Euler's Method

Most differential equations have no formula solution. Numerical methods step forward from a known starting point instead.

The method

From the current point, compute the slope, follow the tangent line for a small step h, and land on a new point. Then repeat.

The formula

yₙ₊₁ = yₙ + h × f(xₙ, yₙ). It is nothing more than a straight-line approximation applied repeatedly.

Where the error comes from

The true solution curves away from the tangent line during each step. That small gap accumulates over many steps.

Smaller steps

Halving the step roughly halves the total error, at the cost of twice as many steps. Accuracy is bought with work.

Better methods exist

Runge-Kutta methods sample the slope several times per step and are far more accurate for the same effort. Euler is the idea in its simplest form.

Step along the tangent

Euler's method starts at the initial condition and steps forward using the slope at the current point, recomputing at each new one. It is repeated linear approximation.

The step

yₙ₊₁ = yₙ + h·f(xₙ, yₙ), with h the step size. The whole method is that line applied repeatedly, which makes it the simplest possible numerical scheme.

Error accumulates

Each step introduces error and later steps build on the drift. Halving the step size roughly halves the total error, which makes Euler first-order accurate and rather slow to converge.

The error has a predictable direction

For a concave-up solution the tangent lies below the curve, so Euler underestimates consistently. Knowing the direction of the bias is often as useful as knowing its size.