An analysis of a uniformly accurate difference method for a singular perturbation problem

Authors:
Alan E. Berger, Jay M. Solomon and Melvyn Ciment

Journal:
Math. Comp. **37** (1981), 79-94

MSC:
Primary 65L10

DOI:
https://doi.org/10.1090/S0025-5718-1981-0616361-0

MathSciNet review:
616361

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Abstract: It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size *h* to: for , *b* and *f* smooth, in (0, 1], and and given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by with the constant *C* independent of *h* and ). This scheme was derived by El-Mistikawy and Werle by a patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.

**[1]**Alan E. Berger, Jay M. Solomon, and Melvyn Ciment,*Higher order accurate tridiagonal difference methods for diffusion convection equations*, Advances in computer methods for partial differential equations, III (Proc. Third IMACS Internat. Sympos., Lehigh Univ., Bethlehem, Pa., 1979), IMACS, New Brunswick, N.J., 1979, pp. 322–330. MR**603482****[2]**Alan E. Berger, Jay M. Solomon, and Melvyn Ciment,*Uniformly accurate difference methods for a singular perturbation problem*, Boundary and interior layers—computational and asymptotic methods (Proc. Conf., Trinity College, Dublin, 1980) Boole, Dún Laoghaire, 1980, pp. 14–28. MR**589348****[3]**Alan E. Berger, Jay M. Solomon, Melvyn Ciment, Stephen H. Leventhal, and Bernard C. Weinberg,*Generalized OCI schemes for boundary layer problems*, Math. Comp.**35**(1980), no. 151, 695–731. MR**572850**, https://doi.org/10.1090/S0025-5718-1980-0572850-8**[4]**T. M. El-Mistikawy & M. J. Werle, "Numerical method for boundary layers with blowing-The exponential box scheme,"*AIAA J.*, v. 16, 1978, pp. 749-751.**[5]**P. P. N. de Groen and P. W. Hemker,*Error bounds for exponentially fitted Galerkin methods applied to stiff two-point boundary value problems*, Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 217–249. MR**556520****[6]**A. F. Hegarty, J. J. H. Miller, and E. O’Riordan,*Uniform second order difference schemes for singular perturbation problems*, Boundary and interior layers—computational and asymptotic methods (Proc. Conf., Trinity College, Dublin, 1980) Boole, Dún Laoghaire, 1980, pp. 301–305. MR**589380****[7]**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****[8]**R. Bruce Kellogg and Alice Tsan,*Analysis of some difference approximations for a singular perturbation problem without turning points*, Math. Comp.**32**(1978), no. 144, 1025–1039. MR**0483484**, https://doi.org/10.1090/S0025-5718-1978-0483484-9**[9]**Jens Lorenz,*Stability and consistency analysis of difference methods for singular perturbation problems*, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 141–156. MR**605505****[10]**John J. H. Miller,*Sufficient conditions for the convergence, uniformly in 𝜖, of a three-point difference scheme for a singular perturbation problem*, Numerical treatment of differential equations in applications (Proc. Meeting, Math. Res. Center, Oberwolfach, 1977) Lecture Notes in Math., vol. 679, Springer, Berlin, 1978, pp. 85–91. MR**515572****[11]**Murray H. Protter and Hans F. Weinberger,*Maximum principles in differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR**0219861****[12]**Steven A. Pruess,*Solving linear boundary value problems by approximating the coefficients*, Math. Comp.**27**(1973), 551–561. MR**0371100**, https://doi.org/10.1090/S0025-5718-1973-0371100-1**[13]**Milton E. Rose,*Weak-element approximations to elliptic differential equations*, Numer. Math.**24**(1975), no. 3, 185–204. MR**0411206**, https://doi.org/10.1007/BF01436591**[14]**Donald R. Smith,*The multivariable method in singular perturbation analysis*, SIAM Rev.**17**(1975), 221–273. MR**0361331**, https://doi.org/10.1137/1017032

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0616361-0

Article copyright:
© Copyright 1981
American Mathematical Society