High-order linearly implicit energy-stable schemes for two-dimensional Navier–Stokes equations in vorticity-velocity formulation

Abstract

In this work, we propose a novel class of high-order, linearly implicit, and energy-stable Runge-Kutta methods for the two-dimensional Navier-Stokes equations in vorticity-velocity formulation, based on the scalar auxiliary variable approach and the Fourier-Galerkin method. Compared to traditional Runge-Kutta methods, the proposed schemes achieve high-order accuracy with a linearly implicit structure at each stage, ensuring that the numerical solutions adhere to the dissipation laws associated with modified enstrophy. To address the nonlinear terms explicitly, the same-stage integrating factor Runge-Kutta methods are employed in the prediction step. We also implement a 2/3 de-aliasing technique for the nonlinear terms to reduce the aliasing error. Furthermore, the integration terms in our numerical scheme are exact due to the Fourier-Galerkin method. Numerical experiments are presented to verify the stability and efficiency of the proposed scheme.