Numerical Method For Singularly Perturbed Parabolic Convection-diffusion Equations

Singularly perturbed parabolic convection-diffusion equations often appear in celestial mechanics,fluid mechanics,quantum mechanics and other fields.Because these equations contain small parameters,the solution of the equation has a boundary layer or an internal layer,and the analytical solution cannot be obtained.Therefore,we often use numerical methods to find their approximate solutions.This paper deals with the numerical solution and error analysis of singularly perturbed parabolic convection-diffusion equations on Bakhvalov-Shishkin grids.The research contents are as follows:In the first part,we solve the initial boundary value problem for the singularly perturbed parabolic convection-diffusion equation on a Bakhvalov-Shishkin grid.The backward Euler method is used to discrete the time variable on a uniform grid,and the midpoint upwind scheme is used to discrete the spatial variable on a Bakhvalov-Shishkin grid.We decompose the truncation error into two parts,one of which satisfies the two-point boundary value problem and the other satisfies the parabolic problem.For the two-point boundary value problem,there is already first-order convergence result.For the parabolic problem,we use the discrete maximum principle to analyze its convergence.Combining the convergence of the two parts,we get that it has order O(M-1)in time and O(N-1)in space.Finally,the theoretical results are verified by numerical examples.In the second part,we solve the initial boundary value problem for the singularly perturbed parabolic convection-diffusion equation in conservative form on the Bakhvalov-Shishkin grid.We give the properties of the solution.The time variable is discretized on the uniform grid by the backward Euler method,and the space variable is discretized on the Bakhvalov-Shishkin grid by the hybrid difference scheme.The barrier function is obtained by Taylor expansion and the properties of the solution.Its error is analyzed according to the discrete comparison principle.It is found that the convergence is O(M-1)order in time direction,and is O(N-2)order convergent in the smooth part and O(N-1)order convergent in the boundary layer part in the spatial direction.Numerical examples show that the theoretical results are correct.