Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data

In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by using the weighted and shifted Grünwald difference (WSGD) formula. Then, a second-order WSGD scheme is obtained after making some initial corrections. Moreover, the error estimates of the proposed time-stepping scheme are rigorously established without the regularity requirement on the exact solution. Finally, some numerical experiments are performed to validate the efficiency and accuracy of the proposed numerical scheme.