AN IMPROVEMENT OF A MATHEMATICAL INTERPRETATION OF MOVING PARTICLE SEMI-IMPLICIT METHOD

Abstract

This paper presents an improvement of a mathematical interpretation of moving particle semi-implicit (MPS) method. The mathematical interpretation leads to a mathematical reformulation of MPS (MRMPS) based on Taylor expansions. The improvement of MRMPS in this paper is featured by solving a system of 9 × 9 (or 5 × 5 for two dimensional settings) equations for the gradient vector and for all the components of the Hessian matrix. Numerical experiments with various types of target functions showed that the improved MRMPS possesses a second-order convergence rate for the relative error of the gradient and a first-order convergence rate for the relative error of the Laplacian, in three-dimensional settings with randomly distributed neighboring particles. Moreover, there is no deterioration of accuracy for realistic particle configurations near free surfaces, where the neighboring particles are distributed not only randomly but also one-sided. Further, the aforementioned accuracy of the improved MRMPS can be obtained by using about 40 to 50 neighboring particles considerably less than conventional particle methods. A simplification for the improved MRMPS is also presented with less computational complexity, solving two 3 × 3 systems instead of one 9 × 9 system, at the cost of losing one order of convergence rate of error.