Multi-domain Decomposition Spectral Method For Burgers Equations On The Whole Line

In 1915,Bateman^[1]put forward Burgers equation for the first time when studying fluid motion.In 1948,Burgers^[2]established turbulence theory based on its model.For many years,plenty of people have been devoted to studying this equation.Burgers equation is a nonlinear partial differential equation which simulates the propagation and reflection of shock wave.It is a kind of the basic partial differential equation in hydrodynamics.It can be regarded as a simple model equation of Navier-Stokes equation in hydrodynamics,a flood mathematical model of shallow water wave problem,a model of modern traffic flow mechanics and other model equations.Burgers equation contains not only nonlinear convective term and diffusive term,but also time independent term.This makes it has the characteristics of the first-order wave equation and the heat transfer equations,that is,the mixed characteristics of Navier-Stokes equation in hydrodynamics.The research on Burgers equation is obviously of academic value,but it is still very difficult to study it,because with the increasement of time,for a given nonlinear Burgers equation with the initial and the boundary conditions,shock wave phenomenon is often accompanied.As we know,the exact solution of nonlinear partial differential equations is often difficult to find,therefore the numerical solution of partial differential equations is naturally very important.But the difficulty is often the challenge and opportunity covered with a layer of coat.It also proves that the research on the numerical method of Burgers equation on the whole line has important theoretical significance and application value.In this paper,we study the composition generalized Laguerre-Legendre spectral method of Burgers equation on the whole line.In the second chapter,we first introduce the theory of one-dimensional Laguerre orthogonal approximation and one-dimensional Legendre orthogonal approximation,and then give some results of generalized Laguerre quasi orthogonal approximation and Legendre quasi orthogonal approximation.Finally,based on the former quasi orthogonal approximation,we establish the composition generalized Laguerre-Legendre approximation theory on the whole line.The approximation theory is the mathematical foundation for Burgers equation on the whole line in the next chapters.In the third chapter,the composition generalized Laguerre-Legendre spectral scheme of nonlinear Burgers equation on the whole line is constructed,and the convergence of the scheme is analyzed with the theory of composition generalized Laguerre-Legendre approximation.Then,the corresponding fully discrete generalized Laguerre-Legendre spectral scheme is proposed by using the Crank-Nicholson discrete scheme with step?in time direction.The numerical simulation is designed and carried out,and the experimental results verify the correctness of the theoretical analysis.Finally,in the last chapter,a brief summary of the whole paper is made and several problems which need to be further studied and solved are put forward.