Results of von Neumann analyses for reproducing kernel semi-discretizations

Abstract

The reproducing kernel particle method (RKPM) has many attractive properties that make it ideal for treating a broad class of physical problems. RKPM may be implemented in a ‘mesh-full’ or a ‘mesh-free’ manner and provides the ability to tune the method, via the selection of a window function and its associated dilation parameter, in order to achieve the requisite numerical performance. RKPM also provides a framework for performing hierarchical computations making it an ideal candidate for simulating multi-scale problems. Although the method has many appealing attributes, it is quite new and its numerical performance is still being quantified with respect to more traditional discretization techniques. In order to assess the numerical performance of RKPM, detailed studies of the method on a series of model partial differential equations has been undertaken. The results of von Neumann analyses for RKPM semi-discretizations of one and two-dimensional, first- and second-order wave equations are presented in the form of phase and group errors. Excellent dispersion characteristics are found for the consistent mass matrix with the proper choice of dilation parameter. In contrast, row-sum lumping the mass matrix is demonstrated to introduce severe lagging phase errors. A ‘higher-order’ mass matrix improves the dispersion characteristics relative to the lumped mass matrix but also yields significant lagging phase errors relative to the fully integrated, consistent mass matrix. Published in 2000 by John Wiley & Sons, Ltd.