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High Order Structure-Preserving Discontinuous Galerkin Methods for Shallow Water Equations and Nonlinear Dirac Equation

Yang, Ruize
http://orcid.org/0000-0003-3837-5329
http://rave.ohiolink.edu/etdc/view?acc_num=osu1650299205496592

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
Discontinuous Galerkin (DG) methods form a class of numerical methods for solving partial differential equations. Combined with proper time discretizations, they have been successfully applied to various problems. This thesis consists of two major parts which construct structure-preserving DG methods for the shallow water equations with friction and the nonlinear Dirac equation. In the first part, we propose a family of second and third order temporal integration methods for systems of stiff ordinary differential equations, and explore their application in solving the shallow water equations with friction. The new temporal discretization methods come from a combination of the traditional Runge-Kutta method (for non-stiff equations) and the exponential Runge-Kutta method (for stiff equations), and are shown to have both the sign-preserving and steady-state-preserving properties. They are combined with the well-balanced DG spatial discretization to solve the nonlinear shallow water equations with non-flat bottom topography and (stiff) friction terms. We have demonstrated that the fully discrete schemes satisfy the well-balanced, positivity-preserving and sign-preserving properties simultaneously. The proposed methods have been tested and validated on several one- and two-dimensional test cases, and good numerical results have been observed. In the second part, we propose a fully-discrete energy-conserving scheme for the nonlinear Dirac equation, by combining the scalar auxiliary variable (SAV) technique with DG discretization. We start by discussing the semi-discrete DG discretization, and show that, with suitable choices of numerical fluxes, the resulting method conserves the charge, energy exactly and preserves the multi-symplectic structure. The optimal error estimate of semi-discrete DG scheme is carried out. We combine it with the energy conserving SAV technique, and demonstrate that the fully-discrete scheme conserves the discrete global energy exactly. Both second order SAV method based on midpoint rule and its high order extension have been studied. The proposed methods have been tested in some numerical experiments, which confirm the optimal rates of convergence and the energy conserving property. Numerical comparison with energy dissipative DG method and Runge-Kutta DG method is also provided to demonstrate that the numerical error of the energy conserving method does not grow significantly in long time simulation.
Yulong Xing (Advisor)
Saleh Tanveer (Committee Member)
Dongbin Xiu (Committee Member)
109 p.
Mathematics

Recommended Citations

Citations

  • Yang, R. (2022). High Order Structure-Preserving Discontinuous Galerkin Methods for Shallow Water Equations and Nonlinear Dirac Equation [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1650299205496592

    APA Style (7th edition)

  • Yang, Ruize. High Order Structure-Preserving Discontinuous Galerkin Methods for Shallow Water Equations and Nonlinear Dirac Equation. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1650299205496592.

    MLA Style (8th edition)

  • Yang, Ruize. "High Order Structure-Preserving Discontinuous Galerkin Methods for Shallow Water Equations and Nonlinear Dirac Equation." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1650299205496592

    Chicago Manual of Style (17th edition)

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This open access ETD is published by The Ohio State University and OhioLINK.