Efficient discontinuous Galerkin (DG) methods for time-dependent fourth order problems

In this thesis, we design, analyze and implement efficient discontinuous Galerkin (DG) methods for a class of fourth order time-dependent partial differential equations (PDEs). The main advantages of such schemes are their provable unconditional stability, high order accuracy, and their easiness for generalization to multi-dimensions for arbitrarily high order schemes on structured and unstructured meshes. These schemes have been applied to two fourth order gradient flows such as the Swift-Hohenberg (SH) equation and the Cahn-Hilliard (CH) equation, which are well known nonlinear models in modern physics. For fourth order PDEs of the form $\partial_t u= -\mathcal{L}^2 u +f$, where $\mathcal{L}$ is an adjoint elliptic operator, the fully discrete DG schemes are constructed in several steps: (a) rewriting the equation as a system of second order PDEs so that $u_t=\mathcal{L}q +f, \quad q=-\mathcal{L}u$; (b) applying the DG discretization to this mixed formulation with central numerical fluxes on interior interfaces and weakly enforcing the specified boundary conditions; and (c) combining a special class of time discretizations, that allows the method to be unconditionally stable regardless of its accuracy. Main contributions of this thesis are as follows: Firstly, we introduce mixed discontinuous Galerkin methods without interior penalty for the spatial DG discretization, and the semi-discrete schemes are shown $L^2$ stable for linear problems, and unconditionally energy stable for nonlinear gradient flows. For the mixed DG method applied to linear problems with periodic boundary conditions, we establish the optimal $L^2$ error estimate of order $O(h^{k+1} +\Delta t^2)$ for polynomials of degree $k$ with the Crank-Nicolson time discretization. In addition, the resulting DG methods can easily handle different boundary conditions. Secondly, for a class of fourth order gradient flow problems, including the SH equation, we combine the so-called \emph{Invariant Energy Quadratization} (IEQ) approach [X. Yang, J. Comput. Phys., 327:294{316, 2016] as time discretization. Coupled with a projection step for the auxiliary variable, both first and second order EQ-DG schemes are shown unconditionally energy stable. In addition, they are linear and can be efficiently solved without resorting to any iteration method. We present extensive numerical examples that support our theoretical results and illustrate the efficiency, accuracy, and stability of our new algorithms. Benchmark problems are also presented to examine the long time behavior of the numerical solutions. Both the theoretical and algorithmic aspects of these methods have potentially wide applications. Progress is made with the IEQ-DG framework to solve the Cahn-Hilliard equation. With the usual penalty in the DG discretization, the resulting EQ-DG schemes are shown to be able to produce free-energy-decaying, and mass conservative solutions, irrespective of the time step and the mesh size. In addition, the schemes are easy to implement, and test cases for the Cahn-Hilliard equation will be reported

A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure

In this paper we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential / capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes the resulting systems of nonlinear algebraic equations are solved with Newton's method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust and efficient. In particular no post-processing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1000 processors

Discontinuous Galerkin method for the spherically reduced BSSN system with second-order operators

We present a high-order accurate discontinuous Galerkin method for evolving the spherically-reduced Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system expressed in terms of second-order spatial operators. Our multi-domain method achieves global spectral accuracy and long-time stability on short computational domains. We discuss in detail both our scheme for the BSSN system and its implementation. After a theoretical and computational verification of the proposed scheme, we conclude with a brief discussion of issues likely to arise when one considers the full BSSN system.Comment: 35 pages, 6 figures, 1 table, uses revtex4. Revised in response to referee's repor