Dissertation

2019

#### Degree Name

Doctor of Philosophy

Mathematics

#### Major

Applied Mathematics

Hailiang Liu

Songting Luo

#### Abstract

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.

Peimeng Yin

en

application/pdf

132 pages

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