Runge-Kutta Method

General Procedure

A Butcher Table for a Runge-Kutta method with $s$ stages has the form:

c A b^{T} = c_{1} c_{2} ⋮ c_{s} a_{11} a_{21} ⋮ a_{s 1} b_{1} a_{12} a_{22} ⋮ a_{s 2} b_{2} \dots \dots ⋱ \dots \dots a_{1 s} a_{2 s} ⋮ a_{ss} b_{s}

The procedure to find $y_{n + 1}$ from $y_{n}$ is:

1. Calculate Intermediate Stages:

q_{i} = h \cdot f (t_{n} + c_{i} h, y_{n} + j = 1 \sum s a_{ij} q_{j}) for i = 1, \dots, s

2. Calculate Final Step:

y_{n + 1} = y_{n} + i = 1 \sum s b_{i} q_{i}

0 2/3 1/3 0 1/4 1/3 2/3 3/4

Procedure:

Stage 1:
$q_{1} = h \cdot f ((t_{n} + c_{1} h), (y_{n} + a_{11} q_{1} + a_{12} q_{2} + a_{13} q_{3} + ...))$

q_{1} = h \cdot f (t_{n}, y_{n} + \frac{1}{3} q_{1} + \frac{1}{3} q_{2})

Stage 2:
$q_{2} = h \cdot f (t_{n} + \frac{2}{3} h, y_{n} + \frac{2}{3} q_{2})$
Final Result:
$y_{n + 1} = y_{n} + b_{1} q_{1} + b_{2} q_{2} + ...$

y_{n + 1} = y_{n} + \frac{1}{4} q_{1} + \frac{3}{4} q_{2}

1/2 1/2 1

Procedure:

Stage 1:
$q_{1} = h \cdot f (t_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} q_{1})$
Final Result:
$y_{n + 1} = y_{n} + q_{1}$

01 1/2 01 1/4 1/6 00 1/4 1/6 000 2/3

Procedure:

Stage 1:
$q_{1} = h \cdot f (t_{n}, y_{n})$
Stage 2:
$q_{2} = h \cdot f (t_{n} + h, y_{n} + q_{1})$
Stage 3:
$q_{3} = h \cdot f (t_{n} + \frac{1}{2} h, y_{n} + \frac{1}{4} q_{1} + \frac{1}{4} q_{2})$
Final Result:
$y_{n + 1} = y_{n} + \frac{1}{6} q_{1} + \frac{1}{6} q_{2} + \frac{2}{3} q_{3}$

Explicit if the Butcher tableau has zeros on and above the main diagonal (lower triangular with zero diagonal)
Implicit if there are non-zero entries on or above the main diagonal

Explicit: Euler-Heun Method The Butcher tableau for the Euler-Heun method is:
$01 01 1/2 00 1/2$
The coefficient matrix is $A = (0100)$ . Since all entries on and above the main diagonal are zero, the method is explicit.
Implicit: Crank-Nicolson Method The Butcher tableau for the Crank-Nicolson method is:
$01 0 1/2 1/2 0 1/2 1/2$
The coefficient matrix is $A = (0 1/2 0 1/2)$ . Because there is a non-zero entry on the main diagonal ( $a_{22} = 1/2$ ), the method is implicit. The calculation for the second stage depends on itself, which requires solving an equation.