Local discontinuous Galerkin methods for the time tempered fractional diffusion equation
It provides rigorous numerical analysis for a class of fractional PDEs with tempered derivatives, which is incremental as it extends existing LDG techniques to a specific fractional operator.
This paper develops and analyzes local discontinuous Galerkin methods for a time-tempered fractional diffusion equation, proving unconditional stability and optimal convergence rates of O(h^{k+1}) for the semi-discrete scheme, with numerical experiments confirming the theory.
In this article, we consider discrete schemes for a fractional diffusion equation involving a tempered fractional derivative in time. We present a semi-discrete scheme by using the local discontinuous Galerkin (LDG) discretization in the spatial variables. We prove that the semi-discrete scheme is unconditionally stable in $L^2$ norm and convergence with optimal convergence rate $\mathcal{O}(h^{k+1})$. We develop a class of fully discrete LDG schemes by combining the weighted and shifted Lubich difference operators with respect to the time variable, and establish the error estimates. Finally, numerical experiments are presented to verify the theoretical results.