An efficient scheme for approximating long-time dynamics of a class of non-linear models

Abstract: We propose a novel, highly efficient, mean-reverting-SAV-BDF2-based, long-time unconditionally stable numerical scheme for a class of finite-dimensional nonlinear models important in geophysical fluid dynamics. The scheme is highly efficient in that only a fixed symmetric positive definite linear problem (with varying right-hand sides) is solved at each time step. The solutions remain uniformly bounded for all time. We show that the scheme accurately captures the long-time dynamics of the underlying geophysical model, with the global attractors and invariant measures of the scheme converging to those of the original model as the step size approaches zero. In our numerical experiments, we adopt an indirect approach, using statistics from long-time simulations to approximate the invariant measures. Our results suggest that the convergence rate of the long-term statistics, as a function of terminal time, is approximately first-order under the Jensen-Shannon metric and half-order under the total variation metric. This implies that extremely long simulations are required to achieve high-precision approximations of the invariant measure (or climate). Nevertheless, the second-order scheme significantly outperforms its first-order counterpart, requiring far less time to reach a small neighborhood of statistical equilibrium for a given step size.