A Finite Volume Method For Two-Dimensional Time Fractional Fokker-Planck Equations

The following two-dimensional time fractional Fokker-Planck equation is consid-ered:（?）with （?）being the Riemann-Liouville fractional derivative of order 1-α:with 0<α<1.We design a finite volume method for two－dimensional time fractional Fokker-Planck equations and analyze its stability and convergence.In the method,the spatial derivatives are discreted using mid-point difference scheme together with first order upwind difference scheme,and the time fractional derivative is discreted using the L₁ approximation.In the proof of the stability and convergence,we show that the coefficient matrix of the discrete system is an M-matrix and further the corresponding finite volume method has monotone property;on this basis,we show the stability and convergence of the method under a discrete L¹ norm.The error bound of our method is O(h + τ^2-α)and if the space mesh is taken sufficiently fine,the error bound can be improved to O(h² + τ^2-α).In the last,we carry out some numerical tests to support our theory.