Defect correction for two-point boundary value problems on nonequidistant meshes

Authors:
J. C. Butcher, J. R. Cash, G. Moore and R. D. Russell

Journal:
Math. Comp. **64** (1995), 629-648

MSC:
Primary 65L10; Secondary 65L12

DOI:
https://doi.org/10.1090/S0025-5718-1995-1284662-1

MathSciNet review:
1284662

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Abstract | References | Similar Articles | Additional Information

Abstract: New finite difference formulae of arbitrary order are derived for the special class of second-order two-point boundary value problems . Variable mesh spacing is possible, and the required accuracy is achieved under a very mild mesh condition. A natural defect correction framework is set up to compute the higher-order approximations.

**[1]**U. Ascher, R. Mattheij, and R. D. Russell,*Numerical solution of boundary value problems for ODEs*, Prentice-Hall, Englewood Cliffs, NJ, 1988. MR**1000177 (90h:65120)****[2]**U. Ascher, J. Christiansen, and R. D. Russell,*Collocation software for boundary value ordinary differential equations*, ACM Trans. Math. Software**7**(1981), 209-222.**[3]**J. W. Barrett, G. Moore, and K. W. Morton,*Optimal recovery in the finite element method, Part*2:*Defect correction for ordinary differential equations*, IMA J. Numer. Anal.**8**(1988), 527-540. MR**975612 (90b:65168)****[4]**K. Bohmer and H. J. Steuer (eds.),*Defect correction methods*:*Theory and applications*, Springer-Verlag, Wien, 1984. MR**782686 (85m:65003)****[5]**J. R. Cash,*Numerical integration of nonlinear two-point boundary value problems using iterated deferred correction*-I.*A survey and comparison of some one-step formulae*, Comput. Math. Appl.**12**(1986), 1029-1048. MR**862027 (87k:65096)****[6]**-,*Numerical integration of nonlinear two-point boundary value problems using iterated deferred correction*-II.*The development and analysis of highly stable deferred correction formulae*, SIAM J. Numer. Anal.**25**(1988), 862-882. MR**954789 (89g:65099)****[7]**-,*Adaptive Runge-Kutta methods for nonlinear two-point boundary value problems with mild boundary layers*, Comput. Math. Appl.**12**(1985), 605-620. MR**795498 (86k:65061)****[8]**-,*High order P-stable formulae for the numerical integration of periodic initial value problems*, Numer. Math.**37**(1981), 355-370. MR**627110 (82j:65044)****[9]**J. R. Cash and A. Singhal,*Mono-implicit Runge-Kutta formulae for the numerical integration of stiff differential equations*, IMA J. Numer. Anal.**2**(1982), 211-227. MR**668593 (83m:65053)****[10]**-,*High order methods for the numerical solution of two-point boundary value problems*, BIT**22**(1986), 184-199. MR**672130 (83m:65062)****[11]**J. R. Cash and M. H. Wright,*A deferred correction method for nonlinear two-point boundary value problems*:*implementation and numerical evaluation*, AT&T Bell Laboratories, Comput. Sci. Report No. 146, Murray Hills, NJ, 1989.**[12]**M. M. Chawla,*A sixth order tri-diagonal finite difference method for nonlinear two-point boundary value problems*, BIT**17**(1977), 128-133. MR**0471328 (57:11063)****[13]**-,*An eighth order tri-diagonal finite difference method for nonlinear two-point boundary value problems*, BIT**17**(1977), 281-285. MR**0483477 (58:3478)****[14]**J. W. Daniel and A. J. Martin,*Numerov's method with deferred corrections for two-point boundary value problems*, SIAM J. Numer. Anal.**14**(1977), 1033-1050. MR**0464599 (57:4526)****[15]**C. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko,*Low dimensional behaviour in the complex Ginzburg-Landau equation*, Nonlinearity**1**(1988), 279-309. MR**937004 (89g:35095)****[16]**R. D. Grigorieff,*On stability constants and condition number of discretization methods*, Report No. 149, Technische Universität Berlin, 1986.**[17]**P. Henrici,*Discrete variable methods in ordinary differential equations*, Wiley, New York, 1962. MR**0135729 (24:B1772)****[18]**M. S. Jolly,*Explicit construction of an inertial manifold for a reaction diffusion equation*, J. Differential Equations**78**(1989), 220-261. MR**992147 (90j:35118)****[19]**H. O. Kreiss,*Difference approximations for boundary and eigenvalue problems for ODEs*, Math. Comp.**26**(1972), 605-624. MR**0373296 (51:9496)****[20]**H. O. Kreiss, T. A. Manteuffel, B. Swartz, B. Wendroff, and A. B. White,*Supra-convergent schemes on irregular grids*, Math. Comp.**47**(1986), 537-554. MR**856701 (88b:65082)****[21]**M. Lentini and V. Pereyra,*A variable order finite difference method for nonlinear multi-point boundary value problems*, Math. Comp.**28**(1974), 981-1004. MR**0386281 (52:7139)****[22]**-,*An adaptive finite difference solver for nonlinear two-point boundary value problems with mild boundary layers*, SIAM J. Numer. Anal.**14**(1977), 91-111. MR**0455420 (56:13658)****[23]**T. A. Manteuffel and A. B. White,*The numerical solution of second-order boundary value problems on nonuniform meshes*, Math. Comp.**47**(1986), 511-535. MR**856700 (87m:65116)****[24]**G. Moore,*Defect correction from a Galerkin viewpoint*, Numer. Math.**52**(1988), 565-582. MR**945100 (89e:65084)****[25]**V. Pereyra, PASVA3:*An adaptive finite difference FORTRAN program for first order nonlinear, ordinary boundary value problems*, Codes for Boundary Value Problems in Ordinary Differential Equations (B. Childs, M. Scott, J. E. Daniel, E. Denman, and P. Nelson, eds.), Springer-Verlag, Berlin, 1979, pp. 67-88.**[26]**R. D. Skeel,*A theoretical framework for proving accuracy results for deferred corrections*, SIAM J. Numer. Anal.**19**(1982), 171-196. MR**646602 (83d:65184)****[27]**R. D. Skeel and M. Berzins,*A method for the spatial discretization of parabolic equations in one space variable*, SIAM J. Sci. Statist. Comput.**11**(1990), 1-32. MR**1032224 (91g:65202)**

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DOI:
https://doi.org/10.1090/S0025-5718-1995-1284662-1

Article copyright:
© Copyright 1995
American Mathematical Society