Collocation methods for boundary value problems on ``long'' intervals

Authors:
Peter A. Markowich and Christian A. Ringhofer

Journal:
Math. Comp. **40** (1983), 123-150

MSC:
Primary 65L10; Secondary 65D07

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679437-X

MathSciNet review:
679437

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an *A*-stable symmetric collocation method. Problems arise, because standard theory predicts the use of many grid points as the length of the interval increases. Using the exponential decay of the 'infinite' solution, an 'asymptotic' a priori mesh-size sequence which increases exponentially, and which therefore only employs a reasonable number of meshpoints, is developed and stability, as the length of the interval tends to infinity, is shown. We also show that the condition number of the collocation equations is asymptotically proportional to the number of meshpoints employed when using this exponentially graded mesh. Using *k*-stage collocation at Gaussian points and requiring an accuracy at the knots implies that the number of meshpoints is as .

**[1]**U. Ascher, J. Christiansen, and R. D. Russell,*A collocation solver for mixed order systems of boundary value problems*, Math. Comp.**33**(1979), no. 146, 659–679. MR**521281**, https://doi.org/10.1090/S0025-5718-1979-0521281-7**[2]**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**[3]**Owe Axelsson,*A class of 𝐴-stable methods*, Nordisk Tidskr. Informationsbehandling (BIT)**9**(1969), 185–199. MR**255059**, https://doi.org/10.1007/bf01946812**[4]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**373328**, https://doi.org/10.1137/0710052**[5]**C. de Boor & R. Weiss, ""SOLVEBLOCK"--A package for almost block diagonal linear systems,"*ACM Trans. Math. Software*, v. 61, 1980, pp. 80-87.**[6]**Frank R. de Hoog and Richard Weiss,*The numerical solution of boundary value problems with an essential singularity*, SIAM J. Numer. Anal.**16**(1979), no. 4, 637–669. MR**537278**, https://doi.org/10.1137/0716049**[7]**Frank R. de Hoog and Richard Weiss,*On the boundary value problem for systems of ordinary differential equations with a singularity of the second kind*, SIAM J. Math. Anal.**11**(1980), no. 1, 41–60. MR**556495**, https://doi.org/10.1137/0511003**[8]**F. R. de Hoog and R. Weiss,*An approximation theory for boundary value problems on infinite intervals*, Computing**24**(1980), no. 2-3, 227–239 (English, with German summary). MR**620090**, https://doi.org/10.1007/BF02281727**[9]**H. B. Keller,*Approximation methods for nonlinear problems with application to two-point boundary value problems*, Math. Comp.**29**(1975), 464–474. MR**371058**, https://doi.org/10.1090/S0025-5718-1975-0371058-7**[10]**Marianela Lentini and Herbert B. Keller,*Boundary value problems on semi-infinite intervals and their numerical solution*, SIAM J. Numer. Anal.**17**(1980), no. 4, 577–604. MR**584732**, https://doi.org/10.1137/0717049**[11]**M. Lentini and V. Pereyra,*An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers*, SIAM J. Numer. Anal.**14**(1977), no. 1, 94–111. MR**455420**, https://doi.org/10.1137/0714006**[12]**Peter A. Markowich,*Analysis of boundary value problems on infinite intervals*, SIAM J. Math. Anal.**14**(1983), no. 1, 11–37. MR**686232**, https://doi.org/10.1137/0514002**[13]**Peter A. Markowich,*A theory for the approximation of solutions of boundary value problems on infinite intervals*, SIAM J. Math. Anal.**13**(1982), no. 3, 484–513. MR**653468**, https://doi.org/10.1137/0513033**[14]**Peter A. Markowich,*Eigenvalue problems on infinite intervals*, Math. Comp.**39**(1982), no. 160, 421–441. MR**669637**, https://doi.org/10.1090/S0025-5718-1982-0669637-6**[15]**J. B. McLeod,*Von Kármán’s swirling flow problem*, Arch. Rational Mech. Anal.**33**(1969), 91–102. MR**239160**, https://doi.org/10.1007/BF00247753**[16]**C. A. Ringhofer,*On Collocation Methods for Singularly Perturbed Boundary Value Problems*, Thesis, TU Vienna, Austria, 1981.**[17]**Robert D. Russell,*Collocation for systems of boundary value problems*, Numer. Math.**23**(1974), 119–133. MR**416074**, https://doi.org/10.1007/BF01459946**[18]**R. D. Russell and J. Christiansen,*Adaptive mesh selection strategies for solving boundary value problems*, SIAM J. Numer. Anal.**15**(1978), no. 1, 59–80. MR**471336**, https://doi.org/10.1137/0715004**[19]**W. Schneider, "A similarity solution for combined forced and free convection flow over a horizontal plate,"*Internat. J. Stat. and Mass Transfer*, v. 22, 1979, pp. 1401-1406.**[20]**Richard Weiss,*The application of implicit Runge-Kutta and collection methods to boundary-value problems*, Math. Comp.**28**(1974), 449–464. MR**341881**, https://doi.org/10.1090/S0025-5718-1974-0341881-2

Retrieve articles in *Mathematics of Computation*
with MSC:
65L10,
65D07

Retrieve articles in all journals with MSC: 65L10, 65D07

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679437-X

Keywords:
Nonlinear boundary value problems,
singular points,
asymptotic properties,
difference equations,
stability of difference equations

Article copyright:
© Copyright 1983
American Mathematical Society