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.

在本文中，我们提出并分析了具有非光滑初始数据的非齐次向后分数式 Feynman-Kac 方程的二阶时间步进数值格式。方程中复杂的参数和时空耦合的黎曼-刘维尔分数阶积分和导数给数值分析和计算带来了挑战。非局部算子通过使用加权和移位 Grünwald 差值 (WSGD) 公式进行近似。然后，经过一些初始修正后得到二阶WSGD方案。此外，所提出的时间步长方案的误差估计是严格建立的，对精确解没有规律性要求。最后，进行了一些数值实验来验证所提出的数值方案的效率和准确性。