High Order Numerical Methods: Entropy Stability and Deterministic Solvers of Stochastic PDEs

Chen, Tianheng

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Year:: 2019
Contributor:: Chen, Tianheng (creator); Shu, Chi-Wang (Advisor); Rozovsky, Boris (Reader); Guzmán, Johnny (Reader); Brown University. Department of Applied Mathematics (sponsor)
Genre:: theses
Subject:: Discontinuous Galerkin Methods; Conservation laws (Mathematics)--Numerical solutions; Polynomial chaos; Stochastic calculus
Extent:: xviii, 195 p.
DOI: https://doi.org/10.26300/7c2r-yw67

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Abstract:: This thesis consists of two diverse topics on high order numerical methods for time-dependent partial differential equations (PDEs). In the first part, we develop a unified framework of entropy stable Discontinuous Galerkin (DG) type methods for systems of hyperbolic conservation laws. The well-known cell entropy inequality of classic DG method is limited to the square entropy and assumes exact integration. Our framework overcomes such limitation by designing DG method satisfying entropy inequalities for any given single entropy function, through suitable numerical quadrature rules. The one-dimensional methodology is based on Legendre Gauss-Lobatto quadrature. The main ingredients are discrete operators with summation-by-parts (SBP) property, the flux differencing technique, and entropy stable fluxes at cell boundary. We then extend the methodology to higher space dimensions by constructing SBP operators for simplicial meshes with Gauss-Lobatto type quadrature points. The further extension to more general quadrature points is achieved through careful modification of boundary terms. A local discontinuous Galerkin (LDG) type treatment is also incorporated to enable the generalization to convection-diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing capability of these entropy stable DG methods. In the second part, we explore the polynomial chaos expansion method for distribution-free stochastic partial differential equations (SPDEs). So far the theory and numerical practice of SPDEs have dealt almost exclusively with Gaussian noise or L\'{e}vy noise. Recently, Mikulevicius and Rozovskii proposed a distribution-free Skorokhod-Malliavin calculus framework that is based on generalized polynomial chaos (gPC) expansion, and is compatible with arbitrary driving noise. We will analyze these newly developed distribution-free SPDEs. We obtain an estimate for the mean square truncation error in the linear case. The convergence rate is exponential with respect to polynomial order and cubic with respect to number of random variables included. Numerical experiments are conducted to exhibit the efficiency of truncated polynomial chaos solutions in approximating moments and distributions. The theoretical convergence rate is also verified by numerical results.
Notes:: Thesis (Ph. D.)--Brown University, 2019

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Chen, Tianheng, "High Order Numerical Methods: Entropy Stability and Deterministic Solvers of Stochastic PDEs" (2019). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://doi.org/10.26300/7c2r-yw67

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Applied Mathematics Theses and Dissertations

Theses and Dissertations for the Applied Mathematics department.
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