A Late-Time Stable Scalar-Only Integral Equation for the Modelling of Transient Scattering by Perfect Conductors

Abstract

Scattering of time-harmonic Maxwell fields by perfect conductors is most commonly modelled by the electric field integral equation. This equations can be rendered numerically stable in both the low frequency and dense grid regimes using a combination of Helmholtz splitting and rescaling using quasi-Helmholtz projectors, and Calderón preconditioning. For simply connected scatterers, the mixed formulation introduced by Buffa et al., which is set entirely on scalar valued spaces, provides an alternative. Advantages are that the spatial discretisation of the Helmholtz components can be chosen independently to better account for solution regularity, that the implementation can be built on top of code designed for the scalar wave equation, and that there is no need to implement the vector-valued Buffa-Christiansen functions for preconditioning. In this contribution, a mixed formulation is derived for time-domain simulations, and appropriate temporal discretisation schemes, informed by the coercivity properties of the frequency domain method, are derived. Simulations show the absence of DC instabilities and high frequency instabilities, supporting the choice of Helmholtz component rescaling and temporal discretization.