Fractional spectral vanishing viscosity method: Application to the quasi-geostrophic equation

We introduce the concept of fractional spectral vanishing viscosity (fSVV) to solve conservations laws that govern the evolution of steep fronts. We apply this method to the two-dimensional surface quasi-geostrophic (SQG) equation. The classical solutions of the inviscid SQG equation can develop finite-time singularities. By applying the fSVV method, we are able to simulate these solutions with high accuracy and long-time integration with relatively low resolution. Numerical diffusion in fSVV can be tuned by the fractional order as needed. Hence, fSVV can also be applied to integer-order conservation laws that exhibit steep solutions and evolving fronts.