| The traditional Fick's law is the basic law to describe the diffusion(heat con-duction)phenomenon.Since this law does not involve the time term,it implies the unreasonable assumption that the propagation velocity is infinite.Therefore,many works have modified the traditional Fick's law,the most famous of which is the Cattaneo model,Compte and Metzler extended the Cattaneo model to the time fractional Cattaneo equation.In this paper,we propose a class of compact finite difference methods for the generalized one-dimensional time fractional Cattaneo equation,and a class of s-patially compact ADI difference methods for the two-dimensional time fractional Cattaneo equation.The compact finite difference method has the advantages of high accuracy,high resolution and low requirement for the number of grid nodes,so it has attracted more and more attention.The compact ADI difference method is applied to high dimensional problems,which effectively reduces the storage and improves the computational efficiency.This paper is divided into four chapters.The first chapter is the introduction,which introduces the physical background and research status of the time fraction-al Cattaneo equation at home and abroad,and briefly describes the main content of the paper.In the second chapter,for one-dimensional time fractional Cattaneo equation,we construct a space fourth-order compact difference scheme,which uses L1 approximation to the time fractional derivative term.Then,the stability and convergence of the scheme are proved.Finally,a numerical example is used to ver-ify the effectiveness of the scheme.In the third chapter,a compact ADI difference method,i.e.alternating direction implicit finite difference scheme,is constructed for the two-dimensional time fractional Cattaneo equation.The stability and con-vergence of the scheme are also proved.Finally,a numerical example is given to verify the effectiveness of the scheme.The fourth chapter is the summary of the whole paper and the future work. |
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