In this paper, we study the following time-fractional Feynman-Kac equation

As is well known, the optimal rate of convergence \(\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) \) with \(\sigma =0\) on graded meshes has been proved in [Stynes et al., SIAM J. Numer. Anal. 55, 1057–1079 (2017)] by L1 scheme. However, there are still some significant differences when \(\sigma >0\). More concretely, it shall drop down to the \(\mathcal {O}\left( \tau ^{\min \{1,~r\alpha \}}\right) \) by the implicit L1 scheme. This motivates us to design the implicit-explicit L1 scheme, which recovers a convergence rate \(\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) \) on graded meshes. Finally, numerical experiments are given to illustrate theoretical results.

在本文中，我们研究以下时间分数 Feynman-Kac 方程

众所周知，最优收敛速度\(\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right) \)与\(\sigma =0\)已在 [Stynes 等人，SIAM J. Numer. 肛门。55 , 1057–1079 (2017)] L 1 计划。但是，当\(\sigma >0\)时仍然存在一些显着差异。更具体地说，它应该通过隐式L 1 方案下降到\(\mathcal {O}\left( \tau ^{\min \{1,~r\alpha \}}\right) \) \)。这促使我们设计隐式-显式L 1 方案，该方案恢复收敛速度\(\mathcal {O}\left( \tau ^{\min \{2-\alpha ,~r\alpha \}}\right ) \)在分级网格上。最后，给出了数值实验来说明理论结果。