A stable second-order splitting method for incompressible Navier–Stokes equations using the scalar auxiliary variable approach

Abstract

We propose a novel second-order fractional-step method for the numerical solution of incompressible Navier–Stokes equations. This fractional-step method consists of two splitting steps and it employs the second-order implicit backward differentiation formula for the time integration. Unlike most of the projection methods for solving incompressible Navier–Stokes equations, the proposed method is free from any numerical inconsistencies generated by the treatment of boundary conditions on the pressure solution. Two pressure-correction strategies including the scalar auxiliary variable approach are proposed to enhance the accuracy of the method. A rigorous stability analysis is also carried out in this study for the considered strategies. Numerical results are presented for three benchmark problems to validate the unconditional stability and to demonstrate the performance of the proposed fractional-step method for solving unsteady incompressible viscous flows. The obtained computational results support our theoretical expectations for an unconditionally stable second-order fractional-step method for the incompressible Navier–Stokes equations.