Abstract

This paper is an exploration of numerical methods for solving initial-value problems for ordinary differential equations. We look in detail at one “simple” problem, namely, the leaky bucket, and use it to explore the notions of explicit marching methods vs. implicit marching methods, interpolation, backward error, and stiffness. While the leaky bucket example is very well known, and these topics are studied in a great many textbooks, we will here emphasize backward error in a way that might be new and that we hope will be useful for your students. Indeed, the paper is intended to be a resource for students themselves. We will also use two techniques not normally seen in a first course, a new one, namely, “optimal backward error,” and an old one, namely, “the method of modified equations.”

Keywords

  1. Torricelli's law
  2. numerical methods
  3. modified equations
  4. optimal backward error

MSC codes

  1. 65L20

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Supplementary Material


PLEASE NOTE: These supplementary files have not been peer-reviewed.


Index of Supplementary Materials

Title of paper: Variations on a Theme of Euler

Authors: Robert M. Corless and Julia E. Jankowski1

File: var_solvers.m

Type: Matlab script

Contents: Plot residuals of the ODE y'= -sqrt(y) for various ODE solvers.


File: forward_error.m

Type: Matlab script

Contents: Compute the optimal backward error for the forward Euler method.


File: forward_deltahat.m

Type: Matlab script

Contents: Compute the optimal backward error for a modified equation using the forward Euler method.


File: backward_error.m

Type: Matlab script

Contents: Compute the optimal backward error for the Euler backward method.


File: backward_deltahat.m

Type: Matlab script

Contents: Compute the optimal backward error for a modified equation using the backward Euler method.


File: modifiedequationTorricelliEuler.mw

Type: Maple Worksheet

Contents: Details of the derivation of the modified equations in section 6.

References

1.
G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, Wiley, New York, 1989.
2.
M. Blasone, F. Dell'Anno, R. D. Luca, O. Faella, O. Fiore, and A. Saggese, Discharge time of a cylindrical leaking bucket, Eur. J. Phys., 36 (2015), 035017, doi:10.1088/0143-0807/36/3/035017.
3.
J. Butcher, Numerical Methods for Ordinary Differential Equations, Wiley, New York, 1987.
4.
R. M. Corless and N. Fillion, A Graduate Introduction to Numerical Methods: From the Viewpoint of Backward Error Analysis, Springer, New York, 2013.
5.
R. M. Corless and J. Jankowski, Revisiting the discharge time of a cylindrical leaking bucket, in preparation (2016).
6.
R. M. Corless and C. Y. Kaya, Minimizing residuals in ODE integration using optimal control theory, in preparation (2016).
7.
R. M. Corless, C. Y. Kaya, and R. H. Moir, Optimal backward error and the Dahlquist test problem, in preparation (2016).
8.
W. H. Enright and W. B. Hayes, Robust and reliable defect control for Runge-Kutta methods, ACM Trans. Math. Software, 33 (2007), 1206041.
9.
D. F. Griffiths and D. J. Higham, Numerical Methods for Ordinary Differential Equations, Springer, New York, 2010.
10.
D. F. Griffiths and J. M. Sanz-Serna, On the scope of the method of modified equations, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 994--1008, doi:10.1137/0907067.
11.
E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer, New York, 2006.
12.
E. Hairer, S. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, New York, 1993, https://books.google.ca/books?id=F93u7VcSRyYC.
13.
J. H. Hubbard and B. H. West, Differential Equations: A Dynamical Systems Approach. Part I. Ordinary Differential Equations, Texts Appl. Math., Springer, New York, 1997.
14.
E. Hubert, Etude algébrique et algorithmique des singularités des équations différentielles implicites, Ph.D. thesis, Institut National Polytechnique de Grenoble, 1997.
15.
L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), pp. 1--22, doi:10.1137/S1064827594276424.
16.
G. Söderlind, L. Jay, and M. Calvo, Stiffness 1952--2012: Sixty years in search of a definition, BIT, 55 (2014), pp. 531--558.
17.
L. N. Trefethen, Numerical analysis, in The Princeton Companion to Mathmatics, T. Gowers, ed., 2010, pp. 604--615.
18.
R. F. Warming and B. J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys., 14 (1974), pp. 159--179.

Information & Authors

Information

Published In

cover image SIAM Review
SIAM Review
Pages: 775 - 792
ISSN (online): 1095-7200

History

Submitted: 24 July 2015
Accepted: 22 March 2016
Published online: 3 November 2016

Keywords

  1. Torricelli's law
  2. numerical methods
  3. modified equations
  4. optimal backward error

MSC codes

  1. 65L20

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