in practiceIn the case of scalar problems, this result is strengthened to obtain the convergence to the entropy weak solution, via the proof of $L_\infty$-boundedness of the solution as well as its consistency with all entropy inequalities. As a main step in the boundedness proof, we show the coercivity of the shock-capturing operator employing new arguments from polynomial inequalities. 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. Starting from a mixed formulation, we handle the difficulties arising from the nonlinearity of the viscous flux by an additional projection. We prove the entropy stability of the corresponding primal form for different treatments of the viscous flux; thus unifying the existing results in the literature as well as establishing the entropy stability for less-analyzed methods. Our analysis is also valid for the case of degenerate diffusion. Considering quasilinear elliptic problems in scalar settings, we prove that the proposed approach for viscous discretization is asymptotically consistent and adjoint consistent. For the special case of strongly monotone and globally Lipschitz problems, we prove the uniqueness and stability of the numerical solution. For this class of operators, we also provethe optimal convergence to the exact solution with respect to mesh size, in both energy and $L_2$ norms. Such optimal convergence rates for asymptotically (adjoint) consistent schemes have been observed before in numerical experiments.