An example of ill-conditioning in the numerical solution of singular perturbation problems

Author:
Fred W. Dorr

Journal:
Math. Comp. **25** (1971), 271-283

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1971-0297142-0

MathSciNet review:
0297142

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Abstract | References | Similar Articles | Additional Information

Abstract: The use of finite-difference methods is considered for solving a singular perturbation problem for a linear ordinary differential equation with an interior turning point. Computational results demonstrate that such problems can lead to very ill-conditioned matrix equations.

**[1]**I. Babuška,*Numerical Stability in the Solution of the Tri-Diagonal Matrices*, Report BN-609, The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., 1969.**[2]**I. Babuška,*Numerical Stability in Problems in Linear Algebra*, Report BN-663, The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., 1970.**[3]**F. W. Dorr,*The Asymptotic Behavior and Numerical Solution of Singular Perturbation Problems with Turning Points*, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1969.**[4]**F. W. Dorr, "The numerical solution of singular perturbations of boundary value problems,"*SIAM J. Numer. Anal.*, v. 7, 1970, pp. 281-313. MR**0267781 (42:2683)****[5]**F. W. Dorr & S. V. Parter, "Singular perturbations of nonlinear boundary value problems with turning points,"*J. Math. Anal. Appl.*, v. 29, 1970, pp. 273-293. MR**0262622 (41:7227)****[6]**F. W. Dorr & S. V. Parter,*Extensions of Some Results on Singular Perturbation Problems With Turning Points*, Report LA-4290-MS, Los Alamos Scientific Laboratory, Los Alamos, N. M., 1969.**[7]**G. H. Golub,*Matrix Decompositions and Statistical Calculations*, Report CS-124, Computer Science Department, Stanford University, Stanford, Calif., 1969.**[8]**D. Greenspan,*Numerical Studies of Two Dimensional, Steady State Navier-Stokes Equations for Arbitrary Reynolds Number*, Report 9, Computer Sciences Department, University of Wisconsin, Madison, Wis., 1967.**[9]**W. D. Murphy, "Numerical analysis of boundary-layer problems in ordinary differential equations,"*Math. Comp.*, v. 21, 1967, pp. 583-596. MR**37**#1089. MR**0225496 (37:1089)****[10]**B. Noble,*Personal Communication*, University of Wisconsin, Madison, Wis., Feb. 19, 1970.**[11]**C. E. Pearson, "On a differential equation of boundary layer type,"*J. Mathematical Phys.*, v. 47, 1968, pp. 134-154. MR**37**#3773. MR**0228189 (37:3773)****[12]**C. E. Pearson, "On non-linear ordinary differential equations of boundary layer type,"*J. Mathematical Phys.*, v. 47, 1968, pp. 351-358. MR**38**#5400. MR**0237107 (38:5400)****[13]**H. S. Price, R. S. Varga & J. E. Warren, "Application of oscillation matrices to diffusion-convection equations,"*J. Mathematical Phys.*, v. 45, 1966, pp. 301-311. MR**34**#7046. MR**0207230 (34:7046)****[14]**H. S. Price & R. S. Varga,*Error Bounds for Semidiscrete Galerkin Approximations of Parabolic Problems With Applications to Petroleum Reservoir Mechanics*, SIAM-AMS Proc., vol. 2, Amer. Math. Soc., Providence, R. I., 1970, pp. 74-94. MR**0266452 (42:1358)****[15]**R. S. Varga,*Matrix Iterative Analysis*, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**28**#1725. MR**0158502 (28:1725)****[16]**J. H. Wilkinson,*The Algebraic Eigenvalue Problem*, Clarendon Press, Oxford, 1965. MR**32**#1894. MR**0184422 (32:1894)**

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DOI:
https://doi.org/10.1090/S0025-5718-1971-0297142-0

Keywords:
Ordinary differential equations,
boundary-value problems,
singular perturbation problems,
finite-difference equations,
matrix equations,
ill-conditioning

Article copyright:
© Copyright 1971
American Mathematical Society