Queste note sono basate sugli appunti fatti con Gianmarco Manzini negli anni 1995-2005

Metodi di Runge-Kutta esercizi

Appunti di calcolo numerico

Autore/Autrice

Affiliazione

Enrico Bertolazzi

University of Trento, Department of Industrial Engineering

Criteri per controllo ordine

\large \color{teal} \left. \begin{aligned} &\color{DarkGreen}\underbrace{ \color{DarkRed}\overbrace{\color{blue}\overbrace{ \underbrace{\color{magenta}\bm{A}\bm{1}=\bm{c}}_{\textrm{consistenza}},\quad \color{blue}\bm{b}\cdot\bm{1}=1, }^{\textrm{ordine 1}}\quad \color{DarkRed}\bm{b}\cdot\bm{c} = \frac{1}{2}}^{\textrm{ordine 2}},\quad \color{DarkGreen}\bm{b}\cdot\bm{c}^2 = \frac{1}{3},\quad \bm{b}\cdot\bm{d} = \frac{1}{6} }_{\textrm{ordine 3}} \\ & \color{teal}\bm{b}\cdot\bm{c}^3 = \frac{1}{4},\quad (\bm{b}\otimes\bm{c})\cdot\bm{d} = \frac{1}{8},\quad \bm{e}\cdot\bm{c}^2 = \frac{1}{12},\quad \bm{e}\cdot\bm{d} = \frac{1}{24} \quad \end{aligned} \right\} _{\color{teal}\textrm{ordine 4}}

\bm{d} = \bm{A}\bm{c}, \qquad \bm{e} = \bm{A}^T\bm{b}, \qquad \bm{c}^2 = \bm{c}\otimes\bm{c},\qquad \bm{c}^3 = \bm{c}\otimes\bm{c}\otimes\bm{c}

Euler Esplicito

Il metodo di Euler esplicito rappresentato utilizzando un tableau di Butcher:

\def\arraystretch{1.5} \begin{array}{c|c} 0 & 0 \\ \hline & 1 \end{array}

• \color{magenta}\textrm{Check consistency}\quad \bm{A}\bm{1}=\bm{c}
  

  \underbrace{(0)}_{\bm{A}}\bm{1} = \underbrace{(0)}_{\bm{c}}

• \color{blue}\textrm{Check ordine 1}\quad \bm{b}\cdot\bm{1}=\bm{1}
  \underbrace{(1)}_{\bm{b}}\cdot\bm{1} = 1

• \color{DarkRed}\textrm{Check ordine 2}\quad\bm{b}\cdot\bm{c} = \frac{1}{2}

  \underbrace{(1)}_{\bm{b}}\cdot \underbrace{(0)}_{\bm{c}} = 0 \color{red}\neq \frac{1}{2}

l’ordine è quindi 1.

Metodo del Punto Medio

\def\arraystretch{1.5} \begin{array}{c|cc} 0 & 0 & 0 \\ 1/2 & 1/2 & 0 \\ \hline & 0 & 1 \end{array}

• \color{magenta}\textrm{Check consistency}\quad \bm{A}\bm{1}=\bm{c}
  

  \underbrace{\begin{pmatrix} 0 & 0 \\ 1/2 & 0 \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} 1\\ 1 \end{pmatrix}}_{\bm{1}} = \underbrace{\begin{pmatrix} 0 \\ 1/2 \end{pmatrix}}_{\bm{c}}

• \color{blue}\textrm{Check ordine 1}\quad \bm{b}\cdot\bm{1}=\bm{1}
  \underbrace{\begin{pmatrix} 0 \\ 1 \end{pmatrix}}_{\bm{b}}\cdot\underbrace{\begin{pmatrix} 1\\ 1 \end{pmatrix}}_{\bm{1}} = 1

• \color{DarkRed}\textrm{Check ordine 2}\quad\bm{b}\cdot\bm{c} = \frac{1}{2}

  \underbrace{\begin{pmatrix} 0 \\ 1 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 1/2 \end{pmatrix}}_{\bm{c}} = \frac{1}{2}

• \color{DarkGreen}\textrm{Check ordine 3}\quad\bm{b}\cdot\bm{c}^2 = \frac{1}{3} \underbrace{\begin{pmatrix} 0 \\ 1 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 1/4 \end{pmatrix}}_{\bm{c}^2} = \frac{1}{4} \color{red}\neq \frac{1}{3}

• \color{DarkGreen}\textrm{Check ordine 3}\quad\bm{b}\cdot\bm{d} = \frac{1}{6} (non serve ordine 2)

l’ordine è quindi 2.

Metodo di Ralston

\def\arraystretch{1.5} \begin{array}{c|cc} 0 & 0 & 0 \\ 2/3 & 2/3 & 0 \\ \hline & 1/4 & 3/4 \end{array}

• \color{magenta}\textrm{Check consistency}\quad \bm{A}\bm{1}=\bm{c}
  

  \underbrace{\begin{pmatrix} 0 & 0 \\ 2/3 & 0 \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} 1\\ 1 \end{pmatrix}}_{\bm{1}} = \underbrace{\begin{pmatrix} 0 \\ 2/3 \end{pmatrix}}_{\bm{c}}

• \color{blue}\textrm{Check ordine 1}\quad \bm{b}\cdot\bm{1}=\bm{1}
  \underbrace{\begin{pmatrix} 1/4 \\ 3/4 \end{pmatrix}}_{\bm{b}}\cdot\underbrace{\begin{pmatrix} 1\\ 1 \end{pmatrix}}_{\bm{1}} = 1

• \color{DarkRed}\textrm{Check ordine 2}\quad\bm{b}\cdot\bm{c} = \frac{1}{2}

  \underbrace{\begin{pmatrix} 1/4 \\ 3/4 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 2/3 \end{pmatrix}}_{\bm{c}} = \frac{1}{2}

• \color{DarkGreen}\textrm{Check ordine 3}\quad\bm{b}\cdot\bm{c}^2 = \frac{1}{3} \underbrace{\begin{pmatrix} 1/4 \\ 3/4 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 4/9 \end{pmatrix}}_{\bm{c}^2} = \frac{1}{3}

• \color{DarkGreen}\textrm{Check ordine 3}\quad\bm{b}\cdot\bm{d} = \frac{1}{6}

  \bm{d}= \underbrace{\begin{pmatrix} 0 & 0 \\ 2/3 & 0 \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} 0\\ 2/3 \end{pmatrix}}_{\bm{c}} = \underbrace{\begin{pmatrix} 0 \\ 0 \end{pmatrix}}

  \underbrace{\begin{pmatrix} 1/4 \\ 3/4 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 0 \end{pmatrix}}_{\bm{d}} = 0 \color{red} \neq \frac{1}{6}

l’ordine è quindi 2.

Metodo del terzo ordine di Runge-Kutta (RK3)

\def\arraystretch{1.5} \begin{array} {c|ccc} 0 & 0 & 0 & 0 \\ 1/2 & 1/2 & 0 & 0 \\ 1 & -1 & 2 & 0 \\ \hline & 1/6 & 2/3 & 1/6 \end{array}

• \color{magenta}\textrm{Check consistency}\quad \bm{A}\bm{1}=\bm{c}
  

  \underbrace{\begin{pmatrix} 0 & 0 & 0 \\ 1/2 & 0 & 0 \\ -1 & 2 & 0 \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}}_{\bm{1}} = \underbrace{\begin{pmatrix} 0 \\ 1/2 \\ 1 \end{pmatrix}}_{\bm{c}}

• \color{blue}\textrm{Check ordine 1}\quad \bm{b}\cdot\bm{1}=\bm{1}
  \underbrace{\begin{pmatrix} 1/6 \\ 2/3 \\ 1/6 \end{pmatrix}}_{\bm{b}}\cdot\underbrace{\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}}_{\bm{1}} = 1

• \color{DarkRed}\textrm{Check ordine 2}\quad\bm{b}\cdot\bm{c} = \frac{1}{2}

  \underbrace{\begin{pmatrix} 1/6 \\ 2/3 \\ 1/6 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 1/2 \\ 1 \end{pmatrix}}_{\bm{c}} = \frac{1}{3}+\frac{1}{6} = \frac{1}{2}

• \color{DarkGreen}\textrm{Check ordine 3}\quad\bm{b}\cdot\bm{c}^2 = \frac{1}{3} \underbrace{\begin{pmatrix} 1/6 \\ 2/3 \\ 1/6 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 1/4 \\ 1 \end{pmatrix}}_{\bm{c}^2} = \frac{1}{6}+\frac{1}{6}=\frac{1}{3}

• \color{DarkGreen}\textrm{Check ordine 3}\quad\bm{b}\cdot\bm{d} = \frac{1}{6}

  \bm{d}= \underbrace{\begin{pmatrix} 0 & 0 & 0 \\ 1/2 & 0 & 0 \\ -1 & 2 & 0 \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} 0\\ 1/2 \\ 1 \end{pmatrix}}_{\bm{c}} = \underbrace{\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}}_{\bm{d}}

  \underbrace{\begin{pmatrix} 1/6 \\ 2/3 \\ 1/6 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}}_{\bm{d}} = \frac{1}{6}

• \color{teal}\textrm{Check ordine 4}\quad\bm{b}\cdot\bm{c}^3 = \frac{1}{4} \underbrace{\begin{pmatrix} 1/6 \\ 2/3 \\ 1/6 \end{pmatrix}}_{\bm{b}}\cdot \underbrace{\begin{pmatrix} 0 \\ 1/8 \\ 1 \end{pmatrix}}_{\bm{c}^3} = \frac{1}{12}+\frac{1}{6}=\frac{1}{4}

• \color{teal}\textrm{Check ordine 4}\quad (\bm{b}\otimes\bm{c})\cdot\bm{d} = \frac{1}{8} \underbrace{\begin{pmatrix} 1/6 \\ 2/3 \\ 1/6 \end{pmatrix}}_{\bm{b}}\otimes \underbrace{\begin{pmatrix} 0 \\ 1/2 \\ 1 \end{pmatrix}}_{\bm{c}} \cdot \underbrace{\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}}_{\bm{d}} = \frac{1}{6}\color{red}\neq \frac{1}{8}

• \color{teal}\textrm{Check ordine 4}\quad\bm{b}\quad \bm{e}\cdot\bm{c}^2 = \frac{1}{12} (non serve ordine 3)

• \color{teal}\textrm{Check ordine 4}\quad\bm{e}\cdot\bm{d} = \frac{1}{24} (non serve ordine 3

l’ordine è quindi 3.

Metodo del quarto ordine con regola 3/8

\def\arraystretch{1.5} \begin{array}{c|cccc} 0 & 0 & 0 & 0 & 0 \\ 1/3 & 1/3 & 0 & 0 & 0 \\ 2/3 & -1/3 & 1 & 0 & 0 \\ 1 & 1 & -1 & 1 & 0 \\ \hline & 1/8 & 3/8 & 3/8 & 1/8 \end{array}

• \color{magenta}\textrm{Check consistency}\quad \bm{A}\bm{1}=\bm{c}
  

  \underbrace{\begin{pmatrix} 0 & 0 & 0 & 0 \\ 1/3 & 0 & 0 & 0 \\ -1/3 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} 1\\ 1\\ 1\\ 1 \end{pmatrix}}_{\bm{1}} = \underbrace{\begin{pmatrix} 0 \\ 1/3 \\ 2/3 \\ 1 \end{pmatrix}}_{\bm{c}}

• \color{blue}\textrm{Check ordine 1}\quad \bm{b}\cdot\bm{1}=1

  \begin{array}{c||cccc||cc} \hline \bm{b} & 1/8 & 3/8 & 3/8 & 1/8 & 1 & \color{green}\textrm{OK} \\ \hline \end{array}

• \color{DarkRed}\textrm{Check ordine 2}\quad\bm{b}\cdot\bm{c} = \frac{1}{2}

  \begin{array}{c||cccc||cc} \hline \bm{b} & 1/8 & 3/8 & 3/8 & 1/8 & 1 & \textrm{OK} \\ \hline \hline \bm{c} & 0 & 1/3 & 2/3 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/8 & 1/4 & 1/8 & 1/2 & \color{green}\textrm{OK} \\ \hline \end{array}

• \color{DarkGreen}\textrm{Check ordine 3}\quad\bm{b}\cdot\bm{c}^2 = \frac{1}{3}

  \begin{array}{c||cccc||cc} \hline \bm{b} & 1/8 & 3/8 & 3/8 & 1/8 & 1 & \textrm{OK} \\ \hline \hline \bm{c} & 0 & 1/3 & 2/3 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/8 & 1/4 & 1/8 & 1/2 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 0 & 1/24 & 1/6 & 1/8 & 1/3 & \color{green}\textrm{OK} \\ \hline \end{array}

• \color{DarkGreen}\textrm{Check ordine 3}\quad\bm{b}\cdot\bm{d} = \frac{1}{6}

  \underbrace{\begin{pmatrix} 0 & 0 & 0 & 0 \\ 1/3 & 0 & 0 & 0 \\ -1/3 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} 0 \\ 1/3 \\ 2/3 \\ 1 \end{pmatrix}}_{\bm{c}} = \underbrace{\begin{pmatrix} 0 \\ 0 \\ 1/3 \\ 1/3 \end{pmatrix}}_{\bm{d}}

  \begin{array}{c||cccc||cc} \hline \bm{b} & 1/8 & 3/8 & 3/8 & 1/8 & 1 & \textrm{OK} \\ \hline \hline \bm{c} & 0 & 1/3 & 2/3 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/8 & 1/4 & 1/8 & 1/2 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 0 & 1/24 & 1/6 & 1/8 & 1/3 & \textrm{OK} \\ \hline \bm{d} & 0 & 0 & 1/3 & 1/3 & & \\ \hline \bm{b}\otimes\bm{d} & 0 & 0 & 1/8 & 1/24 & 1/6 & \color{green}\textrm{OK} \\ \hline \end{array}

• \color{teal}\textrm{Check ordine 4}\quad\bm{b}\cdot\bm{c}^3 = \frac{1}{4}

  \begin{array}{c||cccc||cc} \hline \bm{b} & 1/8 & 3/8 & 3/8 & 1/8 & 1 & \textrm{OK} \\ \hline \hline \bm{c} & 0 & 1/3 & 2/3 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/8 & 1/4 & 1/8 & 1/2 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 0 & 1/24 & 1/6 & 1/8 & 1/3 & \textrm{OK} \\ \hline \bm{d} & 0 & 0 & 1/3 & 1/3 & & \\ \hline \bm{b}\otimes\bm{d} & 0 & 0 & 1/8 & 1/24 & 1/6 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & 0 & 1/72 & 1/9 & 1/8 & 1/4 & \color{green}\textrm{OK} \\ \hline \end{array}

• \color{teal}\textrm{Check ordine 4}\quad (\bm{b}\otimes\bm{c})\cdot\bm{d} = \frac{1}{8}

  \begin{array}{c||cccc||cc} \hline \bm{b} & 1/8 & 3/8 & 3/8 & 1/8 & 1 & \textrm{OK} \\ \hline \hline \bm{c} & 0 & 1/3 & 2/3 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/8 & 1/4 & 1/8 & 1/2 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 0 & 1/24 & 1/6 & 1/8 & 1/3 & \textrm{OK} \\ \hline \bm{d} & 0 & 0 & 1/3 & 1/3 & & \\ \hline \bm{b}\otimes\bm{d} & 0 & 0 & 1/8 & 1/24 & 1/6 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & 0 & 1/72 & 1/9 & 1/8 & 1/4 & \textrm{OK} \\ \hline \bm{b}\otimes\bm{c}\otimes\bm{d} & 0 & 0 & 1/12 & 1/24 & 1/8 & \color{green}\textrm{OK} \\ \hline \end{array}

• \color{teal}\textrm{Check ordine 4}\quad\bm{e}\cdot\bm{c}^2 = \frac{1}{12}

  \dfrac{1}{8} \underbrace{\begin{pmatrix} 1 & 3 & 3 & 1 \end{pmatrix}}_{\bm{b}^T} \underbrace{\begin{pmatrix} 0 & 0 & 0 & 0 \\ 1/3 & 0 & 0 & 0 \\ -1/3 & 1 & 0 & 0 \\ 1 & -1 & 1 & 0 \end{pmatrix} }_{\bm{A}} = \dfrac{1}{8} \underbrace{\begin{pmatrix} 1 & 2 & 1 & 0 \end{pmatrix}}_{\bm{e}^T}

  \begin{array}{c||cccc||cc} \hline \bm{b} & 1/8 & 3/8 & 3/8 & 1/8 & 1 & \textrm{OK} \\ \hline \hline \bm{c} & 0 & 1/3 & 2/3 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/8 & 1/4 & 1/8 & 1/2 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 0 & 1/24 & 1/6 & 1/8 & 1/3 & \textrm{OK} \\ \hline \bm{d} & 0 & 0 & 1/3 & 1/3 & & \\ \hline \bm{b}\otimes\bm{d} & 0 & 0 & 1/8 & 1/24 & 1/6 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & 0 & 1/72 & 1/9 & 1/8 & 1/4 & \textrm{OK} \\ \hline \bm{b}\otimes\bm{c}\otimes\bm{d} & 0 & 0 & 1/12 & 1/24 & 1/8 & \textrm{OK} \\ \hline \bm{e} & 1/8 & 1/4 & 1/8 & 0 & & \\ \hline \bm{c}^2 & 0 & 1/9 & 4/9 & 1 & & \\ \hline \bm{e}\otimes\bm{c}^2 & 0 & 1/36 & 1/18 & 0 & 1/12 & \color{green}\textrm{OK} \\ \hline \end{array}

• \color{teal}\textrm{Check ordine 4}\quad\bm{e}\cdot\bm{d} = \frac{1}{24}

  \begin{array}{c||cccc||cc} \hline \bm{b} & 1/8 & 3/8 & 3/8 & 1/8 & 1 & \textrm{OK} \\ \hline \hline \bm{c} & 0 & 1/3 & 2/3 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/8 & 1/4 & 1/8 & 1/2 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 0 & 1/24 & 1/6 & 1/8 & 1/3 & \textrm{OK} \\ \hline \bm{d} & 0 & 0 & 1/3 & 1/3 & & \\ \hline \bm{b}\otimes\bm{d} & 0 & 0 & 1/8 & 1/24 & 1/6 & \textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & 0 & 1/72 & 1/9 & 1/8 & 1/4 & \textrm{OK} \\ \hline \bm{b}\otimes\bm{c}\otimes\bm{d} & 0 & 0 & 1/12 & 1/24 & 1/8 & \textrm{OK} \\ \hline \bm{e} & 1/8 & 1/4 & 1/8 & 0 & & \\ \hline \bm{c}^2 & 0 & 1/9 & 4/9 & 1 & & \\ \hline \bm{e}\otimes\bm{c}^2 & 0 & 1/36 & 1/18 & 0 & 1/12 & \textrm{OK} \\ \hline \bm{e}\otimes\bm{d} & 0 & 0 & 1/24 & 0 & 1/24 & \color{green}\textrm{OK} \\ \hline \end{array}

l’ordine è quindi è almeno 4. Essendo a 4 stadi ed esplicito l’ordine è al più 4. Quindi l’ordine e 4.

Metodo del Punto Medio Implicito

\def\arraystretch{1.5} \begin{array}{c|c} 1/2 & 1/2 \\ \hline & 1 \end{array}

bisogna riempire la tabella fino a che soddisfa le condizioni di ordine \begin{array}{c||c||cc} \hline \bm{b} & & 1 & \color{LightGrey}\textrm{OK} \\ \hline \hline \bm{c} & & & \\ \hline \bm{b}\otimes\bm{c} & & 1/2 & \color{LightGrey}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & & 1/3 & \color{LightGrey}\textrm{OK} \\ \hline \bm{d} & & & \\ \hline \bm{b}\otimes\bm{d} & & 1/6 & \color{LightGrey}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & & 1/4 & \color{LightGrey}\textrm{OK} \\ \hline \bm{b}\otimes\bm{c}\otimes\bm{d} & & 1/8 & \color{LightGrey}\textrm{OK} \\ \hline \bm{e} & & & \\ \hline \bm{c}^2 & & & \\ \hline \bm{e}\otimes\bm{c}^2 & & 1/12 & \color{LightGrey}\textrm{OK} \\ \hline \bm{e}\otimes\bm{d} & & 1/24 & \color{LightGrey}\textrm{OK} \\ \hline \end{array}

riempimento

\begin{array}{c||c||cc} \hline \bm{b} & 1 & 1 & \color{green}\textrm{OK} \\ \hline \hline \bm{c} & 1/2 & & \\ \hline \bm{b}\otimes\bm{c} & 1/2 & 1/2 & \color{green}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 1/4 & 1/4 & \color{red}\textrm{NO} \\ \hline \bm{d} & & & \\ \hline \bm{b}\otimes\bm{d} & & & \color{LightGrey}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & & & \color{LightGrey}\textrm{OK} \\ \hline \bm{b}\otimes\bm{c}\otimes\bm{d} & & & \color{LightGrey}\textrm{OK} \\ \hline \bm{e} & & & \\ \hline \bm{c}^2 & & & \\ \hline \bm{e}\otimes\bm{c}^2 & & & \color{LightGrey}\textrm{OK} \\ \hline \bm{e}\otimes\bm{d} & & & \color{LightGrey}\textrm{OK} \\ \hline \end{array}

ordine 2.

Metodo di Crank-Nicolson

\def\arraystretch{1.5} \begin{array}{c|cc} 0 & 0 & 0 \\ 1 & 1/2 & 1/2 \\ \hline & 1/2 & 1/2 \end{array}

riempimento \begin{array}{c||cc||cc} \hline \bm{b} & 1/2 & 1/2 & 1 & \color{green}\textrm{OK} \\ \hline \hline \bm{c} & 0 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/2 & 1/2 & \color{green}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 0 & 1/2 & 1/2 & \color{red}\textrm{NO} \\ \hline \bm{d} & & & & \\ \hline \bm{b}\otimes\bm{d} & & & & \color{LightGrey}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & & & & \color{LightGrey}\textrm{OK} \\ \hline \bm{b}\otimes\bm{c}\otimes\bm{d} & & & & \color{LightGrey}\textrm{OK} \\ \hline \bm{e} & & & & \\ \hline \bm{c}^2 & & & & \\ \hline \bm{e}\otimes\bm{c}^2 & & & & \color{LightGrey}\textrm{OK} \\ \hline \bm{e}\otimes\bm{d} & & & & \color{LightGrey}\textrm{OK} \\ \hline \end{array}

ordine 2.

Metodo di Gauss-Legendre del quarto ordine

\def\arraystretch{3} \begin{array}{c|cc} \dfrac{1}{2}-\dfrac{\sqrt3}{6} & \dfrac{1}{4} & \dfrac{1}{4}-\dfrac{\sqrt3}{6} \\ \dfrac{1}{2}+\dfrac{\sqrt3}{6} & \dfrac{1}{4}+\dfrac{\sqrt3}{6} & \dfrac{1}{4} \\ \hline & \dfrac{1}{2} & \dfrac{1}{2} \end{array}

\begin{aligned} \bm{d}&= \underbrace{\begin{pmatrix} \dfrac{1}{4} & \dfrac{1}{4}-\dfrac{\sqrt3}{6} \\ \dfrac{1}{4}+\dfrac{\sqrt3}{6} & \dfrac{1}{4} \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} \dfrac{1}{2}-\dfrac{\sqrt3}{6}\\ \dfrac{1}{2}+\dfrac{\sqrt3}{6} \end{pmatrix}}_{\bm{c}} = \underbrace{\begin{pmatrix} \dfrac{1}{6} -\dfrac{\sqrt{3}}{12} \\ \dfrac{1}{6} +\dfrac{\sqrt{3}}{12} \end{pmatrix}}_{\bm{d}} \\ \bm{e}&= \underbrace{\begin{pmatrix} \dfrac{1}{4} & \dfrac{1}{4}+\dfrac{\sqrt3}{6} \\ \dfrac{1}{4}-\dfrac{\sqrt3}{6} & \dfrac{1}{4} \end{pmatrix} }_{\bm{A}^T} \underbrace{\begin{pmatrix} \dfrac{1}{2}\\ \dfrac{1}{2} \end{pmatrix}}_{\bm{b}} = \underbrace{\begin{pmatrix} \dfrac{1}{4} +\dfrac{\sqrt{3}}{12} \\ \dfrac{1}{4} -\dfrac{\sqrt{3}}{12} \end{pmatrix}}_{\bm{e}} \end{aligned}

riempimento \begin{array}{c||cc||cc} \hline \bm{b} & 1/2 & 1/2 & 1 & \color{green}\textrm{OK} \\ \hline \hline \bm{c} & \dfrac{1}{2}-\dfrac{\sqrt{3}}{6} & \dfrac{1}{2}+\dfrac{\sqrt{3}}{6} & & \\ \hline \bm{b}\otimes\bm{c} & \dfrac{1}{4}-\dfrac{\sqrt{3}}{12} & \dfrac{1}{4}+\dfrac{\sqrt{3}}{12} & 1/2 & \color{green}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & \dfrac{1}{6}-\dfrac{\sqrt{3}}{12} & \dfrac{1}{6}+\dfrac{\sqrt{3}}{12} & 1/3 & \color{green}\textrm{OK} \\ \hline \bm{d} &\dfrac{1}{6} -\dfrac{\sqrt{3}}{12} & \dfrac{1}{6} +\dfrac{\sqrt{3}}{12} & & \\ \hline \bm{b}\otimes\bm{d} & \dfrac{1}{12} -\dfrac{\sqrt{3}}{24} & \dfrac{1}{12} +\dfrac{\sqrt{3}}{24} & 1/6 & \color{green}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & \dfrac{1}{8} - \dfrac{5\sqrt{3}}{72} & \dfrac{1}{8} + \dfrac{5\sqrt{3}}{72} & 1/4 & \color{green}\textrm{OK} \\ \hline \bm{b}\otimes\bm{c}\otimes\bm{d} & \dfrac{1}{16} - \dfrac{5\sqrt{3}}{144} & \dfrac{1}{16} + \dfrac{5\sqrt{3}}{144} & 1/8 & \color{green}\textrm{OK} \\ \hline \bm{e} & \dfrac{1}{4} +\dfrac{\sqrt{3}}{12} & \dfrac{1}{4} -\dfrac{\sqrt{3}}{12} & & \\ \hline \bm{c}^2 & \dfrac{1}{3} - \dfrac{\sqrt{3}}{6} & \dfrac{1}{3} + \dfrac{\sqrt{3}}{6} & & \\ \hline \bm{e}\otimes\bm{c}^2 & \dfrac{1}{24} - \dfrac{\sqrt{3}}{72} & \dfrac{1}{24} - \dfrac{\sqrt{3}}{72} & 1/12 & \color{green}\textrm{OK} \\ \hline \bm{e}\otimes\bm{d} & \dfrac{1}{48} - \dfrac{\sqrt{3}}{144} & \dfrac{1}{48} - \dfrac{\sqrt{3}}{144} & 1/24 & \color{green}\textrm{OK} \\ \hline \end{array}

Lobatto IIIB del quarto ordine

\def\arraystretch{1.5} \begin{array}{c|ccc} 0 & 1/6 & -1/6 & 0 \\ 1/2 & 1/6 & 1/3 & 0 \\ 1 & 1/6 & 5/6 & 0 \\ \hline & 1/6 & 2/3 & 1/6 \end{array}

\begin{aligned} \bm{d}&= \underbrace{\begin{pmatrix} 1/6 & -1/6 & 0 \\ 1/6 & 1/3 & 0 \\ 1/6 & 5/6 & 0 \end{pmatrix} }_{\bm{A}} \underbrace{\begin{pmatrix} 0 \\ 1/2 \\ 1 \end{pmatrix}}_{\bm{c}} = \underbrace{\begin{pmatrix} -1/12 \\ 1/6 \\ 5/12 \end{pmatrix}}_{\bm{d}} \\ \bm{e}&= \underbrace{\begin{pmatrix} 1/6 & 1/6 & 1/6 \\ -1/6 & 1/3 & 5/6 \\ 0 & 0 & 0 \end{pmatrix} }_{\bm{A}^T} \underbrace{\begin{pmatrix} 1/6 \\ 2/3 \\ 1/6 \end{pmatrix}}_{\bm{b}} = \underbrace{\begin{pmatrix} 1/6 \\ 1/3 \\ 0 \end{pmatrix}}_{\bm{e}} \end{aligned}

riempimento \begin{array}{c||ccc||cc} \hline \bm{b} & 1/6 & 2/3 & 1/6 & 1 & \color{green}\textrm{OK} \\ \hline \hline \bm{c} & 0 & 1/2 & 1 & & \\ \hline \bm{b}\otimes\bm{c} & 0 & 1/3 & 1/6 & 1/2 & \color{green}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^2 & 0 & 1/6 & 1/6 & 1/3 & \color{green}\textrm{OK} \\ \hline \bm{d} & -1/12 & 1/6 & 5/12 & \\ \hline \bm{b}\otimes\bm{d} & -1/72 & 1/9& 5/72 & 1/6 & \color{green}\textrm{OK} \\ \hline \hline \bm{b}\otimes\bm{c}^3 & 0 & 1/12 & 1/6 & 1/4 & \color{green}\textrm{OK} \\ \hline \bm{b}\otimes\bm{c}\otimes\bm{d} & 0 & 1/18 & 5/72 & 1/8 & \color{green}\textrm{OK} \\ \hline \bm{e} & 1/6 & 1/3 & 0 & & \\ \hline \bm{c}^2 & 0 & 1/4 & 1 & & \\ \hline \bm{e}\otimes\bm{c}^2 & 0 & 1/12 & 0 & 1/12 & \color{green}\textrm{OK} \\ \hline \bm{e}\otimes\bm{d} & -1/72 & 1/18 & 0 & 1/24 & \color{green}\textrm{OK} \\ \hline \end{array}

ordine almeno 4.

Referenze

Butcher, J. C. 1987. The Numerical analysis of ordinary differential equations : Runge-Kutta and general linear methods. First. Chichester: Wiley.

Cameron, Frank. 2006. «A Matlab package for automatically generating Runge-Kutta trees, order conditions, and truncation error coefficients». ACM Trans. Math. Softw. 32 (2): 274–98. https://doi.org/10.1145/1141885.1141892.