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

TL;DR

This paper introduces an efficient, unconditionally stable numerical scheme for long-time simulation of nonlinear geophysical models, accurately capturing their statistical properties with improved computational performance.

Contribution

The paper presents a novel mean-reverting-SAV-BDF2 scheme that is highly efficient, unconditionally stable, and accurately approximates the long-time dynamics and invariant measures of nonlinear models.

Findings

The scheme remains uniformly bounded for all time.

It converges to the true model's attractors and invariant measures as step size decreases.

The second-order scheme outperforms first-order methods in reaching statistical equilibrium.

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.