The surface quasi-geostrophic (SQG) equation is useful in modeling atmospheric phenomena such as the frontogenesis i.e., the formation of strong fronts between masses of hot and cold air. 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.