On high-order schemes for tempered fractional partial differential equations

TL;DR

This paper introduces high-order numerical schemes for solving tempered fractional PDEs, demonstrating their stability, convergence, and effectiveness through theoretical analysis and numerical experiments.

Contribution

It develops third-order semi-discretized schemes using tempered-WSGD operators for tempered fractional diffusion and Black-Scholes equations, with stability and convergence proofs.

Findings

Third-order accuracy in space for tempered fractional diffusion.

Stable and convergent schemes verified by numerical tests.

Effective methods for tempered Black-Scholes equation.

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.