Long-time stability of implicit-explicit Runge-Kutta methods for two-dimensional incompressible flows
This work provides the first theoretical guarantee of long-time stability for high-order IERK methods in fluid dynamics, addressing a critical gap for practitioners needing reliable long-time simulations.
The paper develops a family of high-order (up to fourth-order) implicit-explicit Runge-Kutta methods for the 2D incompressible Navier-Stokes equations and proves their long-time stability in L^2 and H^1 norms for the first time. The methods require only 5 and 11 order conditions for third- and fourth-order accuracy, respectively, enabling efficient adaptive time-stepping with enlarged step sizes.
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 Hölder inequality with a damping-type multistage Grönwall 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 $H^δ$ norm for some $δ>0$. 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.