Wave Resolution Properties and Weighted Essentially Non-Oscillatory Limiter for Discontinuous Galerkin Methods

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Title
Wave Resolution Properties and Weighted Essentially Non-Oscillatory Limiter for Discontinuous Galerkin Methods
Contributors
Zhong, Xinghui (creator)
Shu, Chi-Wang (Director)
Guzman, Johnny (Reader)
Hesthaven, Jan (Reader)
Brown University. Applied Mathematics (sponsor)
Doi
10.7301/Z0ZW1J6D
Copyright Date
2012
Abstract
This dissertation presents wave resolution properties and weighted essentially non-oscillatory limiter for discontinuous Galerkin methods solving hyperbolic conservation laws. <br/> <br/> In this dissertation, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order $k+2$ superconvergent at Radau points on each element and of order $2k+1$ superconvergent at the downwind point of each element, when using piecewise polynomials of degree $k$. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error <br/> for several fully discrete schemes. <br/> <br/> We also investigate a simple limiter using<br/> weighted essentially non-oscillatory (WENO) methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving conservation laws, with the goal of obtaining a robust<br/> and high order limiting procedure to simultaneously achieve uniform high order accuracy and sharp, non-oscillatory shock transitions. The idea of this limiter is to reconstruct the entire polynomial, instead of reconstructing point values or moments in <br/> the classical WENO reconstructions. That is, the reconstruction polynomial on the target cell is a convex combination of polynomials on this cell and its neighboring cells and the nonlinear weights of the convex combination follow the <br/> classical WENO procedure. The main advantage of this limiter is its simplicity in implementation, especially for multi-dimensional meshes.<br/> <br/>
Keywords
discontinuous Galerkin method
error analysis
superconvergence,fully discretized,points per wavelength,WENO limiter
Error analysis (Mathematics)
Notes
Thesis (Ph.D. -- Brown University (2012)
Extent
xvii, 122 p.

Citation

Zhong, Xinghui, "Wave Resolution Properties and Weighted Essentially Non-Oscillatory Limiter for Discontinuous Galerkin Methods" (2012). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://doi.org/10.7301/Z0ZW1J6D

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