Numerical Investigations of an Implicit Leapfrog Time-Domain Meshless Method

Abstract

Numerical solution of partial differential equations governing time domain simulations in computational electromagnetics, is usually based on grid methods in space and on explicit schemes in time. A predefined grid in the problem domain and a stability step size restriction need. Recently, the authors have reformulated the meshless framework based on smoothed particle hydrodynamics, in order to be applied for time domain electromagnetic simulation. Despite the good spatial properties, the numerical explicit time integration introduces, also in a meshless context, a severe constraint. In this paper, at first, the stability condition is addressed in a general way by allowing the time step increment get away from the minimum points spacing. Then, an alternating direction implicit leapfrog scheme for time evolution is proposed. The unconditional stability of the method is analytically provided and numerically validated. The stability of the method has been proved by avoiding the algebra developments related to the usually adopted von Neumann analysis. Three case studies are investigated by achieving a satisfactory agreement by comparing both numerical and analytical results.