A comparative study of several classes of meshfree methods for solving the Helmholtz equation

Abstract

Meshfree methods have been widely used in physics, materials science and other fields in recent years, because they do not need to rely on grids and are more flexible. A comparative study of several meshfree methods for solving the Helmholtz equation is presented in this paper. The current work proposes a review of the compactly supported radial basis functions collocation method, generalized finite difference method, local radial basis functions collocation method as well as radial basis functions Galerkin method. Meanwhile, the hierarchical radial basis functions collocation method is studied by constructing hierarchical radial basis functions utilizing progressive refinement scattered data sets and scaling compactly supported radial basis functions with variable support radii. The performances of the proposed methods are analyzed by numerical experiments on irregular as well as regular domains. Numerical experiments exhibit that the proposed methods can solve the Helmholtz equation efficiently, however, the hierarchical radial basis functions collocation method not only obtains high precision results, but also reduces the time cost effectively.