Mathematics
The Euler Method and Numerical Solution of Initial Value Problems
Quick fact
The Euler method, introduced by Leonhard Euler in 1768, is one of the oldest numerical algorithms for solving differential equations, yet it remains the basis for many modern simulation techniques. Its simplicity comes at a cost: the error per step shrinks only linearly with the step size, so achieving high accuracy requires very small steps and many iterations.
Why this is interesting
Ever wondered how your GPS predicts your future position, or how a simulation of a virus spread works? Behind many such predictions lies a math problem that can't be solved with a pencil and paper — but a simple 'guess and check' algorithm called the Euler method steps in to save the day.
Read the full explanation
Understanding The Euler Method and Numerical Solution of Initial Value Problems
Imagine you're hiking along a path, but you only know the steepness of the hill at each point. Without a map, you could estimate your path by taking small steps: at your current position, you check the steepness (the slope), walk a short distance in that direction, then re-check the slope, and so on. This is exactly the idea behind the Euler method for solving differential equations. An initial value problem gives you a 'starting point' and a rule for the slope at any point, often written as dy/dt = f(t, y) with y(t₀) = y₀. The Euler method starts at (t₀, y₀), uses the slope f(t₀, y₀) to take a step of size h along a straight line, and arrives at a new point (t₁, y₁). Repeating this process traces out an approximate solution. The smaller the step size h, the closer the polygonal path follows the true curve, but more steps mean more computation.
A deeper explanation
The Euler method is a direct consequence of Taylor's theorem, which says that a smooth function can be approximated by a line over a small interval. In the initial value problem, the true solution y(t) is unknown, but its derivative is given by f(t, y). At each step, we use the current slope to extrapolate linearly: y{n+1} = yn + h f(tn, yn). This is the most basic first-order method. Its local truncation error — the error introduced in a single step — is proportional to h², because it neglects the second derivative term in the Taylor expansion. However, over many steps, the errors accumulate, leading to a global error proportional to h (first-order accuracy). This means that to reduce the error by half, you must halve the step size, which doubles the number of steps. In practice, the method can be unstable if h is too large, especially for stiff equations, leading to wildly inaccurate oscillations. Despite these limitations, the Euler method is the foundation upon which more sophisticated methods like Runge-Kutta and implicit methods are built, and it illustrates the fundamental trade-offs in numerical computing: accuracy, stability, and efficiency.