Contents

23. Initial Value Problems for Ordinary Differential Equations, Part 3: Global Error Bounds for One Step Methods

Updated on March 31 (with some references added on April 14)

References:

  • Section 6.2.1 Local and global truncation error of Sauer

  • Section 5.2 Euler’s Method of Burden&Faires, the subsection Error Bounds for Euler’s Method .

  • Sections 7.1 and 7.2 of Chenney&Kincaid

All the methods seen so far for solving ODE IVP’s are one-step methods: they fit the general form

\[ U_{i+1} = F(t_i, U_i, h) \]

For example, Euler’s Method has

\[ F(t, U, h) = U + h f(t, U), \]

the Explicit Midpoint Method (Modified Euler) has

\[ F(t, U, h) = U + h f(t+h/2, U + hf(t, U)/2) \]

and even the Runge-Kutta method has a similar form, but it is long and ugly.

For these, there is a general result that gives a bound on the globl truncation error (“GTE”) once one has a suitable bound on the local truncation error (“LTE”). This is very useful, because bounds on the LTE are usually far easier to derive.

Theorem

IF, when solving the ODE IVP

\[ \frac{d u}{d t} = f(t, u),\quad u(a) = u_0 \]

on interval \(t \in [a, b]\) by a one step method

one has a bound on the local truncation error

\[ |e_i| = |U_{i+1} - u(t_i+h; t_i, U_i) = |F(t_i, U_i, h) - u(t_i + h; t_i, U_i)| \leq Ch^{p+1} = O(h^{p+1}) \]

and the ODE itself satisfies the Lipschitz Condition that for some constant \(K\),

\[ \left| \frac{\partial F}{\partial u}(t, u) \right| \leq K \]

THEN there is a bound on the global truncation error:

\[ | E_i | = |U_i - u(t_i; a, u_0)| \leq C \frac{e^{K (t_i - a)} - 1}{k} h^p, = O(h^p) \]

So yet again, there is a loss of one factor of \(h\) in going from local to global error, as first seen with the composite rules for definite integrals.

We saw a glimpse of this for Euler’s method, in the section Solving Initial Value Problems for Ordinary Differential Equations, Part 1, where the Taylor’s Theorem error formula canbe used to get the LTE bound

\[ |e_i| \leq C h^2 \text{ where } C = \frac{|u_0 e^{K(b - a)}|}{2} \]

and this leads to to GTE bound

\[ | E_i | \leq \frac{|u_0 e^{K(b - a)}|}{2} \frac{e^{K (t_i - a)} - 1}{k} h. \]

The essence of why it is true is that the Lipshitz condition limits the growth rate of solutions to be no faster than for \(du/dt = KU\), and then the “compounding of errors” is no fast than for that equation, so the argument in Solving Initial Value Problems for Ordinary Differential Equations, Part 1 for getting from a bound on the local trunctation error to a bound on the global truncation error works again with just sight modification.

23.1. Order of accuracy for the basic Runge-Kutta type mehods

  • For Euler’s method, it was stated in section Solving Initial Value Problems for Ordinary Differential Equations, Part 1, (and verified for the test case of \(du/dt = ku\)) that the local truncation error is second order, \(O(h^2)\), and (thus) the global truncation error isfirst order: \(O(h)\):

  • The Explicit (and Implicit) Trapezoid and Midpoint methods, have local truncation error \(O(h^3)\) and so their global truncation error is \(O(h^2)\) — they are second order accurate, just as for the corresponding approximate integration rules.

  • The classical Runge-Kutta method, has local truncation error \(O(h^4)\) and so its global truncation error is \(O(h^4)\) — just as for the composie Simpson’s Rule.

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22. Initial Value Problems for ODEs, Part 2: Runge-Kutta Methods 24. Initial Value Problems for Ordinary Differential Equations, Part 4: Systems of ODEs and Higher Order ODEs