Maximum-norm error analysis of compact difference schemes for the backward fractional Feynman-Kac equation

TL;DR

This paper develops and analyzes high-order compact finite difference schemes for the backward fractional Feynman-Kac equation, providing stability, convergence proofs, and numerical validation for various temporal accuracies.

Contribution

The paper introduces novel compact difference schemes with high-order accuracy for the backward fractional Feynman-Kac equation, including stability and convergence analysis.

Findings

Schemes achieve up to fourth-order spatial accuracy.

Proved stability and convergence in discrete L-infinity norm for q=1.

Numerical examples confirm the effectiveness and accuracy of the schemes.

Abstract

The fractional Feynman-Kac equations describe the distribution of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the discretized schemes for fractional substantial derivatives proposed recently, we construct compact finite difference schemes for the backward fractional Feynman-Kac equation, which has q-th (q=1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. In the case q=1, the numerical stability and convergence of the difference scheme in the discrete L-infinity norm are proved strictly, where a new inner product is defined for the theoretical analysis. Finally, numerical examples are provided to verify the effectiveness and accuracy of the algorithms.