Description
- Abstract:
- This dissertation focuses on studies of two different discontinuous Galerkin (DG) methods for general convection-diffusion equations. One preserves the strict maximum principle for general nonlinear convection-diffusion equations on unstructured triangular meshes, and the other is designed specifically for a class of second order elliptic problems with rough coefficients, where the local oscillating directions vary smoothly in the computational domain.<br/> It is a highly desirable property that a numerical scheme is maximum-principle- satisfying when solving general nonlinear convection-diffusion equations, not only because violating it may produce physically meaningless numerical solutions, but could also lead to numerical instability or ill-posedness. Similar to the pure convection case [75] we propose, under suitable time step constraints, by using Strong Stability Preserving (SSP) time discretizations, and coupling with a simple scaling limiter, DG methods preserve the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Extensions to two-dimensional convection-diffusion equations on triangular meshes are derived without imposing any geometric constraints on the mesh such as angle acuteness.<br/> It is well known that when designing finite element schemes for elliptic equations with rough coefficients, computational efficiency can be improved by properly incorporating local oscillating features into the approximation space. Thus, by choosing a special non-polynomial approximation space, we propose special DG schemes which capture the multi-scale solution without having to resolve the finest scales. Detailed error estimates for two dimensional second order DG methods are derived, and a general guidance on how to construct such non-polynomial basis is discussed.<br/> <br/> This methodology can be applied to various problems and applications including porous medium equations, incompressible Navier-Stokes equation, as well as composite material problems. In addition, the many good properties that DG scheme has make it an ideal method for applications in atmospheric research. After combining it with the H-WENO (Hermite Weighted Essentially Non-Oscillatory) limiter and a Bound-Preserving (BP) filter, we designed a transport equation DG solver for large scale atmospheric modeling on a cubed-sphere geometry, guaranteed to provide non-oscillatory, positivity-preserving solutions.
- Notes:
- Thesis (Ph.D. -- Brown University (2013)
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Citation
Zhang, Yifan,
"Discontinuous Galerkin Methods for Convection Diffusion Equations: Positivity Preserving and Multi-scale Resolution"
(2013).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z01R6NV9
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....