We propose a novel, highly efficient, second-order accurate, long-time unconditionally stable numerical scheme for a class of finite-dimensional nonlinear models that are of importance in geophysical fluid dynamics. The scheme is highly efficient in the sense that only a (fixed) symmetric positive definite linear problem (with varying right hand sides) is involved at each time-step. The solutions to the scheme are uniformly bounded for all time. We show that the scheme is able to capture the long-time dynamics of the underlying geophysical model, with the global attractors as well as the invariant measures of the scheme converge to those of the original model as the step size approaches zero. In our numerical experiments, we take an indirect approach, using long-term statistics 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 using the Jensen-Shannon metric and half-order using the L1 metric. This implies that very long time simulation is needed in order to capture a few significant digits of long time statistics (climate) correct. Nevertheless, the second order scheme's performance remains superior to that of the first order one, requiring significantly less time to reach a small neighborhood of statistical equilibrium for a given step size.

翻译：本文针对地球物理流体动力学中一类重要的有限维非线性模型，提出了一种新颖、高效、二阶精度且长期无条件稳定的数值格式。该格式的高效性体现在每个时间步仅需求解一个（固定的）对称正定线性问题（右侧项可变）。格式的解对所有时间均一致有界。我们证明该格式能够捕捉底层地球物理模型的长期动力学行为，当步长趋于零时，格式的全局吸引子及不变测度均收敛于原始模型。在数值实验中，我们采用间接方法，利用长期统计量逼近不变测度。结果表明：长期统计量的收敛速率（作为终止时间的函数）在Jensen-Shannon度量下近似为一阶，在L1度量下约为半阶。这意味着需要极长时间的模拟才能准确获取长期统计量（气候）的若干有效数字。尽管如此，二阶格式的性能仍优于一阶格式，在给定步长下达到统计平衡态小邻域所需的时间显著更短。