Damping factors for the gap-tooth scheme
An important class of problems exhibits macroscopically smooth behaviour in space and time, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, the gap-tooth scheme has recently been proposed. The scheme approximates the evolution of an unavailable (in closed form) macroscopic equation in a macroscopic domain; it only uses appropriately initialized simulations of the available microscopic model in a number of small boxes. For some model problems, including numerical homogenization, the scheme is essentially equivalent to a finite difference scheme, provided we define appropriate algebraic constraints in each time-step to impose near the boundary of each box. Here, we demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffers that “shield” over relatively short time intervals the dynamics inside each box from boundary effects. We explore and quantify the behavior of these schemes systematically through the numerical computation of damping factors of the corresponding coarse time-stepper, for which no closed formula is available.
KeywordsNumerical Homogenization Impose Dirichlet Boundary Condition Kinetic Monte Carlo Model Explicit Euler Time
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