Finite Difference Methods, Hermite Interpolation and a Quasi-Uniform Spectral Scheme (QUSS).

[摘要] This thesis discusses finite difference approximations, Hermite interpolation and Quasi-Uniform Spectral Schemes. In the discussion of finite difference approximations, an explicit algebraic condition on the grid points is given that explains when a boosted order of accuracy occurs. Moreover, an efficient method for the computation of the finite difference weights is provided. An elegant derivation of the barycentric Hermite interpolant is shown which leads to an alternative derivation of a known method in the literature for the computation of the barycentric weights. This new approach to the construction of the barycentric Hermite interpolant leads to a novel and efficient method for updating the barycentric weights when an additional data item is added.Three different Quasi-Uniform Spectral Schemes are discussed in this thesis. The grids used for interpolation are constructed using the Elliptic, Kosloff-Tal-Ezer (KT) and Theta mapping. The interpolant is a mapped cosine interpolant. Practical advice on how to pick the optimal map parameter is provided. A comparison of the approximation errors of the mapped cosine interpolants for a test function illustrates that the three mappings have no significant difference in terms of their approximation error.

[发布机构] University of Michigan

[效力级别] Finite Difference Methods

[关键词] Numerical Methods;Finite Difference Methods;Spectral Methods;Hermite Interpolation;Mathematics;Engineering;Science;Applied and Interdisciplinary Mathematics