Date of Award


Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Cécile Piret

Committee Member 1

Allan Struthers

Committee Member 2

Argyrios Petras

Committee Member 3

Jiguang Sun


The bulk of this dissertation is mainly composed of four chapters, which are organized as follows: Chapter 1 provides an introduction to the Radial Basis Functions (RBF) method by briefly outlining its historical developments and reviewing the RBF interpolation and the RBF-Finite Difference (FD) methodologies, and their advantages/disadvantages. Chapter 2 describes the Orthogonal Gradients (OGr) method and the Fast OGr method and how these can be used to compute differential operators restricted to hypersurfaces and space curves ($\Gamma$) embedded in R3. We will highlight a challenge of pairing Fast OGr with RBF-FD on nearly flat local clusters and how to overcome it. Numerical results of computing the surface (curve) Laplace-Beltrami operator  $\Delta_{\Gamma}$, will be presented. Chapter 3 presents a novel local parameterization method with RBF-FD for computing arbitrary differential operators restricted to arbitrary manifolds. Numerical results will be presented for computing the Laplace-Beltrami operator of curves embedded in R2. Additionally, results will be presented for a stable, novel methodology of evolving curves via two examples. Chapter 4 is a study of how to couple the spatial differentiation matrices of the RBF-FD method with the Parareal Algorithm, a parallel-in-time integration method. The details of the Parareal algorithm will be reviewed along with potential theoretical speed-up for multi-processor computations. This will be followed by numerical results demonstrating the successful loose and tight couplings of the methodologies. For certain time-dependent partial differential equations the differentiation matrix produced by the RBF-FD method sometimes contains spurious eigenvalues which have a positive real part. We examine how the Parareal algorithm provides an opportunity for mitigating the effects of the spurious eigenvalues in the spectrum.

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Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.