Compact Difference Schemes For Convection-Diffusion Equation And Burger's Equation

Many practical problems in science and engineering come down to the problem of partial differential equations,since the analytical solution is difficult to find,it is of great theoretical and practical significance to study the numerical solution of different types of partial differential equations.Finite difference is one of the basic methods for solving numerical solutions of partial differential equations.The compact difference format has attracted much attention from scholars because of its advantages of fewer grid points and higher precision.In this paper,several compact difference schemes for solving these two equations are given for the convection-diffusion equation and the Burgers equation.The stability and accuracy of the scheme are analyzed by numerical examples.The paper firstly proposes a compact difference scheme for the one-dimensional linear convection diffusion equation.Which is discrete in space and time.The first derivative term is discrete by the fourth-order upwind scheme,and the second derivative term is discrete by the fourth-order center difference scheme.Based on the idea of Taylor expansion and the undetermined coefficient method,the boundary scheme matching the inner point is constructed,so that the truncation error and the truncation error accuracy of the inner point scheme are consistent.Finally analyze the stability of the scheme and verify the accuracy of the scheme.Then further give a combined compact difference scheme.Still for the one-dimensional linear convection diffusion equation,the inner point of the convection term of the equation is discrete by the fifth-order upwind scheme,and the near-boundary point is calculated by the three-point fourth-order differencescheme.The truncation error of the boundary scheme is consistent with the inner point scheme.The diffusion term uses a fourth-order center-difference discrete,and the truncation error of the boundary scheme is fourth-order.The stability of the scheme is verified and the obtained semi-discrete scheme is solved by the third-order Runge-Kutta method in the time direction.The numerical experimental results are compared with the fourth-order implicit scheme,which shows that the scheme error is small and the accuracy is high.For the Burger's equation without viscous term,a compact difference scheme is proposed.The inner point adopts a sixth-order central difference scheme,the near-boundary point adopts a fifth-order difference scheme,and the boundary point adopts a sixth-order scheme that matches the inner point.A hybrid compact difference scheme for solving Burger's equation is obtained.Numerical experiments show that the scheme is stable.