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 dynamic, multiscale viscosity method which enables the solution of such systems with relatively coarse discretizations. The key features of this method are: (1) separate viscosities are applied to the coarse and the fine scale equations; (2) these viscosities are determined as a part of the calculation (dynamically) from a consistency condition which must be satisfied if the resulting numerical solution is optimal in a user-defined sense. In this paper we develop these conditions, and demonstrate how they may be used to determine the numerical viscosities. We apply the proposed method to the one dimensional Burgers equation and note that it yields results that compare favorably with the vanishing spectral viscosity solution.