Numerical Zoo · Differential equation

Euler's decay trap

The differential equation decays peacefully. A bad step size does not have to.

Why numerical analysts keep this one around

Consider y' = -15y with y(0) = 1. The exact solution is exp(-15x), so the physical solution simply decays toward zero. Nothing oscillates and nothing grows.

Euler's method updates y by multiplying by 1 - 15h at every step. That one factor tells us almost everything. If h is small enough, the numerical solution decays. If h is too large, the factor has magnitude greater than one and the numerical solution grows even though the differential equation is trying to kill it.

This is a beautiful teaching example because the instability belongs entirely to the discretization. We do not need a complicated equation to make numerical stability visible.

R setup

Start with the actual object.

f <- function(x, y) {
  -15 * y
}

# Stable-ish
euler(f, x0 = 0, y0 = 1, h = 0.05, n = 20)

# Unstable
euler(f, x0 = 0, y0 = 1, h = 0.2, n = 5)

By hand first

Do enough arithmetic to see the trap.

Euler gives y[n+1] = y[n] + h(-15y[n]) = (1 - 15h)y[n].

For h = 0.05, the multiplier is 0.25. Each step shrinks the magnitude.

For h = 0.2, the multiplier is -2. The sign alternates and the magnitude doubles. The ODE is stable; our numerical scheme is not.

What to look for

Now make it earn its reputation.

Run Euler, midpoint, and RK4 on the same equation.
Keep the total time interval similar while changing h.
The exact solution is easy to add in the Workbench, which makes visual smoothness a poor excuse for skipping an error calculation.

Keep this distinction:

The computer did not misunderstand the differential equation. It followed Euler's update perfectly. That is the problem.

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