Application Studies On Finite Difference Method And Galerkin Finite Element Method For Burgers Equation

Burgers equation is a nonlinear partial differential equation model of studying shock wave propagation and reflection process.As one of the basic partial differential equations in mathematical physics,Burgers equation has been widely applied in many fields such as fluid mechanics,nonlinear acoustics,and gas dynamics.In this thesis,the numerical scheme construction and numerical simulation of Burgers equation based on finite difference method and finite element method are studied.Firstly,the historical development of Burgers equation and the basic mathematical and physical properties of Burgers equation are reviewed,especially the discontinuity and shock wave properties generated by the evolution of Burgers equation.Then,the basic knowledge of vertical linear algebra commonly used in the construction of numerical scheme and numerical simulation is given,including the basic properties of matrix-vector operations.To construct a fast and efficient finite difference scheme for the Burgers equation,we introduce the Hopf-Cole transformation for the one-dimensional Burgers equation.It has achieved the equivalent transformation between the Burgers equation with Dirichlet boundary condition and one-dimensional classic diffusion equation with Neumann boundary condition,thus the difficulty in constructing numerical algorithms caused by nonlinear term in Burgers equation is directly overcome.Specifically,the explicit Euler approach and Crank-Nicolson finite difference scheme for solving Burgers equation are both discussed.Numerical results show that both methods are well-posed under appropriate step size restrictions.Numerical simulations verify the correctness of the theoretical analysis and the validity of the numerical scheme.The Galerkin finite element method of Burgers equation is also discussed in this thesis.The fully discretization scheme based on Galerkin finite element and finite difference method for solving Burgers equation is constructed,where the time-derivative is discretized on uniform mesh,and the spatial-derivative is approximated by B-spline functions.The specific expression of the coefficient matrix of the corresponding linearized equation system is obtained through the selecting of the B-spline function.Similar to the general steps of Galerkin finite element method to solve partial differential equations,the concrete steps of Galerkin finite element method based on Bspline function to solve Burgers equation are given.Finally,we also review the research and applications of the highdimensional Burgers equation in this thesis,including the progresses of Burgers equation in the fields of aerodynamics,traffic flow modeling,shock wave theory viscoelastic flow and turbulence cosmology model and other fields of physics and engineering.The mathematical forms and physical properties of Burgers equation are brief recalled.