Local Discontinuous Galerkin Method for the Backward Feynman-Kac Equation (Advances in Applied Mathematics and Mechanics, 2025)
Anomalous diffusions are ubiquitous in nature, whose functional distributions are governed by the backward Feynman-Kac equation. In this paper, the localdiscontinuous Galerkin (LDG) method is used to solve the 2D backward Feynman-Kacequation in a rectangular domain. The spatial semi-discrete LDG scheme of the equivalent form (obtained by Laplace transform) of the original equation is established. Afterdiscussing the properties of the fractional substantial calculus, the stability and optimal convergence rates of the semi-discrete scheme are proved by choosing an appropriate generalized numerical flux. The $L1$ scheme on the graded meshes is used to dealwith the weak singularity of the solution near the initial time. Based on the theoreticalresults of a semi-discrete scheme, we investigate the stability and convergence of thefully discrete scheme, which shows the optimal convergence rates. Numerical experiments are carried out to show the efficiency and accuracy of the proposed scheme.In addition, we also verify the effect of the central numerical flux on the convergencerates and the condition number of the coefficient matrix.