  This paper provides a finite difference discretization for the backward
Feynman-Kac equation, governing the distribution of functionals of the path for
a particle undergoing both reaction and diffusion [Hou and Deng, J. Phys. A:
Math. Theor., {\bf51}, 155001 (2018)]. Numerically solving the equation with
the time tempered fractional substantial derivative and tempered fractional
Laplacian consists in discretizing these two non-local operators. Here, using
convolution quadrature, we provide a first-order and second-order schemes for
discretizing the time tempered fractional substantial derivative, which doesn't
require the assumption of the regularity of the solution in time; we use the
finite difference method to approximate the two-dimensional tempered fractional
Laplacian, and the accuracy of the scheme depends on the regularity of the
solution on $\bar{\Omega}$ rather than the whole space. Lastly, we verify the
predicted convergence orders and the effectiveness of the presented schemes by
numerical examples.
