Nonlinear Limiters for Discontinuous Galerkin MethodZhao, BingyucreatorZhu, JunReaderFu, GuoshengReaderShu, Chi-WangAdvisorBrown University. Department of Applied Mathematicssponsor2017xv, 93 p.born digitalThesis (Ph. D.)--Brown University, 2017thesesThis dissertation presents parametrized maximum principle preserving (MPP) flux limiters and an artificial diffusion weighed essentially non-oscillatory (WENO) limiter for discontinuous Galerkin (DG) methods solving hyperbolic conservation laws. It is a highly desirable property that a numerical scheme is maximum-principle-satisfying. In this dissertation, we apply the parametrized MPP flux limiters on the implicit DG methods for the linear convection equations. At every final Runge-Kutta stage, we combine the temporal integrated high order numerical flux with a first order one, which preserves the maximum principle, for the cell averages. Based on the same framework in search for the combining parameters, we also introduce another flux limiter by proposing a global optimization problem solved by Simplex algorithm. Numerical tests show that the global flux limiter can achieve the maximum principle for the cell averages, with the original order of accuracy maintained. We also investigate a new limiting procedure based on WENO limiters for the DG methods solving hyperbolic conservation laws, to control spurious numerical oscillations near discontinuities while maintaining uniform high order accuracy in smooth regions. The idea of this limiter is to add an artificial diffusion term containing the difference between the numerical solution and the reconstructed polynomial, to the original variational formulation of the DG method. An analysis in finite difference fashion is provided to demonstrate the added term being numerical viscosity. The main advantage of this method is that it incorporates the limiter into the weak formulation of the DG methods, and has a semi-discrete form. With the parameter suitably chosen, the scheme can achieve its optimal performance. The procedure has been applied in computational fluid dynamics in one and two dimensions to illustrate its good behavior.Discontinuous Galerkin Methodsmaximum principle preserving limiterhyperbolic conservation lawartificial diffusion WENO limiterEnglishRPB2017061610.7301/Z0JQ0ZHKIn CopyrightCollection is open for research.dissertations