Graded-mesh difference schemes for singularly perturbed two-point boundary value problems

Author:
Eugene C. Gartland

Journal:
Math. Comp. **51** (1988), 631-657

MSC:
Primary 65L10; Secondary 34E15

DOI:
https://doi.org/10.1090/S0025-5718-1988-0935072-1

MathSciNet review:
935072

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the numerical approximation by compact finite-difference schemes of differential operators of the form without turning points. The stability of combined with various auxiliary conditions is discussed, and a representation result for solutions of problems involving it is proven. This representation decomposes the solution into a smooth outer component plus a decaying exponential layer term along the lines of the Method of Multiple Scales.

The stability of compact difference analogues of is studied, and a stability result is proven which generalizes earlier work. This result encompasses, for example, discretizations of second-order problems that fail to possess a maximum principle. It allows for standard polynomial-based differences in outer regions (away from boundary layers) with uniform meshes, even though such schemes admit oscillatory solutions.

A family of finite-difference schemes based on an exponentially graded mesh and local polynomial basis functions is discussed. These schemes can be constructed to have arbitrarily high uniform order of convergence. To achieve a scheme of order , roughly *K* times as many points are distributed inside the layer as outside. The high order is achieved by using extra local evaluations of the coefficient functions and source term of the problem. A rigorous discretization error analysis of these schemes, using the established stability and representation results, is given.

Numerical results exhibiting the performance of these schemes are presented and generalizations of the results in the paper are discussed.

**[1]**U. Ascher and R. Weiss,*Collocation for singular perturbation problems. I. First order systems with constant coefficients*, SIAM J. Numer. Anal.**20**(1983), no. 3, 537–557. MR**701095**, https://doi.org/10.1137/0720035**[2]**U. Ascher and R. Weiss,*Collocation for singular perturbation problems. II. Linear first order systems without turning points*, Math. Comp.**43**(1984), no. 167, 157–187. MR**744929**, https://doi.org/10.1090/S0025-5718-1984-0744929-2**[3]**U. Ascher and R. Weiss,*Collocation for singular perturbation problems. III. Nonlinear problems without turning points*, SIAM J. Sci. Statist. Comput.**5**(1984), no. 4, 811–829. MR**765208**, https://doi.org/10.1137/0905058**[4]**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****[5]**Eusebius J. Doedel,*The construction of finite difference approximations to ordinary differential equations*, SIAM J. Numer. Anal.**15**(1978), no. 3, 450–465. MR**0483481**, https://doi.org/10.1137/0715029**[6]**Henning Esser,*Stabilitätsungleichungen für Diskretisierungen von Randwertaufgaben gewöhnlicher Differentialgleichungen*, Numer. Math.**28**(1977), no. 1, 69–100 (German, with English summary). MR**0461926**, https://doi.org/10.1007/BF01403858**[7]**Léonid S. Frank,*Difference singular perturbations. I. A priori estimates*, J. Math. Anal. Appl.**70**(1979), no. 1, 180–235. MR**541069**, https://doi.org/10.1016/0022-247X(79)90085-4**[8]**E. C. Gartland, Jr.,*Strong Stability and a Rrepresentation Result for a Singular Perturbation Problem*, Technical Report AMS 87-1, January, 1987, Department of Mathematics, Southern Methodist University.**[9]**Eugene C. Gartland Jr.,*Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem*, Math. Comp.**48**(1987), no. 178, 551–564, S5–S9. MR**878690**, https://doi.org/10.1090/S0025-5718-1987-0878690-0**[10]**Eugene C. Gartland Jr.,*Strong stability of compact discrete boundary value problems via exact discretizations*, SIAM J. Numer. Anal.**25**(1988), no. 1, 111–123. MR**923929**, https://doi.org/10.1137/0725009**[11]**Rolf Dieter Grigorieff,*Über die Koerzitivität gewöhnlicher Differenzenoperatoren und die Konvergenz von Mehrschrittverfahren*, Numer. Math.**15**(1970), 196–218 (German, with English summary). MR**0273852**, https://doi.org/10.1007/BF02168969**[12]**H. B. Keller,*Approximation methods for nonlinear problems with application to two-point boundary value problems*, Math. Comp.**29**(1975), 464–474. MR**0371058**, https://doi.org/10.1090/S0025-5718-1975-0371058-7**[13]**Heinz-Otto Kreiss,*Difference approximations for boundary and eigenvalue problems for ordinary differential equations*, Math. Comp.**26**(1972), 605–624. MR**0373296**, https://doi.org/10.1090/S0025-5718-1972-0373296-3**[14]**Heinz-Otto Kreiss, N. K. Nichols, and David L. Brown,*Numerical methods for stiff two-point boundary value problems*, SIAM J. Numer. Anal.**23**(1986), no. 2, 325–368. MR**831622**, https://doi.org/10.1137/0723023**[15]**Robert E. Lynch and John R. Rice,*A high-order difference method for differential equations*, Math. Comp.**34**(1980), no. 150, 333–372. MR**559190**, https://doi.org/10.1090/S0025-5718-1980-0559190-8**[16]**Peter A. Markowich and Christian A. Ringhofer,*Collocation methods for boundary value problems on “long” intervals*, Math. Comp.**40**(1983), no. 161, 123–150. MR**679437**, https://doi.org/10.1090/S0025-5718-1983-0679437-X**[17]**Klaus Niederdrenk and Harry Yserentant,*Die gleichmäßige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme*, Numer. Math.**41**(1983), no. 2, 223–253 (German, with English summary). MR**703123**, https://doi.org/10.1007/BF01390214**[18]**Robert E. O’Malley Jr.,*Introduction to singular perturbations*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Applied Mathematics and Mechanics, Vol. 14. MR**0402217****[19]**M. R. Osborne,*Minimising truncation error in finite difference approximations to ordinary differential equations*, Math. Comp.**21**(1967), 133–145. MR**0223107**, https://doi.org/10.1090/S0025-5718-1967-0223107-X**[20]**A. H. Schatz and L. B. Wahlbin,*Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements*, Math. Comp.**33**(1979), no. 146, 465–492. MR**0502067**, https://doi.org/10.1090/S0025-5718-1979-0502067-6**[21]**Donald R. Smith,*Singular-perturbation theory*, Cambridge University Press, Cambridge, 1985. An introduction with applications. MR**812466****[22]**Blair Swartz,*Compact, implicit difference schemes for a differential equation’s side conditions*, Math. Comp.**35**(1980), no. 151, 733–746. MR**572851**, https://doi.org/10.1090/S0025-5718-1980-0572851-X**[23]**Hugh L. Turrittin,*Asymptotic Solutions of Certain Ordinary Differential Equations Associated with Multiple Roots of the Characteristic Equation*, Amer. J. Math.**58**(1936), no. 2, 364–376. MR**1507160**, https://doi.org/10.2307/2371046**[24]**Relja Vulanović,*On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh*, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.**13**(1983), 187–201 (English, with Serbo-Croatian summary). MR**786443**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L10,
34E15

Retrieve articles in all journals with MSC: 65L10, 34E15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0935072-1

Keywords:
Singular perturbations,
boundary value problems,
finite differences,
stability,
uniform methods,
graded meshes

Article copyright:
© Copyright 1988
American Mathematical Society