A meshfree collocation method based on moving Taylor polynomial approximation for high order partial differential equations

Abstract

This paper presents a meshfree collocation method for solving high order partial differential equations (PDEs). The leading numerical difficulty is the approximation of high order derivatives. To make the approximation simple and efficient, a moving Taylor polynomial (MTP) approximation is presented by using movable expansion point for each sub-domain. Derivatives can be derived straightforward from the corresponding Taylor coefficients, which are determined by solving a weighted least squares problem. A distinct feature of the method is its ability to give the derivatives along with the shape function itself without further cost. To ensure the accuracy of high order approximation, stability of the weighted least squares problems for determining the Taylor coefficients is another issue should be addressed. For this purpose, the basis functions are rescaled by the size of window functions, and QR decomposition is adopted to solve the weighted least squares problems. The collocation method based on this MTP approximation does not require any grid or background cell, so it is a truly meshfree method. When solving the linear algebraic system generated by the MTP collocation method, a preconditioned sparse biconjugate gradients stabilized (BICGSTAB) solver is used to accelerate the computation speed. Numerical tests show that the proposed method is much accurate and efficient for high order PDEs.