Abstract
This article is about the numerical solution of initial value problems for systems of ordinary differential equations. At first these problems were solved with a fixed method and constant step size, but nowadays the general-purpose codes vary the step size, and possibly the method, as the integration proceeds. Estimating and controlling some measure of error by variation of step size/method inspires some confidence in the numerical solution and makes possible the solution of hard problems. Common ways of doing this are explained briefly in the article
Similar content being viewed by others
References
P. Bogacki L.F. Shampine (1989) ArticleTitleA 3(2) pair of Runge–Kutta formulas Appl. Math. Letters 2 1–9 Occurrence Handle1025845
M.C. Calvo D.J. Higham J.M. Montijano L. Rández (1997) ArticleTitleStepsize selection for tolerance proportionality in explicit Runge–Kutta codes Advances in Comput. Math. 24 361–382
J. Christiansen R.D. Russell (1978) ArticleTitleAdaptive mesh selection strategies for solving boundary value problems SIAM J. Numer. Anal. 15 59–80 Occurrence Handle57 #11071
J.R. Dormand P.J. Prince (1980) ArticleTitleA family of embedded Runge–Kutta formulae J. Comput. Appl. Math. 6 19–26 Occurrence Handle81g:65098 Occurrence Handle10.1016/0771-050X(80)90013-3
J.R. Dormand P.J. Prince (1984) ArticleTitleGlobal error estimation with Runge–Kutta methods IMA J. Numer. Anal. 4 169–184 Occurrence Handle85k:65058
J.R. Dormand P.J. Prince (1985) ArticleTitleGlobal error estimation with Runge–Kutta methods II IMA J. Numer. Anal. 5 481–497 Occurrence Handle87f:65079
W.H. Enright (2000) ArticleTitleContinuous numerical methods for ODEs with defect control J. Comp. Appl. Math. 125 159–170 Occurrence Handle0982.65078 Occurrence Handle2002h:65119 Occurrence Handle10.1016/S0377-0427(00)00466-0
W.H. Enright (1989) ArticleTitleA new error–control for initial value solvers Appl. Math. Comput. 31 588–599 Occurrence Handle90b:65138 Occurrence Handle10.1016/0096-3003(89)90123-9
W.H. Enright H. Hayashi (1997) ArticleTitleA delay differential equation solver based on a continuous Runge–Kutta method with defect control Numer. Alg. 16 349–364 Occurrence Handle1617169
W.H. Enright P.H. Muir (1996) ArticleTitleRunge–Kutta software with defect control for boundary value ODEs SIAM J. Sci. Comput. 17 479–497 Occurrence Handle1374292 Occurrence Handle10.1137/S1064827593251496
E. Fehlberg (1970) ArticleTitleKlassische Runge–Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme Computing 6 61–71 Occurrence Handle0217.53001 Occurrence Handle43 #5728 Occurrence Handle10.1007/BF02241732
C.W. Gear (1971) Numerical Initial Value Problems in Ordinary Differential Equations Prentice-Hall Englewood Cliffs, NJ
K. Gustafsson (1991) ArticleTitleControl theoretic techniques for stepsize selection in explicit Runge–Kutta methods ACM Trans. Math. Softw. 17 533–554 Occurrence Handle0900.65256 Occurrence Handle1140040 Occurrence Handle10.1145/210232.210242
K. Gustafsson M. Lundh G. Söderlind (1988) ArticleTitleA PI stepsize control for the numerical solution of ordinary differential equations BIT 18 270–287
E. Hairer G. Wanner (1991) Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems Springer Berlin
D.J. Higham (1989) ArticleTitleRobust defect control with Runge–Kutta schemes SIAM J. Numer. Anal. 26 1175–1183 Occurrence Handle0682.65033 Occurrence Handle91e:65096
D.J. Higham (1991) ArticleTitleRunge–Kutta defect control using Hermite–Birkhoff interpolation SIAM J. Sci. Stat. Comput. 12 991–999 Occurrence Handle0745.65051 Occurrence Handle92g:65081 Occurrence Handle10.1137/0912053
L. Jay (1998) ArticleTitleStructure preservation for constrained dynamics with super partitioned additive Runge–Kutta methods. SIAM J Sci. Comput. 20 416–446 Occurrence Handle99g:65081
J. Kierzenka L.F. Shampine (2001) ArticleTitleA BVP solver based on residual control and the Matlab PSE ACM Trans. Math. Softw. 27 299–316 Occurrence Handle2003f:65137 Occurrence Handle10.1145/502800.502801
L.F. Shampine (1973) ArticleTitleLocal extrapolation in the solution of ordinary differential equations Math. Comp. 27 91–97 Occurrence Handle0254.65052 Occurrence Handle48 #10135
L.F. Shampine (1975) Stiffness and non-stiff differential equation solvers L. Collatz (Eds) et al. Numerische Behandlung von Differentialgleichungen, ISNM 27 Birkhauser Basel 287–301
L.F. Shampine (1980) What everyone solving differential equations numerically should know I. Gladwell D.K Sayers (Eds) Computational Techniques for Ordinary Differential Equations Academic London 1–17
L.F. Shampine (1985) ArticleTitleThe step sizes used by one-step codes for ODEs Appl. Numer. Math. 1 95–106 Occurrence Handle0552.65058 Occurrence Handle86e:65099
L.F. Shampine (1989) Tolerance proportionality in ODE codes A. Bellen (Eds) et al. Numerical Methods for Ordinary Differential Equations Springer Berlin 118–136
L.F. Shampine (1994) Numerical Solution of Ordinary Differential Equations Chapman & Hall New York
L.F. Shampine (2002) ArticleTitleVariable order Adams codes Comp. Maths. Applics. 44 749–761 Occurrence Handle1035.65076 Occurrence Handle2003f:65128
Shampine, L. F. Solving ODEs and DDEs with residual control, http://faculty.smu.edu/lshampin/residuals.pdf
L.F. Shampine L.S. Baca (1984) ArticleTitleError estimators for stiff differential equations J. Comp. Appl. Math. 11 197–207 Occurrence Handle86c:65069 Occurrence Handle10.1016/0377-0427(84)90020-7
L.F. Shampine M.K. Gordon (1975) Numerical Solution of Ordinary Differential Equations: the Initial Value Problem W. H. Freeman and Co. San Francisco
L.F. Shampine M.W. Reichelt (1997) ArticleTitleThe Matlab ODE suite SIAM J. Sci. Comput. 18 1–22 Occurrence Handle97k:65307 Occurrence Handle10.1137/S1064827594276424
L.F. Shampine H.A. Watts (1976) ArticleTitleGlobal error estimation for ordinary differential equations ACM Trans. Math. Softw. 2 172–186 Occurrence Handle54 #1621
L.F. Shampine H.A. Watts (1976) ArticleTitleAlgorithm 504, GERK: global error estimation for ordinary differential equations. ACM Trans Math. Softw. 2 200–203 Occurrence Handle54 #1621
L.F. Shampine H.A. Watts (1977) The art of writing a Runge–Kutta code, Part I J.R Rice (Eds) Mathematical Software III Academic New York 257–275
L.F. Shampine H.A. Watts (1979) ArticleTitleThe art of writing a Runge–Kutta code, II Appl. Math. Comp. 5 93–121 Occurrence Handle10.1016/0096-3003(79)90001-8
L.F. Shampine A. Witt (1995) ArticleTitleA simple step size selection algorithm for ODE codes J. Comp. Appl. Math. 58 345–354 Occurrence Handle1341766 Occurrence Handle10.1016/0377-0427(94)00007-N
L.F. Shampine A. Witt (1995) ArticleTitleControl of local error stabilizes integrations J. Comp. Appl. Math. 62 333–351 Occurrence Handle96g:65070 Occurrence Handle10.1016/0377-0427(94)00108-1
H.J. Stetter (1980) Tolerance proportionality in ODE-codes R März (Eds) Seminarberichte No. 32 Humboldt University Berlin 109–123
H.A. Watts (1984) ArticleTitleStep size control in ordinary differential equation solvers Trans. Soc. Comput. Simul. 1 15–25
J.A. Zonneveld (1964) Automatic Numerical Integration Mathematisch Centrum Amsterdam
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shampine, L.F. Error Estimation and Control for ODEs. J Sci Comput 25, 3–16 (2005). https://doi.org/10.1007/s10915-004-4629-3
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10915-004-4629-3