Optimal stabilization and time step constraints for the forward Euler-Local Discontinuous Galerkin method applied to fractional diffusion equations

Highlights: • A new stabilization term proportional to $h^{1-\alpha }$, where α is the fractional derivative order, is proposed. • Optimal rates of convergence are theoretically shown for the proposed penalization. • Using a von Neumann analysis, stability conditions for the forward Euler-Local Discontinuous Galerkin method are obtained. • The CFL condition number is studied with respect to the approximation degree and a penalization parameter. • The use of high order strong stability preserving Runge-Kutta methods is illustrated on 1D and 2D problems. -- Abstract: A time dependent model problem with the Riesz or the Riemann-Liouville fractional differential operator of order $1<\alpha <2$ is considered. By penalyzing the primary variable of the minimal dissipation Local Discontinuous Galerkin (mdLDG) method with a term of order $h^{1-\alpha }$ and using a von Neumann analysis, stability conditions proportional to $h^{\alpha }$ are derived for the forward Euler method and both fractional operators in one dimensional domains. The CFL condition is numerically studied with respect to the approximation degree and the stabilization parameter. Our analysis and computations carried out using explicit high order strong stability preserving Runge-Kutta schemes reveal that the proposed penalization term is suitable for high order approximations and explicit time advancing schemes when α is close to one. A series of numerical experiments in 1D and 2D problems are presented to validate our theoretical results and those not covered by the theory.