Two Three-Level Difference Schemes For Two-Dimensional Diffusion Equations

Many problems in science and engineering can be described by partial differential equations.Solving partial differential equations can discover the development and change laws of problems.However,due to the complexity of practical problems,it is difficult or even impossible to find the exact solution of the equation.Therefore,the research on numerical solution of partial differential equations has important application value and practical significance.In this study,two three-layer difference schemes for solving the two-dimensional diffusion equation with constant coefficients are presented,and the stability and error accuracy of the difference schemes are analyzed by numerical examples.Then,a three-layer 15-point difference scheme is applied to solve the two-dimensional convection-diffusion equation,and the feasibility of the method is analyzed.Firstly,a three-layer difference scheme is presented for the two-dimensional diffusion equation with constant coefficients,which is solved by the method of undetermined coefficients.The scheme is discretized in time and space,respectively.The Taylor expansion idea and Fourier method are used to transform the coefficient matrix,solve the stability conditions of the difference scheme,select appropriate coefficients to determine the difference scheme,and verify the stability and numerical accuracy of the difference scheme with numerical experiments.Secondly,still aiming at the two-dimensional diffusion equation with constant coefficients,a three-layer difference scheme with fixed coefficients is proposed,which can also be regarded as a generalization of the Crank Nicolson scheme on three time layers.Discrete in time and space,transform the coefficient matrix with Taylor expansion idea and Fourier method,calculate the growth matrix,and judge that the difference scheme is unconditionally stable with Von Neumann condition.The stability and accuracy of the scheme are verified by numerical experiments.Finally,a three-layer 15-point difference scheme is applied to solve the two-dimensional convection-diffusion equation.The standard difference scheme is often ineffective for the convection-diffusion equation due to the existence of convection term.Exponential transformation is used to transform the equation to eliminate the convection term in the equation,and anti-exponential transformation is used to solve the flow diffusion equation with three-layer 15-point difference scheme.Then,the Fourier method is used to transform the coefficient matrix to obtain the growth matrix.The Von Neumann condition is used to judge that the difference scheme is unconditionally stable.The numerical example shows that the method is feasible.