Analysis of some difference approximations for a singular perturbation problem without turning points

Authors:
R. Bruce Kellogg and Alice Tsan

Journal:
Math. Comp. **32** (1978), 1025-1039

MSC:
Primary 65L10

DOI:
https://doi.org/10.1090/S0025-5718-1978-0483484-9

MathSciNet review:
0483484

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Abstract: Some three point difference schemes are considered for a singular perturbation problem without turning points. Bounds for the discretization error are obtained which are uniformly valid for all *h* and . The degeneration of the order of the schemes at is considered.

**[1]**Fred Dorr,*The numerical solution of singular perturbations of boundary value problems*, SIAM J. Numer. Anal.**7**(1970), 281–313. MR**0267781**, https://doi.org/10.1137/0707021**[2]**A. M. Il′in,*A difference scheme for a differential equation with a small parameter multiplying the highest derivative*, Mat. Zametki**6**(1969), 237–248 (Russian). MR**0260195****[3]**K. E. Barrett,*The numerical solution of singular-perturbation boundary-value problems*, Quart. J. Mech. Appl. Math.**27**(1974), 57–68. MR**0343628**, https://doi.org/10.1093/qjmam/27.1.57**[4]**L. R. Abrahamsson, H. B. Keller, and H. O. Kreiss,*Difference approximations for singular perturbations of systems of ordinary differential equations*, Numer. Math.**22**(1974), 367–391. MR**0388784**, https://doi.org/10.1007/BF01436920**[5]**Joseph E. Flaherty and R. E. O’Malley Jr.,*The numerical solution of boundary value problems for stiff differential equations*, Math. Comput.**31**(1977), no. 137, 66–93. MR**0657396**, https://doi.org/10.1090/S0025-5718-1977-0657396-0**[6]**Murray H. Protter and Hans F. Weinberger,*Maximum principles in differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR**0219861****[7]**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502****[8]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[9]**V. A. GUSHCHIN &. V. V. SHCHENNIKOV, "A monotonic difference scheme of secondorder accuracy,"*U.S.S.R. Computational Math. and Math. Phys.*, v. 14, 1974, pp. 252-256.

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DOI:
https://doi.org/10.1090/S0025-5718-1978-0483484-9

Article copyright:
© Copyright 1978
American Mathematical Society