Acceleration of Lattice Boltzmann Models through State Extrapolation: a Reaction-Diffusion Example

Recently, several methods were proposed to accelerate a time integrator that uses a time step that is much smaller than the dominant slow time scales of the dynamics of the system. In this paper, we apply these methods to accelerate the lattice Boltzmann BGK model for the one-dimensional FitzHugh-Nagumo reaction-diffusion system. We compare the projective method proposed by Gear and Kevrekidis to the related multistep scheme which we developed in an earlier paper. It is shown that both methods lead to a comparable acceleration error, which is small compared to the discretisation error of the lattice Boltzmann model itself. Therefore, a substantial speedup can be obtained, essentially without accuracy loss. Furthermore, it is shown that the accuracy obtained with these acceleration schemes is better than the accuracy of a lattice Boltzmann model with a larger time step. Finally, we illustrate that it is straightforward to combine these acceleration methods with traditional time integration tools such as adaptive step size control.