Finite Difference Methods For One-Dimensional Burgers Equation

In computational fluid dynamics,Burgers equation is an important nonlinear partial differential equation.The equation exists in both convection and diffusion states.It preserves the basic characteristics of the Navier-Stokes equation and is a simplified model for solving complex fluid dynamics problems.Since its introduction in 1915,theoretical physicists,computational mathematicians and engineers have been concerned.In this paper,based on the finite difference method,the numerical solution of Burgers equation is discussed.The theoretical results are verified by an example by using the basic theory of parabola partial differential equation and the principle of numerical analysis,combined with MATLAB programming.The full text is divided into five parts.The first chapter introduces the research background and research value of Burgers equation,and points out the main work and significance of this paper.In the second chapter,the Hopf-Cole transform is used to transform the Burgers equation into the heat conduction equation,and the Crank-Nicolson scheme is used to solve the equation.The virtual node is introduced to deal with the Robin mixed boundary condition.Simpson integral is used to improve the accuracy of initial value and make the numerical solution reach the second order precision in both time and space.In order to further improve the spatial direction accuracy,the compact difference operator is used to achieve the fourth order accuracy without increasing the node.In the third chapter,the second-order accuracy in time is obtained by Taylor expansion,and the stable discrete calculation is realized by combining the C-N scheme with the nonlinear term processing,by bypassing the classical explicit scheme of forward difference.By using the Richardson method,the fourth order precision of time and space is realized.In the fourth chapter,numerical examples are given to verify the effectiveness of the difference schemes in the second and third chapters,and the convergence order is verified,and the accuracy of the two numerical methods is analyzed and compared.In the fifth chapter,two numerical solutions of Burgers equation proposed in this paper are briefly summarized.The future research direction is established in solving the high dimensional Burgers equation and the high precision method.