3 Euler’s Method

Lecture 3

Published

January 28, 2026

Overview

Basics of Euler’s method,
Examples,
Error estimates

Note

I encourage you to play around with the juptyer notebook for this lecture - you can copy the code with the this notebook button on the side of this page.

Warning

These notes are mainly a record of what we discussed and are not a substitute for attending the lectures and reading books! If anything is unclear/wrong, let me know and I will update the notes.

Tip

This lecture is mostly based on Burden, Faires, and Burden (2015) section 5.2

Recall that we are considering IVPs: seek u :[0,T] \to \mathbb R such that

\begin{align} &u(0) = u_0 \nonumber\\ &u'(t) = f\big( t, u(t) \big). \tag{IVP} \end{align}

We will suppose that this equation is well-posed. The numerical schemes we introduce in this section will approximate the solution u to (\text{IVP}) on a mesh \{t_j\}_{j=0}^n \subset [0,T]. The approximation at time t_j will be written u_j so that u_j \approx u(t_j). We will consider the error on the mesh: \max_j |u(t_j) - u_j |.

3.1 Euler’s method

We now fix an equispaced mesh t_j = j\frac{T}{n}. We will sometimes call h := \frac{T}n the step size.

Replacing the derivative in (\text{IVP}) with the forward difference

\begin{align} u'(t_j) \approx \frac{u_{j+1} - u_j}{h} \end{align}

yields Euler’s method

\begin{align} u_{j+1} = u_j + h f(t, u_j) \tag{Euler} \end{align}

Note

Last time, we saw that by integrating (\text{IVP}) we obtain

\begin{align} u(t_{j+1}) = u(t_{j}) + \int_{t_j}^{t_{j+1}} f\big( s, u(s) \big) \mathrm{d}s. \end{align}

Approximating this integral with the rectangular rule and replacing u(t_j) with u_j also yields Euler’s method.

That is, we approximate the differential equation with a difference equation.

We implement Euler’s method here:

function Euler( u0, f, T, n )
    h = T/n     
    t = 0:h:T    
    u = zeros(n+1)
    u[1] = u0
    for j = 1:n
        u[j+1] = u[j] + h * f( t[j], u[j] )
    end
    return u
end

Euler (generic function with 1 method)

3.2 Examples

We now apply Euler’s method to each of the examples from Lecture 2:

Example 3.1 (Exponential growth) We consider u' = \mu u with u(0) = u_0 > 0. Recall that this problem is well-posed with unique solution u(t) = u_0 e^{\mu t}. We approximate this solution using Euler’s method:

μ = 2
f(t, u) = μ * u
u0, T, n = 1, 2, 4

u = Euler( u0, f, T, n )
scatter( 0:T/n:T, u, 
    label="Euler (n=$n)",
    title="Exponential (u' = u)")

n=100
u2 = Euler( u0, f, T, n )
scatter!( 0:T/n:T, u2, 
    label="Euler (n=$n)" )

plot!(t-> u0 * exp(μ * t),
    label="exact solution")

In the following, we plot the errors as a function of n:

Exercise 3.1 Show that Euler’s method applied to u' = u with u_0 > 0 always underestimates the function: i.e. we have u(t_j)\geq u_j for all j (equality is only achieved for j=0).

Example 3.2 (Logistic equation) Recall that the logistic equation takes the form

\begin{align} &u(0) = u_0, \\ &u'(t) = \mu u(t) - \kappa u(t)^2 \quad \text{on } (0,T). \end{align}

Last time, we saw that this problem is well-posed with solution

\begin{align} u(t) = \frac{\mu}{\kappa u_0 (e^{\mu t}-1) + \mu } u_0 e^{\mu t} \end{align}

u0, μ, κ, T, n = 1, 1., .05, 15, 5
f2(t,u) = μ * u - κ * u^2
u = Euler( u0, f2, T, n )
u_exact(t) = ( μ/( κ*u0*(exp(μ*t) - 1) + μ) ) * u0 * exp( μ * t )

plot(u_exact, 0, T, 
    title=L"u'=\mu u(t) - \kappa u(t)^2", legend=false,
    xlabel=L"t", ylabel=L"u(t)")

scatter!( 0:T/n:T, u, 
    label="Euler (n=$n)")

n=100
u2 = Euler( u0, f2, T, n )
scatter!( 0:T/n:T, u2, 
   label="Euler (n=$n)" )

Heres the error as a function of n:

Exercise 3.2 Now consider the logistic equation as above (with \mu = 1 and \kappa=0.05) but with u_0 = 100. What do expect to happen in Euler’s method when (i) n < 20, (ii) n = 20, and (iii) n > 20?

Example 3.3 Consider the following (well-posed) linear IVP

\begin{align} &u(0) = 0, \\ &u'(t) = t^2 - u \sin(t) \quad \text{on } (0,10) \end{align}

We apply Euler’s method to this equation:

# using the built-in method
f3(u,p,t) = t^2 - u * sin(t)
T = 10
u0, tspan = 0., (0., T)
ivp = ODEProblem(f3, u0, tspan)
sol = solve(ivp, Tsit5());

plot(sol,
    title=L"u'=f(t,u)", legend=false,
    xlabel=L"t",  ylabel=L"u(t)")

# Euler 
n=5
u = Euler( u0, (t,u)->f3(u,0,t), T, n )
scatter!(0:T/n:T, u)

n=50
u2 = Euler( u0, (t,u)->f3(u,0,t), T, n )
scatter!(0:T/n:T, u2)

3.3 Error estimates

We will now prove that, under some assumptions on u, Euler’s method does indeed converge with rate O(h).

Burden, Faires, and Burden (2015) chapter 5.2, some of chapter 5.3

TO-DO

Read: Burden, Faires, and Burden (2015) chapter 5.1, 5.2, 5.3 - please come to the lecture next week with any questions you have,
Assignment 1: Due today,
(Sign-up to office hours if you would like)

Burden, Richard L., Douglas J. Faires, and Annette M. Burden. 2015. Numerical Analysis. 10th ed. CENGAGE Learning.