Higher Order Time-Implicit Bounds Preserving Discontinuous Galerkin Discretizations for Partial Differential Equations

This dissertation discusses higher-order accurate time implicit Discontinuous Galerkin (DG) discretizations for several classes of nonlinear Partial Differential Equations (PDEs). The two main topics considered are bounds preserving limiters combined with Diagonally Implicit Runge Kutta (DIRK) methods, and novel efficient higher-order accurate semi-implicit Spectral Deferred Correction (SDC) DG discretizations, including error estimates.
In Chapter 2, positivity constraints are imposed on time implicit Local Discontinuous Galerkin (LDG) discretizations of degenerate parabolic equations using Lagrange multipliers. In addition, mass conservation of the positivity limited solution is ensured by imposing a mass conservation equality constraint. This results in a mixed complementarity problem, which is expressed by the Karush Kuhn Tucker (KKT) equations. We prove unique solvability and entropy stability of the KKT DIRK LDG discretizations. Finally, numerical results are shown which illustrate the higher order accuracy and entropy dissipation of the KKT DIRK LDG discretizations.
In Chapter 3, we develop higher order accurate bounds preserving time implicit DG
discretizations for the chemically reactive Euler equations. We use a fractional step method, which separates the convection and reaction steps. In order to ensure that the density and pressure are nonnegative, and mass fractions are in the range between zero and one, the KKT limiter is adopted to construct bounds preserving DIRK DG discretizations. In order to deal with the stiff source terms in chemically reactive flows, we use Harten’s subcell resolution technique in the reaction step. Numerical examples demonstrate that the KKT DIRK DG discretization preserves the physical bounds on the solution, and compares well with exact solutions and accurate reference solutions .
In Chapter 4 , we prove stability and error estimates for second and third order accurate semi implicit SDC LDG discretizations of the Allen Cahn equation. For the numerical discretization of this parabolic equation, implicit time integration methods result in a nonlinear system of equation. We first prove the unique solvability of the implicit SDC LDG discretizations through a standard fixed point argument in finite dimensional space. Next, by a careful selection of the test functions, stability and error estimates for second and third order accurate time implicit SDC LDG discretizations are obtained. Also, numerical examples are presented that illustrate the theoretical results.