Xiaoming Wang

We propose a linear variable‐step exponential time‐differencing method for the incompressible Navier Stokes equations in vorticity streamfunction formulation on a two‐dimensional periodic box. The method consists of a second‐order scheme and an embedded first‐order variant, yielding a natural mechanism for adaptive time stepping and a posteriori error control. Each time step requires only uniquely solvable linear problems: two heat equation solves, efficiently handled by Fourier methods in the periodic setting, and one linear scalar auxiliary‐variable equation, evaluated via Laplace transform and Talbot's numerical inverse transform. The construction combines the ETD framework, a mean‐reverting scalar auxiliary variable (mr‐SAV), and second‐order extrapolation of the nonlinear term. The mean‐reverting correction enables long‐time stability while preserving full linearity, distinguishing the method from related mr‐SAV schemes that require nonlinear algebraic solves. We prove unconditional long‐time stability: for uniformly bounded forcing, the discrete vorticity remains bounded in for all Reynolds numbers and time‐step sizes. Numerical experiments demonstrate second‐order accuracy, long‐time stability, and effective adaptive error control.

Within the Darcy-Boussinesq framework for convection in multilayered porous media, we investigate the singular limit in which the thickness of one layer tends to zero. We establish that the solution of the full system converges to that of the corresponding limiting model with one fewer layer. The convergence is established in two complementary senses: (i) Strong $ L^{2} $-convergence over arbitrary finite time intervals, and (ii) upper semi-continuity of the global attractors describing the large-time asymptotic behavior.

We analyze a variable-step extension of a family of arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes recently developed by the authors. We prove that the schemes are unconditionally stable in the sense that a modified energy-representing a slight perturbation of the original energy-decreases monotonically over time, provided the nonlinearity is Lipschitz continuous in some appropriate sense. Moreover, we establish optimal-order convergence under mild conditions on the time-step size and local time-step ratio. Numerical experiments on the thin film epitaxial growth model without slope selection, employing a novel variable-step second-order scheme, validate the theoretical findings as well as its potential in developing highly efficient time-adaptive solution.

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.

We investigate the sharp material interface limit of the Darcy–Boussinesq model for convection in layered porous media with diffused material interfaces, which allow a gradual transition of material parameters between different layers. We demonstrate that as the thickness of these transition layers approaches zero, the conventional sharp interface model with interfacial boundary conditions, commonly adopted by the fluids community, is recovered under the assumption of constant porosity. Our results validate the widely used sharp interface model by bridging it with the more physically realistic case of diffused material interfaces. This limiting process is singular and involves a boundary layer in the velocity field. Our analysis requires delicate estimates for elliptic and parabolic equations with discontinuous coefficients and the subtle validation of the boundary layer.

We investigate the stability of two families of three‐level two‐step schemes that extend the classical second‐order backward differentiation formula (BDF2) and second‐order Adams–Moulton (AM2) schemes. For a free parameter restricted to an appropriate range that covers the classical case, we show that both the generalized BDF2 and the generalized AM2 schemes are A‐stable. We also introduce the concept of uniform‐in‐time stability which characterizes a scheme's ability to inherit the uniform boundedness over all time of the solution of damped and forced equation with the force uniformly bounded in time. We then demonstrate that A‐stability and uniform‐in‐time stability are equivalent for an arbitrary three‐level two‐step schemes. Next, these two families of schemes are utilized to construct efficient and unconditionally stable IMEX schemes for systems that involve a damping term, a skew symmetric term, and a forcing term. These novel IMEX schemes are shown to be uniform‐in‐time energy stable in the sense that the norm of any numerical solution is bounded uniformly over all time, provided that the forcing term is uniformly bounded time, the skew symmetric term is dominated by the dissipative term, together with a mild time‐step restriction. Numerical experiments verify our theoretical results. They also indicate that the generalized schemes could be more accurate and/or more stable than the classical ones for suitable choice of the parameter.