Advances in Time-Multi-Scale Simulations: When Time Becomes Multidimensional

International audienceIf we are interested in solving a transient problem with the time scales ranging from the femto-second to the seconds, where the time scales are not separable, the time axis could be transformed into a 2D domain with a fast and a slow time scales, the finest one defined in a small interval and the slowest one in the whole time interval of interest. This procedure can be extended to ND by introducing intermediate time scales. In 5 dimensions and considering one thousand time steps by direction we could solve a problem with a characteristic time of a femto-second in some seconds of CPU: the complexity remains of the order of thousands because $(10^3)^5=10^{15}$ if the curse of dimensionality can be avoided. The proper generalized decomposition allows circumventing the redoubtable curse of dimensionality. When the different scales can be separated, different times can be introduced and the associated equations solved simultaneously without the necessity to proceed by introducing an asymptotic development. This contribution proposes some new routes for accounting for multi-scale models involving both separated and non-separated time scales