Date of Award
8-2024
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
School
Mathematics and Natural Sciences
Committee Chair
Dr. Huiqing Zhu
Committee Chair School
Mathematics and Natural Sciences
Committee Member 2
Dr. Haiyan Tian
Committee Member 2 School
Mathematics and Natural Sciences
Committee Member 3
Dr. C.S. Chen
Committee Member 3 School
Mathematics and Natural Sciences
Committee Member 4
Dr. James Lambers
Committee Member 4 School
Mathematics and Natural Sciences
Abstract
A localized Hermite method of approximate particular solutions (LHMAPS) is presented in this dissertation. This method is designed to improve the accuracy of the localized method of approximate particular solutions (LMAPS) for solving partial differential equations. LMAPS is a strong-form method that defines local radial basis function approximations to the solution values at a group of collocation points before applying the differential operator to this approximation function. Based on the LMAPS's local scheme, LHMAPS seeks Hermite-type local approximations based on both function values and derivatives. Numerical experiments validate the superior accuracy of the proposed method to the LMAPS by solving the 2D Poisson equation, Helmholtz equation, and convection-reaction-diffusion equation. Also, when the convection term is dominating in the convection-reaction-diffusion problem, we employ the up-winding technique to control the oscillatory nature of the numerical solution. Again, numerical results show that the proposed method has better accuracy than LMAPS and RBF-HFD.
Copyright
2024, Kwesi Acheampong
Recommended Citation
Acheampong, Kwesi, "Localized Hermite Method of Approximate Particular Solutions" (2024). Dissertations. 2276.
https://aquila.usm.edu/dissertations/2276