Local Discontinuous Galerkin Method For The Backward Feynman-Kac Equation

The phenomenon of anomalous diffusion is very common in nature,and its functional distribution can be described by the backward Feynman-Kac equation.In this paper,the local discontinuous Galerkin method is used to solve the 2D backward Feynman-Kac equation in a rectangular region.The LDG spatial semi-discrete scheme of the equivalent form of the original equation which is obtained by Laplace transform is established.After proving the properties of the generalized time fractional derivative,the stability and optimal convergence ratesO(h^k+1)of the semi-discrete scheme are proved by choosing an appropriate generalized numerical flux.Next,the1 scheme on the graded meshes is used to deal with the weak singularity of the solution near the initial time.Based on the theoretical results in a semi-discrete scheme,we investigated the stability and convergence of the fully discrete scheme,which shows the optimal convergence ratesO(h^k+1+T^min{2-?,??} of the proposed method.Numerical results 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 convergence rates and the condition number of the coefficient matrix.