A family of long-time stable implicit-explicit Runge-Kutta methods up to fourth order is constructed and proven stable in L2 and H1 for 2D incompressible Navier-Stokes via Hölder and multistage Grönwall inequalities plus induction.
Wang , An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations, Numer
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Long-time stability of implicit-explicit Runge-Kutta methods for two-dimensional incompressible flows
A family of long-time stable implicit-explicit Runge-Kutta methods up to fourth order is constructed and proven stable in L2 and H1 for 2D incompressible Navier-Stokes via Hölder and multistage Grönwall inequalities plus induction.