The Method of Fundamental Solutions for One-Dimensional Wave Equations

A meshless numerical algorithm is developed for the solutions of one-dimensional wave equations in this paper. The proposed numerical scheme is constructed by the Eulerian-Lagrangian method of fundamental solutions (ELMFS) together with the D’Alembert formulation. The D’Alembert formulation is used to avoid the difficulty to constitute the linear algebraic system by using the ELMFS in dealing with the initial conditions and time-evolution. Moreover the ELMFS based on the Eulerian-Lagrangian method (ELM) and the method of fundamental solutions (MFS) is a truly meshless and quadrature-free numerical method. In this proposed wave model, the one-dimensional wave equation is reduced to an implicit form of two advection equations by the D’Alembert formulation. Solutions of advection equations are then approximated by the ELMFS with exceptionally small diffusion effects. We will consider five numerical examples to test the capability of the wave model in finite and infinite domains. Namely, the traveling wave propagation, the time-space Cauchy problems and the problems of vibrating string, etc. Numerical validations of the robustness and the accuracy of the proposed method have demonstrated that the proposed meshless numerical model is a highly accurate and efficient scheme for solving one-dimensional wave equations.