On the spectral vanishing viscosity method for periodic fractional conservation laws

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 quasigeostrophic (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 ...

Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods

We consider the viscous Burgers equation with a fractional nonlinear term as a model involving non-local nonlinearities in conservation laws, which, surprisingly, has an analytical solution obtained by a fractional extension of the Hopf-Cole transformation. We use this model and its inviscid limit to develop stable spectral and discontinuous Galerkin spectral element methods by employing the co...

Spectral Vanishing Viscosity Method For Nonlinear Conservation Laws

Computer Methods in Applied Mechanics and Engineering 80 (1990) 197-208 North-holland Shock Capturing by the Spectral Viscosity Method*

A main disadvantage of using spectral methods for nonlinear conservation laws lies in the formation of Gibbs phenomenon, once spontaneous shock discontinuities appear in the solution. The global nature of spectral methods then pollutes the unstable Gibbs oscillations over all the computational domain, and the lack of entropy dissipation prevents convergences in these cases. In this paper, we di...

A dynamic multiscale viscosity method for the spectral approximation of conservation laws

We consider the spectral approximation of a conservation law in the limit of small or vanishing viscosities. In this regime, the continuous solution of the problem is known to develop sharp spatial and temporal gradients referred to as shocks. Also, the standard Fourier–Galerkin solution is known to break down if the mesh parameter is larger than the shock width. In this paper we propose a new ...