On high-order schemes for tempered fractional partial differential equations

Abstract: In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Gr\"unwald difference (tempered-WSGD) operators for the tempered fractional diffusion equation. We also show stability and convergence analysis for the fully discrete scheme based a Crank--Nicolson scheme in time. A third-order scheme for the tempered Black--Scholes equation is also proposed and tested numerically. Some numerical experiments are carried out to confirm accuracy and effectiveness of these proposed methods.