(Nodal) Discontinuous Galerkin Methods For The Fractional Cattaneo Type Equation

Heat conduction,diffusion and ultrasonic propagation are ubiquitous in nature and-widely used in many fields.Therefore,as fractional Cattaneo type equations describing these pheno-mena,people have paid much attention to them.In this paper,the finite difference and(nodal)discontinuous Galerkin method is used to study the higher order numerical scheme of time fractional Cattaneo type equations.(1)Several kinds of difference schemes for the time tempered fractional Cattaneo equation are established.The time fractional derivative is approximated by linear L1 interpol-ation and quadratic L2-1σ interpolation,and the spatial derivative is approximated by second-order central difference.Two kinds of implicit difference schemes are constructed,and the numerical scheme is unconditionally stable by using Fourier analysis method.(2)The fully discrete finite element scheme of time fractional heat conduction-transfer equation is constructed.Continuous Galerkin method was used for spatial dispersion.The time derivative was discretized by backward Euler,linear L1 interpolation,second-order backward differential formula,weighted shift Grünwald-Letnikov difference and Crank-Nicolson time step method,and four kinds of full discrete schemes were obtained.The semi-discrete and full discrete finite element schemes and theoretical analysis were given.The numerical comparison of four kinds of fully discrete schemes is carried out to verify the numerical accuracy of the schemes.(3)Considering the local discontinuous Galerkin schemes of time-fractional ultrasonic propagation equation,the original model was coupled into a low-order system,and the local discontinuous Galerkin method was used for spatial dispersion to construct a semi-discrete format.The time derivative is discretized by the second-order weighted shift Grünwald-Letnikov difference operator and linear L1 interpolation formula,and two fully discrete schemes are obtained.The theoretical analysis of numerical schemes is given respectively.The error and convergence order of the two schemes are verified by numerical examples.(4)We construct the full discrete nodal discontinuous Galerkin schemes for the time fract-ional Cattaneo equation.Two semi-discrete schemes are established and analyzed theoretically by using the low order coupling of spatial derivatives,and the nodal discontinuous Galerkin method.The time derivative uses Euler,linear L1 and L2 interpolation,second-order backward differential formula,weighted shift Grünwald-Letnikov difference and Crank-Nicolson discretization is used to obtain five kinds of nodal discontinuous Galerkin schemes,and the stability and error estimates are given.