自共轭常微分方程奇异摄动问题一致收敛的差分格式
The Uniformly Convergent Difference Schemes for a Singular Perturbation Problem of a Self-Adjoint Ordinary Differential Equation
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摘要: 本文对自共轭常微分方程奇异摄动问题,构造一族带拟合因子的差分格式,用不同于[1]的方法,通过对格式截断误差的分析,给出差分格式解一致收敛于微分方程解的充分条件;由此提出几个具体的差分格式,在较弱的条件下,给出较高的一致收敛阶,并将它们应用于例子,给出数值结果.Abstract: In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of lite difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.
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[1] Doolan,E.P,J.J.H.Miller and W.H.A.Schilders,Uniform Numerical Methods for Problemswith Initial and Boundary Layers,Dublin,Boole Press(1980). [2] 林鹏程、郭雯,《奇异摄动问题数值解法》,讲义. [3] Il'in,A.M.,Difference scheme for a differential equation with a small parameter affecting highest derivative.Math.Notes,6(1969),596-692. [4] Protter,M.A.and H.F.Weinberger,Maximum Principles in Differential Equations.Prentice Hall.Englewood.Cliffs.N.J.(1967). [5] Kellogg,R.B.and A.Tsan,Analysis of some difference approximations for a singular perturbation problem without turning points,Math.Comp.32(1978),1025-1039. -
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