Compact, implicit difference schemes for a differential equation's side conditions

Author:
Blair Swartz

Journal:
Math. Comp. **35** (1980), 733-746

MSC:
Primary 65L10; Secondary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1980-0572851-X

MathSciNet review:
572851

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Abstract: Lynch and Rice have recently derived compact, implicit (averaged-operator) difference schemes for the approximate solution of an *m*th order linear ordinary differential equation under *m* separated side conditions. We construct here a simpler form for a compact, implicit difference scheme which approximates a more general side condition. We relax the order of polynomial exactness required for such approximate side conditions. We prove appropriate convergence rates of the approximate solution (and its first divided differences) to (those of) the solution, even, of multi-interval differential equations. Appropriate, here, means *k*th order convergence for schemes whose interior equations are exact for polynomials of order and whose approximation of a side condition of order *l* is exact for polynomials of order . We also prove the feasibility of shooting (and of multiple shooting) based on initial divided differences. The simplicity of the proofs is based upon the simplicity of form of the approximating side conditions, together with the crucial stability result of Lynch and Rice for their interior difference equations under divided-difference initial data.

**[H]**Herbert B. Keller,*Numerical solution of two point boundary value problems*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. Regional Conference Series in Applied Mathematics, No. 24. MR**0433897****[H]**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**[R]**E. LYNCH & J. R. RICE [1975],*The HODIE Method*:*A Brief Introduction with Summary of Computational Properties*, Dept. of Comput. Sci. Report #170, Purdue Univ., West Lafayette, Ind.**[R]**Robert E. Lynch and John R. Rice,*High accuracy finite difference approximation to solutions of elliptic partial differential equations*, Proc. Nat. Acad. Sci. U.S.A.**75**(1978), no. 6, 2541–2544. MR**496774****[R]**E. LYNCH & J. R. RICE [1978b], "The performance of the HODIE method for solving elliptic partial differential equations," in*Recent Developments in Numerical Analysis*(C. de Boor, Ed.), Proc. of an MRC Conf., Academic Press, New York.**[R]**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**[M]**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**[B]**Blair Swartz and Burton Wendroff,*The comparative efficiency of certain finite element and finite difference methods for a hyperbolic problem*, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Springer, Berlin, 1974, pp. 153–163. Lecture Notes in Math., Vol. 363. MR**0431741****[B]**Blair Swartz and Burton Wendroff,*The relative efficiency of finite difference and finite element methods. I. Hyperbolic problems and splines*, SIAM J. Numer. Anal.**11**(1974), 979–993. MR**0362952**, https://doi.org/10.1137/0711076**[B]**Carl de Boor (ed.),*Mathematical aspects of finite elements in partial differential equations*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Publication No. 33 of the Mathematics Research Center, The University of Wisconsin-Madison. MR**0349031****[E]**Eusebius J. Doedel,*Some stability theorems for finite difference collocation methods on nonuniform meshes*, BIT**20**(1980), no. 1, 58–66. MR**569977**, https://doi.org/10.1007/BF01933586**[H]**H. B. Keller and V. Pereyra,*Difference methods and deferred corrections for ordinary boundary value problems*, SIAM J. Numer. Anal.**16**(1979), no. 2, 241–259. MR**526487**, https://doi.org/10.1137/0716018**[M]**M. R. Osborne,*Collocation, difference equations, and stitched function representations*, Topics in numerical analysis, II (Proc. Roy. Irish Acad. Conf., Univ. College, Dublin, 1974) Academic Press, London, 1975, pp. 121–132. MR**0411183****[R]**Robert S. Stepleman,*Tridiagonal fourth order approximations to general two-point nonlinear boundary value problems with mixed boundary conditions*, Math. Comp.**30**(1976), no. 133, 92–103. MR**0408259**, https://doi.org/10.1090/S0025-5718-1976-0408259-6

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DOI:
https://doi.org/10.1090/S0025-5718-1980-0572851-X

Article copyright:
© Copyright 1980
American Mathematical Society