Time marching schemes have been considered for a long time very delicate when applied to time domain integral equations arising from various wave problems. Stability issue has been and seems to remain at the centre of investigations ,. Attempts to reach stability include use of implicit schemes, spatial and temporal averaging procedures during the time stepping, or try to extrapolate the beginning of the response signals using different techniques as Prony's or autoregressive models. Other authors get round the difficulty by working on MFIE or CFIE rather than EFIE. This type of approach only delays the onset of the instability and/or seriously compromise the precision of the results. Recent studies are dedicated to improve the precision by using high order approximation  and leapfrogging , based on extrapolation techniques for band-limited signals. Thus complete time-space variational approximation leads to unconditionally stable and precise schemes. Several PhD thesis have been dedicated to study of this theory and its application to different situations in electromagnetism, acoustics and elastodynamics, under supervision of J.C. Nédélec and A. Bachelot. A rigorous functional framework and the first 3D electromagnetic EFIE stable computation without any averaging trick have been published in the thesis of Isabelle Terrasse . Since then, delayed potentials have been applied and studied in many situations, and accelaration techniques have been transposed from previous frequency domain studies . We present a unified presentation of the full variational theory. The resolution process for this system of equations involve at each time-step the resolution of a sparse system (easily done) for computing the current time step, and the computation of a convolution product for taking into account the influence of this new time step on the future time steps. This convolution product is the most time consuming part of the algorithm. It can be speeded-up through the use of a multipole algorithm, the time-domain equivalent of the widely accepted frequency domain fast multipole algorithm. The harmonic FMM is used to compute matrix vector product in order to solve BEM through an iterative solver, the transient FMM computes the convolution product at each time step : in both cases, it replaces an O(n) algorithm by an O(nlog(n)) method.
This is joint work with Guillaume Sylvand and Toufic Abboud.
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