Devising efficient numerical methods for oscillating patterns in reaction–diffusion systems

Abstract

In this paper, we consider the numerical approximation of a reaction–diffusion system 2D in space whose solutions are patterns oscillating in time or both in time and space. We present a stability analysis for a linear test heat equation in terms of the diffusion $d$ and of the reaction timescales given by the real and imaginary parts $\alpha$ and $\beta$ of the eigenvalues of $J\left(P_e\right)$, the Jacobian of the reaction part at the equilibrium point $P_e$. Focusing on the case $\alpha =0,\beta \ne 0$, we obtain stability regions in the plane $(\xi ,\nu )$, where $\xi =\lambda (h;d)h_t$, $\nu =\beta h_t$, $h_t$ time stepsize, $\lambda$ lumped diffusion scale depending also from the space stepsize $h$ and from the spectral properties of the discrete Laplace operator arising from the semi-discretization in space. In space we apply the Extended Central Difference Formulas (ECDFs) of order $p=2,4,6$. In time we approximate the diffusion part in implicit way and the reaction part by a selection of integrators: the Explicit Euler and ADI methods, the symplectic Euler and a partitioned Runge–Kutta method that are symplectic in the absence of diffusion. Hence, by estimating $\lambda$, for each method we derive stepsize restrictions $h_t⪅F_{met}(h;d,\beta ,p)$ in terms of the stability curve $F_{met}$ depending on diffusion and reaction timescales and from the approximation order in space. For the same schemes, we provide also a dispersion error analysis. We present numerical simulations for the test heat equation and for the Lotka–Volterra PDE system with solutions oscillating only in time for the presence of a centre-type dynamics. In these cases, the implicit-symplectic schemes provide the best choice. We solve also the Schnakenberg model with spatial patterns oscillating in space and time in the presence of an attractive limit cycle due to the Turing–Hopf instability. In this case, all schemes attain closed orbits in the phase space, but the Explicit ADI method is the best choice from the computational point of view.