High Order Conservative Finite Difference/Fourier Spectral Methods for Inviscid Surface Quasi-Geostrophic Flows (Communications in Computational Physics, 2023)
In this paper, we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasi-geostrophic equations, which include a spectral fractional Laplacian determining thevorticity for the transport velocity of the potential temperature. The fractional Laplacian is approximated by a Fourier-Galerkin spectral method, while the time evolutionof the potential temperature is discretized by a high order conservative finite difference scheme. Weighted essentially non-oscillatory (WENO) reconstructions are alsoconsidered for comparison. Due to a low regularity of problems involving such afractional Laplacian, especially in the critical or supercritical regime, directly applyingthe Fourier spectral method leads to a very oscillatory transport velocity associatedwith the gradient of the vorticity, e.g. around smooth extrema. Instead of using anartificial filter, we propose to reconstruct the velocity from the vorticity with centraldifference discretizations. Numerical results are performed to demonstrate the goodperformance of our proposed approach.