Correction of BDFk for fractional Feynman-Kac equation with L\'{e}vy flight

TL;DR

This paper develops correction formulas for BDF convolution quadrature to solve fractional Feynman-Kac equations with Lévy flights, achieving high-order convergence even with nonsmooth data, and provides a detailed convergence analysis supported by numerical experiments.

Contribution

It introduces the first convergence analysis and numerical verification of correction BDF$k$ schemes for space fractional evolution equations with Lévy flights.

Findings

Achieves $k$th-order convergence with nonsmooth data.

Provides detailed convergence analysis for correction BDF$k$ scheme.

Numerical experiments confirm effectiveness of the method.

Abstract

In this work, we present the correction formulas of the $k$-step BDF convolution quadrature at the starting $k-1$ steps for the fractional Feynman-Kac equation with L\'{e}vy flight. The desired $k$th-order convergence rate can be achieved with nonsmooth data. Based on the idea of [{\sc Jin, Li, and Zhou}, SIAM J. Sci. Comput., 39 (2017), A3129--A3152], we provide a detailed convergence analysis for the correction BDF$k$ scheme. The numerical experiments with spectral method are given to illustrate the effectiveness of the presented method. To the best of our knowledge, this is the first proof of the convergence analysis and numerical verified the sapce fractional evolution equation with correction BDF$k$.