High Order Spectral Methods On Unstructured Grids
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High-Order/Spectral Methods on Unstructured Grids
We present an ab initio development of a convergent high-order accurate scheme for the solution of linear conservation laws in geometrically complex domains. As our main example we present a detailed development and analysis of a scheme suitable for the time-domain solution of Maxwell's equations in a three-dimensional domain. The fully unstructured spatial discretization is made possible by the use of high-order nodal basis, employing multivariate Lagrange polynomials defined on the triangles and tetrahedra. Careful choices of the unstructured nodal grid points ensure high-order/spectral accuracy, while the equations themselves are satisfied in a discontinuous Galerkin form with the boundary conditions being enforced weakly through a penalty term. Accuracy, stability, and convergence of the semi-discrete approximation to Maxwell's equations is established rigorously and bounds on the global divergence error are provided. Concerns related to efficient implementations are discussed in detail. This sets the stage for the presentation of examples, verifying the theoretical results, as well as illustrating the versatility, flexibility, and robustness when solving two-and three- dimensional benchmarks in computational electromagnetic. Pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated.
Spectral and High Order Methods for Partial Differential Equations
Author: Jan S. Hesthaven
language: en
Publisher: Springer Science & Business Media
Release Date: 2010-10-29
The book contains a selection of high quality papers, chosen among the best presentations during the International Conference on Spectral and High-Order Methods (2009), and provides an overview of the depth and breadth of the activities within this important research area. The carefully reviewed selection of the papers will provide the reader with a snapshot of state-of-the-art and help initiate new research directions through the extensive bibliography.
Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids
This thesis focuses on the development of high-order finite volume methods and discontinuous Galerkin methods, and presents possible solutions to a number of important and common problems encountered in high-order methods, such as the shock-capturing strategy and curved boundary treatment, then applies these methods to solve compressible flows.