An example of illconditioning in the numerical solution of singular perturbation problems
Author:
Fred W. Dorr
Journal:
Math. Comp. 25 (1971), 271283
MSC:
Primary 65L05
MathSciNet review:
0297142
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The use of finitedifference 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 illconditioned matrix equations.
 [1]
I. Babuška, Numerical Stability in the Solution of the TriDiagonal Matrices, Report BN609, 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 BN663, 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]
Fred
Dorr, The numerical solution of singular perturbations of boundary
value problems, SIAM J. Numer. Anal. 7 (1970),
281–313. MR 0267781
(42 #2683)
 [5]
F.
W. Dorr and S.
V. Parter, Singular perturbations of nonlinear boundary value
problems with turning points, J. Math. Anal. Appl. 29
(1970), 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 LA4290MS, Los Alamos Scientific Laboratory, Los Alamos, N. M., 1969.
 [7]
G. H. Golub, Matrix Decompositions and Statistical Calculations, Report CS124, Computer Science Department, Stanford University, Stanford, Calif., 1969.
 [8]
D. Greenspan, Numerical Studies of Two Dimensional, Steady State NavierStokes Equations for Arbitrary Reynolds Number, Report 9, Computer Sciences Department, University of Wisconsin, Madison, Wis., 1967.
 [9]
W.
D. Murphy, Numerical analysis of boundarylayer
problems in ordinary differential equations, Math. Comp. 21 (1967), 583–596. MR 0225496
(37 #1089), http://dx.doi.org/10.1090/S00255718196702254969
 [10]
B. Noble, Personal Communication, University of Wisconsin, Madison, Wis., Feb. 19, 1970.
 [11]
Carl
E. Pearson, On a differential equation of boundary layer type,
J. Math. and Phys. 47 (1968), 134–154. MR 0228189
(37 #3773)
 [12]
Carl
E. Pearson, On nonlinear ordinary differential equations of
boundary layer type., J. Math. and Phys. 47 (1968),
351–358. MR 0237107
(38 #5400)
 [13]
Harvey
S. Price, Richard
S. Varga, and Joseph
E. Warren, Application of oscillation matrices to
diffusionconvection equations, J. Math. and Phys. 45
(1966), 301–311. MR 0207230
(34 #7046)
 [14]
Harvey
S. Price and Richard
S. Varga, Error bounds for semidiscrete Galerkin approximations of
parabolic problems with applications to petroleum reservoir mechanics,
Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos.
Appl. Math., Durham, N.C., 1968), SIAMAMS Proc., Vol. II, Amer. Math.
Soc., Providence, R.I., 1970, pp. 74–94. MR 0266452
(42 #1358)
 [15]
Richard
S. Varga, Matrix iterative analysis, PrenticeHall Inc.,
Englewood Cliffs, N.J., 1962. MR 0158502
(28 #1725)
 [16]
J.
H. Wilkinson, The algebraic eigenvalue problem, Clarendon
Press, Oxford, 1965. MR 0184422
(32 #1894)
 [1]
 I. Babuška, Numerical Stability in the Solution of the TriDiagonal Matrices, Report BN609, 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 BN663, 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. 281313. 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. 273293. MR 0262622 (41:7227)
 [6]
 F. W. Dorr & S. V. Parter, Extensions of Some Results on Singular Perturbation Problems With Turning Points, Report LA4290MS, Los Alamos Scientific Laboratory, Los Alamos, N. M., 1969.
 [7]
 G. H. Golub, Matrix Decompositions and Statistical Calculations, Report CS124, Computer Science Department, Stanford University, Stanford, Calif., 1969.
 [8]
 D. Greenspan, Numerical Studies of Two Dimensional, Steady State NavierStokes Equations for Arbitrary Reynolds Number, Report 9, Computer Sciences Department, University of Wisconsin, Madison, Wis., 1967.
 [9]
 W. D. Murphy, "Numerical analysis of boundarylayer problems in ordinary differential equations," Math. Comp., v. 21, 1967, pp. 583596. 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. 134154. MR 37 #3773. MR 0228189 (37:3773)
 [12]
 C. E. Pearson, "On nonlinear ordinary differential equations of boundary layer type," J. Mathematical Phys., v. 47, 1968, pp. 351358. MR 38 #5400. MR 0237107 (38:5400)
 [13]
 H. S. Price, R. S. Varga & J. E. Warren, "Application of oscillation matrices to diffusionconvection equations," J. Mathematical Phys., v. 45, 1966, pp. 301311. 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, SIAMAMS Proc., vol. 2, Amer. Math. Soc., Providence, R. I., 1970, pp. 7494. MR 0266452 (42:1358)
 [15]
 R. S. Varga, Matrix Iterative Analysis, PrenticeHall, 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)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65L05
Retrieve articles in all journals
with MSC:
65L05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197102971420
PII:
S 00255718(1971)02971420
Keywords:
Ordinary differential equations,
boundaryvalue problems,
singular perturbation problems,
finitedifference equations,
matrix equations,
illconditioning
Article copyright:
© Copyright 1971 American Mathematical Society
