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 small compared to the dominant slow time scales of the dynamics of the system. In this paper, we apply these methods to accelerate a lattice Boltzmann model for the one-dimensional FitzHugh–Nagumo reaction–diffusion system. We compare the projective method [C.W. Gear, I.G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM Journal on Scientific Computing 24 (4) (2003) 1091–1106] proposed by Gear and Kevrekidis to the related multistep scheme [C. Vandekerckhove, D. Roose, K. Lust, Numerical stability analysis of an acceleration scheme for step size constrained time integrators, Journal of Computational and Applied Mathematics 200 (2) (2007) 761–777] that we developed in an earlier paper. It is shown that the acceleration related error obtained with both methods is comparable and small compared to the discretization 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 the lattice Boltzmann model with a larger time step. Finally, we illustrate that it is straightforward to combine the acceleration methods with traditional time integration tools such as adaptive step size control.