A variable order finite difference method for nonlinear multipoint boundary value problems

Lentini, M.; Pereyra, V.

doi:10.1090/S0025-5718-1974-0386281-4

A variable order finite difference method for nonlinear multipoint boundary value problems
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by M. Lentini and V. Pereyra PDF

Math. Comp. 28 (1974), 981-1003 Request permission

Abstract:

An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied earlier by H. B. Keller. Variable order is provided through deferred corrections, while a built-in natural asymptotic estimator is used to automatically refine the mesh in order to achieve a required tolerance. Extensive numerical experimentation and a FORTRAN program are included.

References

A. C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation, J. Comput. Phys. 6 (1970), 378–391. MR 275685, DOI 10.1016/0021-9991(70)90037-9
P. G. Ciarlet, M. H. Schultz, and R. S. Varga, Numerical methods of high-order accuracy for nonlinear boundary value problems. I. One dimensional problem, Numer. Math. 9 (1966/67), 394–430. MR 221761, DOI 10.1007/BF02162155
J. C. Falkenberg, A method for integration of unstable systems of ordinary differential equation subject to two-point boundary conditions, Nordisk Tidskr. Informationsbehandling (BIT) 8 (1968), 86–103. MR 239766, DOI 10.1007/bf00144993
Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
R. J. Herbold, M. H. Schultz, and R. S. Varga, The effect of quadrature errors in the numerical solution of boundary value problems by variational techniques, Aequationes Math. 3 (1969), 247–270. MR 261798, DOI 10.1007/BF01817445
James F. Holt, Numerical solution of nonlinear two-point boundary problems by finite difference methods, Comm. ACM 7 (1964), 366–373. MR 0168123, DOI 10.1145/512274.512291
J. W. Jerome and R. S. Varga, Generalizations of spline functions and applications to nonlinear boundary value and eigenvalue problems, Theory and Applications of Spline Functions (Proceedings of Seminar, Math. Research Center, Univ. of Wisconsin, Madison, Wis., 1968) Academic Press, New York, 1969, pp. 103–155. MR 0239328
Herbert B. Keller, Numerical methods for two-point boundary-value problems, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0230476
Herbert B. Keller, Accurate difference methods for linear ordinary differential systems subject to linear constraints, SIAM J. Numer. Anal. 6 (1969), 8–30. MR 253562, DOI 10.1137/0706002
Herbert B. Keller, A new difference scheme for parabolic problems, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 327–350. MR 0277129
Herbert B. Keller, Accurate difference methods for nonlinear two-point boundary value problems, SIAM J. Numer. Anal. 11 (1974), 305–320. MR 351098, DOI 10.1137/0711028
Milton Lees, Discrete methods for nonlinear two-point boundary value problems, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 59–72. MR 0202323

Correcciones diferidas para problemas de contorno en sistemas de ecuaciones diferenciales ordinarias de primer orden

J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
M. R. Osborne, On shooting methods for coundary value problems, J. Math. Anal. Appl. 27 (1969), 417–433. MR 245209, DOI 10.1016/0022-247X(69)90059-6
Victor Pereyra, Iterated deferred corrections for nonlinear operator equations, Numer. Math. 10 (1967), 316–323. MR 221760, DOI 10.1007/BF02162030
Victor Pereyra, Iterated deferred corrections for nonlinear boundary value problems, Numer. Math. 11 (1968), 111–125. MR 225498, DOI 10.1007/BF02165307
Victor Pereyra, Highly accurate numerical solution of casilinear elliptic boundary-value problems in $n$ dimensions, Math. Comp. 24 (1970), 771–783. MR 288970, DOI 10.1090/S0025-5718-1970-0288970-5

High Order Finite Difference Solution of Differential Equations

Victor Pereyra, Variable order variable step finite difference methods for nonlinear boundary value problems, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Lecture Notes in Math., Vol. 363, Springer, Berlin, 1974, pp. 118–133. MR 0458918
F. M. Perrin, H. S. Price, and R. S. Varga, On higher-order numerical methods for nonlinear two-point boundary value problems, Numer. Math. 13 (1969), 180–198. MR 255069, DOI 10.1007/BF02163236
S. M. Roberts and J. S. Shipman, The Kantorovich theorem and two-point boundary value problems, IBM J. Res. Develop. 10 (1966), 402–406. MR 202325, DOI 10.1147/rd.105.0402
S. M. Roberts, J. S. Shipman, and W. J. Ellis, A perturbation technique for nonlinear two-point boundary value problems, SIAM J. Numer. Anal. 6 (1969), 347–358. MR 255070, DOI 10.1137/0706032
Josef Stoer and Roland Bulirsch, Einführung in die Numerische Mathematik. II, Heidelberger Taschenbücher, Band 114, Springer-Verlag, Berlin-New York, 1973. Unter Berücksichtigung von Vorlesungen von F. L. Bauer. MR 0400617
J. M. Varah, On the solution of block-tridiagonal systems arising from certain finite-difference equations, Math. Comp. 26 (1972), 859–868. MR 323087, DOI 10.1090/S0025-5718-1972-0323087-4