Author : Jiawei Sun
Release : 2022
Genre : Differential equations, Partial
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Discontinuous Galerkin Method for Boussinesq System and Stochastic Maxwell Equations by : Jiawei Sun

Download or read book Discontinuous Galerkin Method for Boussinesq System and Stochastic Maxwell Equations written by Jiawei Sun. This book was released on 2022. Available in PDF, EPUB and Kindle. Book excerpt: Partial differential equations (PDEs) are commonly used for characterizing physical phenomena such as shallow water equations, water wave problems, electromagnetism, etc. This dissertation contains two parts dedicated to the advancement of numerical methods for two types of PDEs: the abcd-Boussinesq system and the stochastic Maxwell equations. We use discontinuous Galerkin (DG) method as our tool for numerically solving these equations. For the first part, we considered the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, BBM-BBM system, Bona-Smith system etc. We proposed local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this project is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a,b,c,d. Numerical examples are also provided to demonstrate the convergence rates and to show that the proposed method has the capability to simulate the head-on collision and finite time blow-up behavior well. In the second part, firstly we considered one and multi-dimensional stochastic Maxwell equations with additive noise. Such system can also be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. We designed high order DG methods for the stochastic Maxwell equations with additive noise, and we showed that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure in the discrete level. Optimal error estimate of the semi-discrete DG method is also obtained. The fully discrete methods are derived from coupling with symplectic temporal discretization. One- and two-dimensional numerical results are provided to demonstrate the performance of optimal error estimates and linear growth of the discrete energy. Secondly, we studied stochastic Maxwell equation with general multiplicative noise, which is a generalization of the additive noise. High order DG method is also applied to the one and two dimensional cases. Discrete version of stability of numerical solutions is derived. We analyzed the convergence order of our numerical scheme and then obtained optimal error estimate. The full discretization combines DG scheme in space and strong Taylor scheme in time. Numerical implementations are provided to demonstrate the optimal convergence rate in both one and two dimensional cases.