A highly efficient second-order accurate long-time dynamics preserving scheme for some geophysical fluid models

TL;DR

This work develops a highly efficient, second-order ($BDF2$) time-stepping scheme that preserves the long-time statistical properties of geophysical fluid models by embedding the original system into a mean-reverting Scalar Auxiliary Variable framework. The key innovation is the mean-reverting SAV mechanism, which yields a fixed SPD linear system per step and unconditional long-time stability, enabling uniform-in-time bounds and rigorous convergence of global attractors and invariant measures as the time step vanishes. The authors apply the method to NSE, MHD, and both damped-driven barotropic and continuously stratified QG models, proving unconditionally stable behavior in appropriate fractional Sobolev spaces and establishing asymptotic consistency between the discrete and continuous systems. Numerical experiments on NSE and CQG demonstrate second-order accuracy, long-time stability, and the mean-reverting property, along with complex dynamics such as bursting, highlighting the practical utility and robustness of the approach for climate-scale simulations.

Abstract

We develop and analyze a highly efficient, second-order time-marching scheme for infinite-dimensional nonlinear geophysical fluid models, designed to accurately approximate invariant measures-that is, the stationary statistical properties (or climate) of the underlying dynamical system. Beyond second-order accuracy in time, the scheme is particularly well suited for long-time simulations due to two key features: it requires solving only a fixed symmetric positive-definite linear system with constant coefficients at each step; and it guarantees long-time stability, producing uniformly bounded solutions in time for any bounded external forcing, regardless of initial data. For prototypical models such as the barotropic quasi-geostrophic equation, the method preserves dissipativity, ensuring that numerical solutions remain bounded in a function space compactly embedded in the phase space as time tends to infinity. Leveraging this property, we rigorously prove convergence of both global attractors and invariant measures of the discrete system to those of the continuous model in the vanishing time-step limit. A central innovation of the method is a mean-reverting scalar auxiliary variable (mr-SAV) formulation, which preserves the dissipative structure of externally forced systems within an appropriate phase space. For the infinite-dimensional models considered, we additionally employ fractional-order function spaces to establish compactness of numerical solutions in topologies compatible with the phase space.

Figures (6)

• Figure 1: The $L^2$ norm (enstrophy) and $H^1$ norm (palinstrophy) of the vorticity as a function of time by the mr-SAV-BDF2 scheme with 256 Fourier modes and $k=0.01, 0.005, 0.0025$ respectively.
• Figure 2: The time evolution of $|q_n-1|$ by the SAV-BDF2 scheme without mean-reverting $\gamma=0$ and $k=0.002$(left), with mean-reverting $\gamma=1000$ and $k=0.002$ (right), respectively. The spin-up $T_s=100$ is calculated by a first order semi-explicit method with $k=10^{-5}$.
• Figure 3: The real part of the Fourier coefficient of mode $e^{iy}$ as a function of time. $Re=25.7715, k=0.001$ with 256 Fourier modes.
• Figure 4: The palinstrophy ($L^2$ norm of gradient of vorticity) as a function of time. $Re=25.7715, k=0.001$ with 256 Fourier modes.
• Figure 5: Time interval between bursts

Theorems & Definitions (21)

• Theorem 2.1
• proof
• Remark 2.1
• Proposition 3.1
• Lemma 3.1
• proof
• Corollary 3.1
• Theorem 3.1
• proof
• Remark 3.1