Numerical Methods for ODEs - Introduction to Scientific Computing

Discretization¶

We consider the initial value problem:

\frac{du}{dt} = f(t, u), \quad u(t_0) = u_0

(1)

where $u$ may be a vector. We assume $f$ is Lipschitz continuous, guaranteeing a unique solution.

The solution $u(t)$ is a continuous function, but computers work with discrete data. We discretize the time interval $[t_0, t_e]$ into a lattice:

t_0 < t_1 < t_2 < \cdots < t_N = t_e

(2)

with step sizes $h_n = t_{n+1} - t_n$ . For simplicity, we often use uniform spacing $h_n = h$ .

A discretization method associates to each lattice a lattice function:

u_n \approx u(t_n), \quad n = 0, 1, \ldots, N

(3)

We store only the values $\{u_0, u_1, \ldots, u_N\}$ —a finite amount of data representing the continuous solution.

The simplest methods replace derivatives with finite differences:

\frac{du}{dt} \approx \frac{u_{n+1} - u_n}{h}

(4)

This immediately gives forward Euler: $u_{n+1} = u_n + h f(t_n, u_n)$ .

The Error Framework for ODEs¶

The error analysis for ODE solvers mirrors the structure we have seen throughout the course:

Concept	ODE Numerics
Per-step accuracy	Local truncation error $\tau_n = O(h^p)$
Global error	$E_n = u_n - u(t_n)$
Problem sensitivity	Condition number $e^{LT}$ (from Lipschitz constant $L$ )
Stability	Amplification factor $\lvert 1 + h\lambda \rvert \leq 1$

Every numerical method introduces local truncation error at each step. The central question: does this error accumulate controllably, or does it explode?

\text{Global error} \;\leq\; \underbrace{e^{LT}}_{\text{problem sensitivity}} \;\times\; \underbrace{O(h^p)}_{\text{per-step accuracy}}

(5)

The factor $e^{LT}$ depends on the problem (its Lipschitz constant $L$ and integration time $T$ ). Whether the method amplifies or damps these errors depends on the amplification factor, which must stay bounded for the method to be useful. This is the stability requirement.

For stiff problems, where $|\lambda|$ is large, explicit methods require impractically small step sizes for stability even when accuracy would permit larger steps. Implicit methods handle such problems gracefully.

Learning Outcomes¶

After completing this chapter, you should be able to:

L3.1: Derive forward Euler from the forward difference approximation and interpret it geometrically.
L3.2: Derive backward Euler from the backward finite difference and from integral approximations (right rectangle rule).
L3.3: Compute the local truncation error $\tau_n$ and distinguish it from the one-step error $h\tau_n$ .
L3.4: State the Lipschitz condition and explain its role in convergence of Euler’s method.
L3.5: Derive the condition number $e^{LT}$ of an initial value problem from Grönwall’s inequality.
L3.6: State and interpret the convergence theorem for forward Euler: the global error satisfies $|E_n| = O(h)$ when $f$ is Lipschitz and the one-step error is $O(h^2)$ .
L3.7: Determine the stability region of forward Euler and compute the step-size restriction $h \leq 2/|\lambda|$ . Explain that different ODE methods have stability regions of different sizes, and that larger stability regions allow larger step sizes for stiff problems.
L3.8: Recognize stiff problems and explain why they require methods with large stability regions or adaptive step sizes.
L3.9: Given guidance, derive higher-order methods (Runge–Kutta, Rosenbrock) and adaptive time-stepping schemes by Taylor expanding the solution, matching terms to cancel lower-order error, and explaining how this yields higher-order accuracy. (Practiced on homework—you are not expected to derive these from scratch, but should be able to carry out the derivation in a guided setting.)