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Type of Resource (primo): dissertations
Name: Personal
Name: Personal
Name: Personal
Name: Personal
Name: Corporate
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Note: thesis: Thesis (Ph. D.)--Brown University, 2022
Genre (aat): theses
Abstract: This thesis contains two parts, including the development of a modified total variation bounded (TVB) limiter applied to the discontinuous Galerkin (DG) methods, and the application of the local DG (LDG) method to solve the carpet cloak model. The DG method was initially proposed by Reed and Hill to solve the neutron transport problem. Later, Cockburn and Shu introduced the Runge-Kutta DG (RKDG) methods for solving the linear and nonlinear hyperbolic partial differential equations (PDEs), and the LDG methods for solving the time-dependent convection-diffusion systems, which stimulated the rapid development and application of the DG methods. The DG method is widely used in numerical solution of partial differential equations because of its nice features, such as the flexible h-p adaptivity, easy handling of the complicated geometry, easy handling of hanging nodes and adaptivity, and high parallel efficiency. Although the DG method has many good properties, for problems containing strong shocks, the DG method often needs to be supplemented by a limiter to control spurious oscillations and to ensure nonlinear stability. The TVB limiter is a popular choice and can maintain the original high order accuracy of the DG scheme in smooth regions and keep a sharp and non-oscillatory discontinuity transition, when a certain TVB constant $M$ is chosen adequately. For scalar conservation laws, suitable choice of this constant $M$ can be based on solid mathematical analysis. However, for nonlinear hyperbolic systems, there is no rigorous mathematical guiding principle for the determination of this constant, and numerical experiments often use {\em ad hoc} choices based on experience and through trial and error. Our first topic is to develop a TVB constant artificial neural network (ANN) based estimator by constructing a multi-layer perceptron (MLP) model. We generate the training data set by constructing piecewise smooth functions containing local maxima, local minima, and discontinuities. By using the supervised learning strategy, the MLP model is trained offline. The proposed method gives the TVB constant $M$ with robust performance to capture sharp and non-oscillatory shock transitions while maintaining the original high order accuracy in smooth regions. Numerical results using this new estimator in the TVB limiter for DG methods in one and two dimensions are given, and its performance is compared with the classical {\em ad hoc} choices of this TVB constant. In the second part, we introduce the leap-frog LDG methods to solve the carpet cloak model. We prove the stability of the semi-discrete scheme, the sub-optimal error estimate for unstructured meshes, and the optimal error estimate for tensor-product meshes. Then, the fully discrete scheme is stated and the stability is proved. Finally, the numerical accuracy tests on rectangular and triangular meshes are given respectively, and the results of numerical simulations of the wave propagation in the carpet cloak model using the DG scheme are presented.
Subject (fast) (authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00875497")
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