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 concept of spectral vanishing viscosity (SVV). For the global spectral method, SVV is very effective and the computational cost is O(N2), which is essentially the same as for the standard Burgers equation. We also develop a local discontinuous Galerkin (LDG) spectral element method to improve the accuracy around discontinuities, and we again stabilize the LDG method with the SVV operator. Finally, we solve numerically the inviscid fractional Burgers equation both with the spectral and the spectral element LDG methods. We study systematically the stability and convergence of both methods and determine the effectiveness of each method for different parameters.