An automatic continuation strategy for the solution of singularly perturbed nonlinear boundary value problems

ACM Transactions on Mathematical Software

Volumn 27, Issue 2, 2001, Pages 245-266

  Cash, J R ^a   Moore, G ^a   Wright, R W ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Algorithms; Automatic continuation; G.1.7 Numerical Analysis : Ordinary Differential Equations Boundary value problems; G.4 Mathematics of Computing : Mathematical Software; Mesh selection; Nonlinear boundary value problems; Performance; Theory

Indexed keywords

EID: 0037855750     PISSN: 00983500     EISSN: None     Source Type: Journal    
DOI: 10.1145/383738.383742     Document Type: Article

References (23)

1. ASCHER, U. AND RUSSELL, R. D. 1981. Reformulation of boundary value problems into "standard" form. SIAM Rev. 23, 2, 238-254.
2. ASCHER, U., CHRISTIANSEN, J., AND RUSSELL, R. D. 1979. COLSYS - A collocation code for boundary-value problems. In Codes for Boundary Value Problems in Ordinary Differential Equations. Springer Lecture Notes in Computer Science, vol. 76. Springer-Verlag, New York, NY, 164-185.
3. ASCHER, U., CHRISTIANSEN, J., AND RUSSELL, R. 1981. Collocation software for boundary value ODEs. ACM Trans. Math. Softw. 7, 2 (June), 209-222.
4. ASCHER, U., MATTHEIJ, R., AND RUSSELL, R. D. 1988. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice-Hall, New York, NY.
5. BADER, G. AND ASCHER, U. 1987. A new basis implementation for a mixed order boundary value ODE solver. SIAM J. Sci. Stat. Comput. 8, 4 (July), 483-500.
6. BADER, G. AND KUNKEL, P. 1989. Continuation and collocation for parameter-dependent boundary value problems. SIAM J. Sci. Stat. Comput. 10, 1 (Jan.), 72-88.
7. BASHIRALI, Z., CASH, J. R., AND SILVA, H. H. M. 1998. Lobatto deferred correction for stiff two point boundary value problems. Comput. Math. Appl. 36, 56-69.
8. CASH, J. R. AND WRIGHT, M. H. 1990. Implementation issues in solving nonlinear equations for two-point boundary value problems. Computing 45, 1, 17-37.
9. CASH, J. R. AND WRIGHT, M. H. 1991. A deferred correction method for nonlinear two-point boundary value problems: Implementation and numerical evaluation. SIAM J. Sci. Stat. Comput. 12, 4 (July), 971-989.
10. CASH, J. R. AND WRIGHT, R. W. 1998. Continuous extensions of deferred correction schemes for the numerical solution of nonlinear two-point boundary value problems. Appl. Numer. Math. 28, 2-4, 227-244.
11. CASH, J. R., MOORE, G., AND WRIGHT, R. W. 1995. An automatic continuation strategy for the solution of singularly perturbed linear two-point boundary value problems. J. Comput. Phys. 122, 2 (Dec.), 266-279.
12. COLE, J. D. 1983. Perturbation Methods in Applied Mathematics. Blaisdell Publishing Co., London, UK.
13. DOEDEL, E. J. 1981. AUTO: A program for the automatic bifurcation analysis of autonomous systems. Congressus Numerantium 30, 265-284.
14. DEUFLHARD, P. 1979. A stepsize control for continuation methods and its special application to multiple shooting techniques. Numer. Math. 33, 115-146.
15. ENRIGHT, W. H. AND MUIR, P. H. 1996. Runge-Kutta software with defect control for boundary value ODEs. SIAM J. Sci. Comput. 17, 2, 479-497.
16. HEMKER, P. W. 1977. A numerical study of stiff two point boundary value problems. MC Tract 80. Math. Centrum, Amsterdam.
17. KELLER, H. B. 1977. Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory. Academic Press, Inc., New York, NY, 359-371.
18. PEARSON, C. E. 1968. On nonlinear ordinary differential equations of boundary layer type. J. Math. Phys. 47, 351-358.
19. SEYDEL, R. 1991. Tutorial on continuation. Int. J. Bifurcation Chaos 1, 1, 3-11.
20. SHISHKIN, G. I. 1988. Grid approximation of singularly perturbed parabolic equations with internal layers. Sov. J. Num. Anal. Math. Model. 3, 1, 393-407.
21. WASOW, W. 1970. The capriciousness of singular perturbations. Nieuw Arch. Wisk. 18, 190-210.
22. WRIGHT, R. W. 1995. An automatic continuation strategy for the numerical solution of stiff two-point boundary value problems. Ph.D. Dissertation. University of London, London, UK.
23. WRIGHT, R. W., CASH, J. R., AND MOORE, G. 1994. Mesh selection for stiff two-point boundary value problems. Num. Alg. 7, 205-224.