High Order Invariant Domain Preserving Finite Volume Schemes for Nonlinear Hyperbolic Conservation Laws

In this dissertation we develop high order invariant domain preserving schemes for general hyperbolic systems. The schemes are based on the general central schemes of formally second, third and fourth order accuracy. The invariant domain property is modified as the quasiconcave constraint and is enforced via a so-called convex limiting technique. There are two classes of schemes developed. One is based on the invariant domain satisfying nonlinear reconstruction and the other method is made to be invariant domain preserving via the convex flux limiting. The main theoretical results are Theorem 4.3.1 and Theorem 4.3.2. The convex limiting process could sufficiently reduce the oscillations of the numerical solutions at discontinuities like shocks, while it does not deteriorate the order of the underlying central scheme. The numerical performance of the methods is tested on a variety of benchmark problems

A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations

High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a positivity limiting strategy for entropy-stable discontinuous Galerkin discretizations constructed by blending high order solutions with a low order positivity-preserving discretization. The proposed low order discretization is semi-discretely entropy stable, and the proposed limiting strategy is positivity preserving for the compressible Euler and Navier-Stokes equations. Numerical experiments confirm the high order accuracy and robustness of the proposed strategy