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 Gr\"{u}nwald 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.

翻译：本文针对具有非光滑初值的非齐次反向分数阶Feynman-Kac方程，提出并分析了一种二阶时间步进数值格式。方程中的复参数及时间-空间耦合的Riemann-Liouville分数阶实质积分与导数，给数值分析与计算带来了挑战。本文采用加权移位Grünwald差分(WSGD)公式对非局部算子进行逼近，并通过初始校正得到二阶WSGD格式。此外，在不要求精确解正则性的条件下，严格建立了所提时间步进格式的误差估计。最后通过数值实验验证了该数值格式的高效性与精度。