A Space–Time Spectral Method for Nonlinear Fractional Convection–Diffusion Equations with Viscosity Terms

We develop a high-order space-time spectral method for nonlinear convection–diffusion equations with a Riemann–Liouville time-fractional derivative and a spectrally defined space-fractional Laplacian. The spatial discretization uses a Fourier spectral method that diagonalizes the fractional Laplacian under periodic boundary conditions. The temporal discretization employs a Petrov–Galerkin method based on generalized Jacobi functions which capture the initial singularity exactly. The nonlinear convection term is treated pseudo-spectrally, and the resulting algebraic system is solved with a damped Newton iteration. Rigorous error analysis proves exponential convergence in both space and time. Numerical experiments for various fractional orders confirm the spectral accuracy. Simulations of the fractional Burgers equation demonstrate that increasing the viscosity enhances diffusion and stabilizes the solution, while a nonlinear coefficient that significantly exceeds the viscosity leads to error growth over long time intervals. The method provides an efficient and accurate tool for simulating anomalous transport phenomena.