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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size *h* to: $\varepsilon {u_{xx}} + b(x){u_x} = f(x)$ for $0 < x < 1,b > 0$, *b* and *f* smooth, $\varepsilon$ in (0, 1], and $u(0)$ and $u(1)$ given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by $C{h^2}$ with the constant *C* independent of *h* and $\varepsilon$). This scheme was derived by El-Mistikawy and Werle by a ${C^1}$ 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.

- 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** - 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** - 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**, DOI https://doi.org/10.1090/S0025-5718-1980-0572850-8
T. M. El-Mistikawy & M. J. Werle, "Numerical method for boundary layers with blowing-The exponential box scheme," - 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** - 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** - 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**260195** - 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**483484**, DOI https://doi.org/10.1090/S0025-5718-1978-0483484-9 - 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** - John J. H. Miller,
*Sufficient conditions for the convergence, uniformly in $\varepsilon $, 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** - Murray H. Protter and Hans F. Weinberger,
*Maximum principles in differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR**0219861** - Steven A. Pruess,
*Solving linear boundary value problems by approximating the coefficients*, Math. Comp.**27**(1973), 551–561. MR**371100**, DOI https://doi.org/10.1090/S0025-5718-1973-0371100-1 - Milton E. Rose,
*Weak-element approximations to elliptic differential equations*, Numer. Math.**24**(1975), no. 3, 185–204. MR**411206**, DOI https://doi.org/10.1007/BF01436591 - Donald R. Smith,
*The multivariable method in singular perturbation analysis*, SIAM Rev.**17**(1975), 221–273. MR**361331**, DOI https://doi.org/10.1137/1017032

*AIAA J.*, v. 16, 1978, pp. 749-751.

Retrieve articles in *Mathematics of Computation*
with MSC:
65L10

Retrieve articles in all journals with MSC: 65L10

Additional Information

Article copyright:
© Copyright 1981
American Mathematical Society