Numerical algorithm for the space-time fractional Fokker-Planck system with two internal states
Daxin Nie, Jing Sun, Weihua Deng
TL;DR
This work addresses the numerical solution of a space-time fractional Fokker-Planck system with two internal states, where fractional Laplacians $(-\Delta)^{s_i}$ couple the states through a Markov-transition matrix $\mathbf{M}$. The authors develop a robust numerical framework combining finite element discretization for the spatial fractional operators with an $L_1$ time-stepping scheme for the Riemann-Liouville derivatives ${}_0D^{1-\alpha_i}_t$, complemented by a thorough regularity and error analysis. They establish a priori estimates for nonsmooth and smooth initial data, derive semidiscrete and fully discrete error bounds without assuming high regularity, and confirm these findings with extensive numerical experiments. The results yield provably convergent schemes for space-time fractional diffusion with internal-state coupling, applicable to models of anomalous diffusion with power-law jump distributions.
Abstract
The fractional Fokker-Planck system with multiple internal states is derived in [Xu and Deng, Math. Model. Nat. Phenom., $\mathbf{13}$, 10 (2018)], where the space derivative is Laplace operator. If the jump length distribution of the particles is power law instead of Gaussian, the space derivative should be replaced with fractional Laplacian. This paper focuses on solving the two state Fokker-Planck system with fractional Laplacian. We first provide a priori estimate for this system under different regularity assumptions on the initial data. Then we use $L_1$ scheme to discretize the time fractional derivatives and finite element method to approximate the fractional Laplacian operators. Furthermore, we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions. Finally, the effectiveness of the designed scheme is verified by numerical experiments.