Upwind-and-shifted numerical scheme for fractional convection equation

TL;DR

This paper develops a stable, high-order numerical scheme for fractional convection equations, effectively simulating Levy flights and their probability density functions.

Contribution

It introduces an upwind-and-shifted finite difference scheme with order 3−α for space fractional convection equations, improving stability and accuracy.

Findings

The scheme is unconditionally stable for α in (0,1).

Numerical examples demonstrate the scheme's effectiveness.

Simulations accurately reproduce Levy flight probability densities.

Abstract

Fundamental solution of a space fractional convection equation of order $\alpha$ is the probability density function of L\'{e}vy flights with long-tailed $\alpha$-stable jump length distribution. By studying an upwind second-order implicit finite difference scheme for the equation with $\alpha\in(0,1)$, an upwind-and-shifted scheme with order $3-\alpha$ is obtained in this paper, and the scheme is shown to be unconditionally stable for a wide range of $\alpha$. Numerical examples, including simulations on a probability density function, are presented showing the effectiveness of the numerical schemes.