Lab 22: Runge-Kutta and Predictor-Corrector Lab

Type y′ = f(x, y), the start, h and the end. Tick the methods to compare; the table lists the steps of the first one (RK4 by default).

Two equations x′ = f(x, y), y′ = g(x, y). Use the conserved quantity to judge the method.

RK4 and the spring

Try this

Run it. Compare y(0.1) with Problem 1.
Halve h in the spring: how much closer to 1 is the RK4 energy?
Change f to y - x with y = 2 (Problem 3).

Adams–Bashforth–Moulton

Try this

Run it. Compare the value at x = 0.4 with Problem 2.
How does the error estimate compare with the true error?
Change f to -y (exact e^(-x)) and run to x = 1.

Q1 Write the fourth-order Runge–Kutta formulas.

k_1 = hf(x_n, y_n)

,

k_2 = hf(x_n + \frac h2, y_n + \frac{k_1}{2})

,

k_3 = hf(x_n + \frac h2, y_n + \frac{k_2}{2})

,

k_4 = hf(x_n + h, y_n + k_3)

,

y_{n+1} = y_n + \frac16(k_1 + 2k_2 + 2k_3 + k_4)

.

Q2 Why are Runge–Kutta methods preferred to the Taylor series method?

They reach the same order using only values of

f

, so no derivatives of

f

have to be derived and coded for each new equation.

Q3 State Milne's predictor and corrector formulas.

Predictor

y_{n+1} = y_{n-3} + \frac{4h}{3}(2f_n - f_{n-1} + 2f_{n-2})

; corrector

y_{n+1} = y_{n-1} + \frac h3(f_{n-1} + 4f_n + f_{n+1})

(Simpson's rule).

Q4 Why are predictor–corrector methods not self-starting?

Their formulas use values at several previous points (

y_{n-3}, \dots, y_n

); at the start only

y_0

is known, so the first few values must come from a one-step method such as RK4.

Q5 What is a stiff differential equation?

One with components that decay at very different rates, so that explicit methods must use a step size set by the fastest (already negligible) component to stay stable, e.g.

y' = -50(y - \cos x)

. Implicit methods remove this restriction.