Algorithm: Explicit Euler Method

Given: a time step h, an initial condition (t₀, y₀), and a maximum number of iterations N.

For 0 ≤ n ≤ N:

• y_n+1 = y_n + h f(t_n, y_n)
• t_n+1 = t_n + h
• Display t_n+1 and y_n+1

Stop.

Example

Consider the differential equation (from Fortin & Pierre):

y′(t) = −y(t) + t + 1

with the initial condition y(0) = 1. Therefore, t₀ = 0 and y₀ = 1, and we choose a time step h = 0.1.

The function is:

f(t, y) = −y + t + 1

We apply the explicit Euler method to obtain successive approximations of y(0.1), y(0.2), y(0.3), … denoted y₁, y₂, y₃, …

First step

y₁ = y₀ + h f(t₀, y₀) = 1 + 0.1 f(0, 1) = 1 + 0.1(−1 + 0 + 1) = 1

Second step

y₂ = y₁ + h f(t₁, y₁) = 1 + 0.1 f(0.1, 1) = 1 + 0.1(−1 + 0.1 + 1) = 1.01

Third step

y₃ = y₂ + h f(t₂, y₂) = 1.01 + 0.1 f(0.2, 1.01) = 1.01 + 0.1(−1.01 + 0.2 + 1) = 1.029

The following table shows the first ten steps, along with the analytical solution y(t) = e^−t + t, which allows us to compare the numerical and exact values and observe the growth of the error.

Explicit Euler Method: Numerical Results

Definition

A numerical method for solving differential equations is called a one-step method if it has the form:

y_n+1 = y_n + h φ(t_n, y_n)

where φ is any function. Such an expression is called a difference equation.

A method is called a multistep method if computing y_n+1 requires the numerical solution at earlier times t_n−1, t_n−2, t_n−3, …