Beyond the mesh handling Maxwell's curl equations with an unconditionally leapfrog stable scheme

TL;DR

This paper introduces an unconditionally stable, meshless electromagnetic simulation method that combines leapfrog time integration with an implicit scheme, overcoming traditional grid and stability constraints in time domain simulations.

Contribution

The authors develop a novel meshless, unconditionally stable scheme for electromagnetic simulations by integrating an implicit time-stepping method with a meshless spatial discretization, enabling larger time steps.

Findings

Unconditionally stable in time without grid constraints

Accurate simulation of open spatial problems with PML

Effective boundary treatment with consistency restoring approach

Abstract

Numerical solution of equations governing time domain simulations in computational electromagnetics, is usually based on grid methods in space and on explicit schemes for the time evolution. A predefined grid in the problem domain and a stability step size restriction must be accepted. Evidence is given that efforts need for overcoming these heavy constraints. Recently, the authors developed a meshless method to avoid the connective laws among the points scattered in the problem domain. Despite the good spatial properties, the numerical explicit integration used in the original formulation of the method provides,also in a meshless context, spatial and time discretization strictly interleaved and mutually conditioned. Afterwards, in this paper the stability condition is firstly addressed in a general way by allowing the time step increment get away from the minimum points spacing. Meanwhile, a formulation of the alternating direction implicit scheme for the evolution in time is combined with the meshless solver. The formulation preserves the leapfrog marching on in time of the explicit integration scheme. The new method, not constrained by a gridding in space and unconditionally stable in time, is numerical assessed by different numerical simulations. Perfect matching layer technique is used in simulating open spatial problems; otherwise, a consistency restoring approach is introduced in treating truncation at finite boundary and irregular points distribution. Three case studies are investigated by achieving a satisfactory agreement comparing both numerical and analytical results.