The Compact Finite Difference Method For Variable Coefficient Cattaneo Equation

Traditional Fick's law and Fourier's law are widely used in the study of diffusion and heat conduction problems,but this implies the assumption that the propagation velocity is infinite,which is non-physical.Cattaneo proposes a heat conduction model characterized by finite propagation velocity.At present,some studies on Cattaneo model are mainly focused on the constant coefficient and studies have found that the diffusion coefficient or heat conduction coefficient in some important diffusion equations and heat conduction equations will change with time or space.For example,heat conduction or liquid flow processes in porous media,seismic wave propagation,etc.So the research on the variable coefficient Cattaneo equation is also of great significance.The numerical solution to it has important theoretical and practical significance,so it has been paid more and more attention and research by more and more scholars.There are many numerical methods for solving partial differential equations.Compact finite difference method is a kind of high precision finite difference method,which has the advantages of high precision,high resolution and low demand on mesh nodes.It has received the attention of many scholars.Based on the important physical background of Cattaneo equation with variable coefficients,a compact difference scheme with high precision is constructed for this kind of equation,which makes this kind of equation better applied in scientific and engineering calculation.This paper is divided into four chapters.The first chapter introduces the physical background and research status of the variable coefficient Cattaneo equation,and briefly describes the main content and structural framework of the paper.In second chapter,a compact difference scheme is proposed for Cattaneo equation with variable coefficients of integer order.The stability and convergence of the scheme are analyzed by using energy method.The validity of the scheme is verified by numerical experiments.The Cattaneo equation with constant coefficient and the Cattaneo equation with variable coefficient are compared,and the effects of relaxation parameters are tested.In the third chapter,a compact difference scheme is proposed for the fractional Cattaneo equation with variable coefficients,and the energy method is also used for theoretical analysis.The validity of the scheme is verified by comparing the real solution with the numerical solution of the specific problem.Finally,the relaxation parameters and different are tested.The fourth chapter is a summary of the whole work and the prospect of future work.