Improved Kernel-based Meshless Method for Solving Partial Differential Equations

Abstract

In the last decades the development of meshless computational methods for solving Boundary Value Problems (BVPs) have been quite successful. Compared with traditional mesh-dependent Finite Difference Method (FDM), Finite Volume Method (FVM), and Finite Element Method (FEM), the use of smooth kernels in these meshless methods have the distinct advantages of higher efficiency, faster convergence, and easier implementation. Among these kernel-based meshless methods, the Radial Basis Function Collocation Method (RBFCM) was firstly devised and is now most widely adopted to solve various kinds of BVPs. The RBFCM, however, suffers from the well known ill-conditioning problem due to its full coefficient matrix and hence hinders its application to solve large scale problems.
In this dissertation, we investigate the recently developed Localized Radial Basis Function Collocation Method (LRBFCM) and extend it to solve several larger scale problems. For illustration, the numerical simulation of solution to a thermodriven fluid-flow model defined in a square cavity is firstly obtained by using the LRBFCM with explicit time iteration scheme. Combining with the upwind scheme, the LRBFCM is then applied to solve a convection-dominated model to obtain a stable and oscillation-free solution. For higher accuracy, the upwind scheme is later replaced by partial upwind scheme. Furthermore, the LRBFCM is firstly applied to solve space-time fractional partial differential equations, which have drawn much attention from researchers and engineers.
Finally, we adapt and modify a newly developed integration-based method to obtain an integration-based meshless kernel method to solve stiff problems with boundary layers and shock waves. Instead of collocating the RBF-kernels into the differential operator in RBFCM and LRBFCM, we apply the method of integration by parts to transform the original partial differential equation into an equivalent integral form whose numerical approximation is obtained stably by using the RBF-kernels. Several numerical examples are given to illustrate the accuracy and efficiency of this state-of-the-art integration-based meshless computational method.

Date of Award	1 Sept 2015
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Yiu Chung Benny HON (Supervisor)