On the behaviour of time discretisations of the electric field integral equation

We derive a separation of variables solution for time-domain electromagnetic scattering from a perfectly conducting infinite flat plate. The time dependent part of the equations are then used as a model problem in order to study the effects of various time discretisations on the full scattering problem. We examine and explain how exponential and polynomial instabilities arise in the approximation schemes, and show that the time averaging which is often used in an attempt to stabilise solutions of the full problem acts to destabilise some of the schemes. Our results show that two of the time discretisations can produce good results when coupled with a space-exact approximation, and indicate that they will be useful when coupled with an accurate enough spatial approximation.