Advancements in Numerical Modeling: High-Order Methods for Fractional Initial Value Problems and Meshfree Solvers for Partial Differential Equations

Graduate Thesis Or Dissertation

Citations

Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/cv43nz38x

Abstract

This dissertation presents four topics which broadly fall into the category of computational mathematics. A method is described for performing quadrature with up to 10th order accuracy for functions when one or both end points of the integration interval do not coincide with any of the equispaced grid points. This method can be utilized, for example, when functions to be integrated feature discontinuities at locations that can be separately determined (at or in between the equispaced grid points). The same underlying quadrature routine is then used to develop a solver for initial value problems on fractional differential equations. Order 6 convergence is achieved for problems with minimal smoothness assumptions. An algorithm which efficiently subsamples quasi-uniform, variable density node sets is then presented. This subsampling algorithm is able to preserve the local densities of the original node set. Finally a meshfree PDE solver is presented which utilizes the subsampling algorithm to implement a multilevel solver with RBF-FD. Efficiencies in the algorithm are gained through domain decomposition techniques.

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