T-7 · Differential equations

Laws of change.

A differential equation says how a thing changes, and a solution is what it becomes. Each equation here has its answer in closed form, so every numerical method is measured against the truth. The error shows its order, a solution reaches infinity in finite time, and an oscillator shows which integrators keep energy and which do not.

y′ = y

Growth in proportion to size: the exponential, the function that is its own derivative.

exact solution	y = y₀ e^(x − x₀)
through	(x₀, y₀) = (−2, 0.5)
y(−1), exact	1.3591409142

At x₁ = −1, in 2 steps

method

y(x₁)

error

Euler

1.125

2.3e−1

Heun

1.3203125

3.9e−2

Runge–Kutta 4

1.3586731

4.7e−4

Halving the step

A method of order p cuts the error at x₁ by 2ᵖ each time the step halves: Euler 1, Heun 2, Runge–Kutta 4.

steps	error	observed order
4	1.4e−1	—
8	7.6e−2	0.86
16	4.0e−2	0.92
32	2.1e−2	0.96
64	1.0e−2	0.98
128	5.3e−3	0.99
256	2.6e−3	0.99

Euler (1)Heun (2)Runge–Kutta 4 (4)