Long-time stability of implicit-explicit Runge-Kutta methods for two-dimensional incompressible flows

TL;DR

This paper develops and analyzes high-order implicit-explicit Runge-Kutta methods for 2D incompressible flows, proving their long-time stability and suitability for adaptive time-stepping in fluid simulations.

Contribution

It introduces the first long-time stability analysis for high-order IERK schemes applied to 2D Navier-Stokes equations, with a simplified set of order conditions.

Findings

Established long-time stability in L2 and H1 norms.

Constructed efficient third- and fourth-order IERK schemes with reduced order conditions.

Demonstrated suitability for adaptive time-stepping with larger step sizes.

Abstract

High-order adaptive time-stepping algorithms are of significant practical value and theoretical interest for accelerating long-time fluid-flow simulations and resolving complex dynamical behaviors. While several high-order implicit-explicit schemes have been proposed in the literature, their long-time stability properties remain largely unexplored. We develop a family of long-time stable implicit-explicit Runge-Kutta (IERK) methods, up to fourth-order temporal accuracy, for the two-dimensional incompressible Navier-Stokes equations in vorticity-stream function formulation. By combining a convolution-type H\"{o}lder inequality with a damping-type multistage Gr\"{o}nwall inequality, we establish a unified analytical framework that proves long-time stability in both the $L^2$ and $H^1$ norms. A key component of the analysis is a mathematical-induction argument that ensures stage-wise boundedness of the vorticity in the $H^\delta$ norm for some $\delta>0$. To the best of our knowledge, this is the first work to establish large-time stability results for high-order IERK algorithms for the two-dimensional incompressible Navier-Stokes equations. Our IERK schemes employ stiffly accurate diagonally implicit Runge-Kutta approximations for the linear diffusive term together with explicit Runge-Kutta approximations for the nonlinear advection term. By exploiting the specific structure of the Navier-Stokes model, we derive a reduced set of order conditions-requiring only 5 and 11 conditions for the third- and fourth-order methods, respectively, in contrast to the classical 6 and 18-allowing the construction of a parameterized family of efficient schemes. These IERK methods are particularly well suited for adaptive time-stepping, as they permit significantly enlarged step sizes in actual computations.