The method of particular solutions with polynomial basis functions for solving axisymmetric problems

Abstract

In this paper, we extend the previous work of Chen et al. (Numer Methods Partial Differential Eq 21: 349–367, 2005) on the two-step method of particular solutions (MPS) for solving the Poisson equation with an axisymmetric forcing term and boundary conditions in an axisymmetric geometry to general differential equations and time-dependent problems using the one-step MPS. Polynomial basis functions are sufficient for the proposed approach instead of using Chebyshev polynomials. Furthermore, no boundary method is required for solving the homogeneous equation which is required in the two-step approach. In the solution process of the two-step MPS, we only require the closed form particular solution of the Laplacian or Helmholtz equation with respect to the monomial basis functions. The proposed approach is more simplified compared to the previous work and also allows us to solve a large class of partial differential equations including those with variable coefficients. We further extend the proposed approach to time-dependent problems using the Houbolt method, which is a third order time marching finite difference scheme. In the numerical implementation, we compare the results using reduced axisymmetric equations and the original 3D equations. Numerical results show the high simplicity, accuracy, and efficiency of the proposed numerical method.