论文标题
向后Feynman-kac方程的本地不连续的盖尔金方法
Local discontinuous Galerkin method for the Backward Feynman-Kac Equation
论文作者
论文摘要
异常的扩散本质上是普遍存在的，其功能分布受落后Feynman-KAC方程的控制。在本文中，局部不连续的盖尔金（LDG）方法用于在矩形域中求解2D向后Feynman-kac方程。建立了原始方程的等效形式的（通过拉普拉斯变换获得）的空间半混凝土LDG方案。在讨论了分数实质积分的属性之后，通过选择适当的通用数值通量来证明，稳定性和最佳收敛率$ O（H^{K+1}）$可以证明。分级网格的$ L1 $方案用于处理初始时间附近解决方案的弱点。基于半混凝土方案的理论结果，我们研究了完全离散方案的稳定性和收敛性，该方案显示了最佳收敛速率$ O（H^{k+1}+τ^{\ min \ min \ min \ min \ {2-α，γδ\}}}}）$。进行数值实验以显示所提出的方案的效率和准确性。此外，我们还验证了中央数值通量对系数矩阵的收敛速率和条件数的影响。
Anomalous diffusions are ubiquitous in nature, whose functional distributions are governed by the backward Feynman-Kac equation. In this paper, the local discontinuous Galerkin (LDG) method is used to solve the 2D backward Feynman-Kac equation in a rectangular domain. The spatial semi-discrete LDG scheme of the equivalent form (obtained by Laplace transform) of the original equation is established. After discussing the properties of the fractional substantial calculus, the stability and optimal convergence rates $O(h^{k+1})$ of the semi-discrete scheme are proved by choosing an appropriate generalized numerical flux. The $L1$ 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 of a semi-discrete scheme, we investigate the stability and convergence of the fully discrete scheme, which shows the optimal convergence rates $O(h^{k+1}+τ^{\min\{2-α,γδ\}})$. 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 convergence rates and the condition number of the coefficient matrix.
