Description
- Abstract:
- This thesis consists of two topics on discontinuous Galerkin (DG) finite element methods for time-dependent problems. In the first part of the thesis, we analyze the stability of explicit time stepping methods with the DG spatial discretization for linear conservation laws. For the second-order and the third-order schemes, Lax-Wendroff DG methods are analyzed. We firstly note that a class of Lax-Wendroff DG methods is equivalent with the Runge-Kutta DG methods after one full time step. It is then shown that the stability of the fully discretized schemes depends on the choice of numerical fluxes of intermediate variables in time discretizations. Finally, for stable schemes in one dimension, optimal error estimates are given for both the solution and its first-order time derivative. For the fourth-order method, we focus on the Runge-Kutta time discretization. It is shown that the numerical scheme is two-step strongly stable under an appropriate time step constraint. This result applies not only to DG methods, but also to other spatial discretizations that preserve the semi-negativity of the continuum equation. In the second part, we aim at designing an entropy stable high-order DG method for gradient flow problems. The positivity (non-negativity) of the solution is also expected in various applications. It plays a crucial role for the well-definedness and the dissipative property of the entropy functional. By adopting the Gauss-Lobatto quadrature rule in the local DG scheme and with a suitable choice of numerical fluxes, our semi-discrete scheme is entropy stable for scalar problems without non-smooth interaction kernels. The scheme also features with weak positivity with the Euler forward time discretization. Hence by applying a scaling limiter and with the strong-stability-preserving Runge-Kutta method, the scheme is able to produce non-negative solutions. The method can also be applied to two-dimensional problems on Cartesian meshes. Numerical examples are given to confirm the high-order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics. Finally, the extension to cross-diffusion gradient flow systems is also discussed.
- Notes:
- Thesis (Ph. D.)--Brown University, 2018
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Citation
Sun, Zheng,
"Discontinuous Galerkin Methods: Stability of Time Discretizations and Applications to Gradient Flows"
(2018).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/ty11-d104
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....