On the spectral vanishing viscosity method for periodic fractional conservation laws

TL;DR

This paper introduces a spectral vanishing viscosity method for periodic fractional conservation laws, proving convergence, spectral accuracy, and computational efficiency, with numerical validation on fractional Burgers' equation.

Contribution

It extends spectral vanishing viscosity techniques to fractional conservation laws, including non-local operators, and proves convergence and efficiency of the method.

Findings

Method converges to entropy solutions

Retains spectral accuracy

Reduces computational cost for fractional terms

Abstract

We introduce and analyze a spectral vanishing viscosity approximation of periodic fractional conservation laws. The fractional part of these equations can be a fractional Laplacian or other non-local operators that are generators of pure jump L\'{e}vy processes. To accommodate for shock solutions, we first extend to the periodic setting the Kru\v{z}kov-Alibaud entropy formulation and prove well-posedness. Then we introduce the numerical method, which is a non-linear Fourier Galerkin method with an additional spectral viscosity term. This type of approximation was first introduced by Tadmor for pure conservation laws. We prove that this {\em non-monotone} method converges to the entropy solution of the problem, that it retains the spectral accuracy of the Fourier method, and that it diagonalizes the fractional term reducing dramatically the computational cost induced by this term. We also derive a robust $L^1$-error estimate, and provide numerical experiments for the fractional Burgers' equation.