Scalar auxiliary variable approache for the surface quasi-geostrophic equation

Abstract

We present a numerical method for approximating the two-dimensional surface quasi-geostrophic equation. We first reformulate the equation into an equivalent system by using the scalar auxiliary variable approach. Then we propose first-order and second-order discretization schemes for solving the new surface quasi-geostrophic system in time. Furthermore, we prove that both of these schemes based on the scalar auxiliary variable approach satisfy an unconditionally energy stability property. Since the method gives two linear equations with constant coefficients in each time step. We have an efficient approach and accurate for solving these spacial fractional diffusion equations. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for inviscid problems and the problems with small viscosity.