Local Discontinuous Galerkin Methods For The Time(Tempered) Fractional Diffusion Equation

The fractional calculus has a wide application in many fields,and using the fractional differential equation to characterize the anomalous diffusion of particles is a typical application of fractional calculus.Because of historical dependence and non-local property of fractional calculus,solving fractional differential equations has great difficulties such as higher computat-ional complexity and more computer storage.In general,high order algorithm and fast algorithm are effective ways to overcome these difficulties.Local discontinuous Galerkin finite element method attracts people's attention because of its extreme flexibility and high accuracy properties.In this paper,combining the finite difference method and the local discontinuous Galerkin method,the high order numerical methods are presented for the time tempered fractional diffusion equation and time fractional diffusion equation,and the theoretical analysis of the corresponding numerical schemes are established.The main contents are as follows:(1)High order numerical schemes are developed to solve the time tempered fractional diffusion equation.First,we present a semi-discrete scheme by using the local discontinuous Galerkin method for the spatial derivative,the semi-discrete scheme with(k+1)order accuracy in spatial direction,where k denotes the most degree of polynomial basis functions in finite element space.Based on the q-th(q = 1,2,3,4)tempered and weighted and shifted Lubich's difference operators,we then establish a class of high order fully discrete schemes with convergent rate O(?q +hk+1),the stability and convergence of the semi-discrete scheme and fully discrete schemes are proved in L2 norm.Finally,numerical examples and numerical simulations are presented to verify the effectiveness of the numerical schemes.(2)The two kinds of numerical schemes with second order accuracy in time direction,are presented for the time fractional diffusion equation.We first develop a spatial semi-discrete scheme by using the local discontinuous Galerkin method,and give the theoretical analysis for the semi-discrete scheme.For the time direction,the second order weighted and shifted Grunwald difference operator,Crank-Nicolson difference method and the second order backward difference scheme are used to approximate fractional integral and the classical first order derivative.The convergence rate of the fully discrete scheme ? and scheme ? is O(?2 +hk+1),and the detailed theoretical analysis of the two fully discrete schemes are proved in L2 norm.Finally,a numerical example is used to verify the validity of our numerical schemes.(3)High order and fast algorithm is established for the time fractional diffusion equation.Combining the fast algorithm of the Caputo time fractional derivative and the high order local discontinuous Galerkin method,we develop a fully discrete scheme to effectively solve the time fractional diffusion equation.We prove that the fully discrete scheme is unconditionally stable in L2 norm and convergence with O(?t2-? + hk+1 + ?).