Combining the gap-tooth scheme with projective integration: Patch dynamics

An important class of problems exhibits macroscopically smooth behaviour in space and time, while only a microscopic evolution law is known, which describes effects on fine space and time scales. A simulation of the full microscopic problem in the whole space-time domain can therefore be prohibitively expensive. In the absence of a simplified model, we can approximate the macroscopic behaviour by performing appropriately initialized simulations of the available microscopic model in a number of small spatial domains ("boxes") over a relatively short time interval. Here, we show how to obtain such a scheme, called "patch dynamics," by combining the gap-tooth scheme with projective integration. The gap-tooth scheme approximates the evolution of an unavailable (in closed form) macroscopic equation in a macroscopic domain using simulations of the available microscopic model in a number of small boxes. The projective integration scheme accelerates the simulation of a problem with multiple time scales by taking a number of small steps, followed by a large extrapolation step. We illustrate this approach for a reaction-diffusion homogenization problem, and comment on the accuracy and efficiency of the method.