The Numerical Solution of a Singularly Perturbed Problem for Quasilinear Parabolic Differential Equation

Su Yu-cheng; Shen Quan

Volume 13 Issue 6

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Article Navigation > Applied Mathematics and Mechanics > 1992 > 13(6): 479-488

Su Yu-cheng, Shen Quan. The Numerical Solution of a Singularly Perturbed Problem for Quasilinear Parabolic Differential Equation[J]. Applied Mathematics and Mechanics, 1992, 13(6): 479-488.

Citation:

Su Yu-cheng, Shen Quan. The Numerical Solution of a Singularly Perturbed Problem for Quasilinear Parabolic Differential Equation[J]. Applied Mathematics and Mechanics, 1992, 13(6): 479-488.

Citation:

Su Yu-cheng, Shen Quan. The Numerical Solution of a Singularly Perturbed Problem for Quasilinear Parabolic Differential Equation[J]. Applied Mathematics and Mechanics, 1992, 13(6): 479-488.

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The Numerical Solution of a Singularly Perturbed Problem for Quasilinear Parabolic Differential Equation

Department of Mathematics, Nanjing University, Nanjing

Received Date: 1991-07-25
Publish Date: 1992-06-15

Abstract

Abstract

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation,we construct a linear three-level finite difference scheme on a nonuniform grid.The uniform convergence in the sense of discrete L² norm is proved and numerical examples are presented.
- quasilinear parabolic difTerential equation,
- singular perturbation,
- linear three-level difference scheme,
- uniform convergence

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References(11)

References

[1]	Oleinik,O.A.and T.D.Ventcel,The first boundary problem and the Cauchy problem for quasi-linear equations of parabolic type,Mat.Sb.N.S.,41,83(1957),105-128.
[2]	Vishik,M.I.and L.A.Lyusternik,Regular degeneration and boundary layer for linear differential equations with small parameter,Usp.Mat.Nauk,12(1957),3-122.
[3]	苏煌城.《奇异摄动中的边界层校正法》.上海科技出版社(1983).
[4]	Trenogin,V.A.,On the asymptotic character of solutions of near-linear parabolic equations with a parabolic boundary layer,Usp.Mat.Nauk,16,1(1961).163-169.
[5]	Bakhvalov,N.S.,On the optimization of the methods for solving boundary value problems in the presence of a boundary layer,Zh.Vychisl.Mat.i Mat.Fiz.,9(1969),841-859.
[6]	Shishkin,G.I.,Solution of a boundary value problem for an elliptic equation with small parameter multiplying the highest derivatives,ibid,86,7(1986),1019-1031.
[7]	Vulanovic,R.,On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh,Zb.rad.Prir-mat.Fak.Univ.u Novom Sadu,Ser.Mat.,13(1983),187-201.
[8]	Lees,M.,A linear three-level difference scheme for quasilinear parabolic equation,Math.Comp.,20(1966),516-522.
[9]	戴伟忠、陈传淡.二维非线性抛物型微分方程的一个三层差分格式,计算数学,11(1)(1989)1-9.
[10]	Meck,P.C.and J.Norbury,Two-stage,two-level finite difference schemes for non-linear parabolic equations,IMA J.Numer.Anal.,2(1982),335-370.
[11]	萨马尔斯基、安德烈耶夫,《椭圆型方程差分方法》(中译本),科学出版社(1984).

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