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\begin_body

\begin_layout Standard
Dados un problema de valores iniciales
\begin_inset Formula 
\[
\left\{ \begin{aligned}\dot{x}(t) & =f(t,x(t)),\\
x(a) & =x_{0}
\end{aligned}
\right.
\]

\end_inset

y una solución aproximada 
\begin_inset Formula $(t_{i},\omega_{i})_{i=0}^{n}$
\end_inset

 del problema, llamamos 
\series bold
solución local
\series default
 
\begin_inset Formula $z_{i}$
\end_inset

 a la solución del problema
\begin_inset Formula 
\[
\left\{ \begin{aligned}\dot{z}_{i}(t) & =f(t,z_{i}(t)),\\
z_{i}(t_{i}) & =\omega_{i}.
\end{aligned}
\right.
\]

\end_inset

Como 
\series bold
teorema
\series default
, si 
\begin_inset Formula $f$
\end_inset

 es lipschitziana de constante 
\begin_inset Formula $k>0$
\end_inset

 y existe 
\begin_inset Formula $\varepsilon>0$
\end_inset

 tal que 
\begin_inset Formula $\Vert z_{i}(t_{i+1})-\omega_{i+1}\Vert\leq\varepsilon(t_{i+1}-t_{i})$
\end_inset

 para cada 
\begin_inset Formula $i\in\{0,\dots,n-1\}$
\end_inset

, entonces
\begin_inset Formula 
\[
\Vert y(t_{n})-\omega_{n}\Vert\leq e^{k(t_{n}-a)}\Vert y(t_{0})-\omega_{0}\Vert+\frac{e^{k(t_{n}-a)}-1}{k}\varepsilon.
\]

\end_inset

Si el método es un 
\series bold
método en diferencias
\series default
, uno de la forma 
\begin_inset Formula $t_{i+1}=t_{i}+h_{i}$
\end_inset

 y 
\begin_inset Formula $\omega_{i+1}=\omega_{i}+h_{i}Ø(t_{i},\omega_{i},h_{i})$
\end_inset

, y si 
\begin_inset Formula $z_{i}(t_{i})\cong x(t_{i})\cong\omega_{i}$
\end_inset

, el 
\series bold
criterio de error local
\series default
 para algún 
\begin_inset Formula $\varepsilon>0$
\end_inset

 e 
\begin_inset Formula $i\in\{0,\dots,n-1\}$
\end_inset

 consiste en que
\begin_inset Formula 
\[
\Vert\tau_{i+1}(h_{i})\Vert:=\frac{\Vert x(t_{i+1})-x(t_{i})-h_{i}Ø(t_{i},x(t_{i}),h_{i})\Vert}{h_{i}}\leq\varepsilon.
\]

\end_inset

Si 
\begin_inset Formula $z_{i}(t_{i})\approxeq x(t_{i})\approxeq\omega_{i}$
\end_inset

, se tiene
\begin_inset Formula 
\[
\Vert\tau_{i+1}(h_{i})\Vert\approx\cong\frac{\Vert z_{i}(t_{i+1})-\omega_{i+1}\Vert}{h}.
\]

\end_inset

Queremos ajustar el paso automáticamente para mantener el error local dentro
 de ciertos límites y economizar en número de cálculos.
\end_layout

\begin_layout Section
Extrapolación de Richardson
\end_layout

\begin_layout Standard
Como 
\series bold
teorema
\series default
, si el método en diferencias 
\begin_inset Formula $\omega_{i+1}=\omega_{i}+h_{i}Ø(t_{i},\omega_{i},h_{i})$
\end_inset

 verifica en cada paso que 
\begin_inset Formula $z_{i}(t_{i+1})=\omega_{i+1}+cz_{i}^{(k+1)}(t_{i})h^{k+1}+O(h^{k+2})$
\end_inset

 para ciertos 
\begin_inset Formula $c\in\mathbb{R}$
\end_inset

 y 
\begin_inset Formula $k\in\mathbb{N}$
\end_inset

, sean 
\begin_inset Formula $h\coloneqq h_{i}$
\end_inset

 y 
\begin_inset Formula $(t_{i+1},Y)$
\end_inset

 el resultado de dar dos pasos desde 
\begin_inset Formula $(t_{i},\omega_{i})$
\end_inset

 con el método de paso fijo 
\begin_inset Formula $\xi_{j+1}=\xi_{j}+\frac{h}{2}Ø(t_{j},\xi_{j},\frac{h}{2})$
\end_inset

, entonces 
\begin_inset Formula 
\[
z_{i}(t_{i+1})=Y+2cz_{i}^{(k+1)}(t_{i})\left(\frac{h}{2}\right)^{k+1}+O(h^{k+2}).
\]

\end_inset


\end_layout

\begin_layout Standard
En estas condiciones:
\end_layout

\begin_layout Enumerate
\begin_inset Formula 
\[
z_{i}(t_{i+1})=\frac{2^{k}Y-\omega_{i+1}}{2^{k}-1}+O(h^{k+2}).
\]

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Multiplicando por 
\begin_inset Formula $2^{k}$
\end_inset

 el resultado del teorema y restando la fórmula de la hipótesis,
\begin_inset Formula 
\begin{align*}
(2^{k}-1)z_{i}(t_{i+1}) & =2^{k}Y+2^{k+1}cz_{i}^{(k+1)}(t_{i})\left(\frac{h}{2}\right)^{k+1}-cz_{i}^{(k+1)}(t_{i})h^{k+1}-\omega_{i+1}+O(h^{k+2})\\
 & \overset{2^{k}\left(\frac{h}{2}\right)^{k+1}=h^{k+1}}{=}2^{k}Y-\omega_{i+1}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula 
\[
z_{i}(t_{i+1})-\omega_{i+1}=\frac{2^{k}}{2^{k}-1}(Y-\omega_{i+1})+O(h^{k+2}).
\]

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Restando 
\begin_inset Formula $\omega_{i+1}$
\end_inset

 a ambas partes de lo anterior.
\end_layout

\end_deeper
\begin_layout Standard
Para un 
\begin_inset Formula $\varepsilon>0$
\end_inset

, queremos que 
\begin_inset Formula $\Vert z_{i}(t_{i+1})-\omega_{i+1}\Vert<\varepsilon h_{i}$
\end_inset

.
 El siguiente es un método práctico para dar un paso de tamaño adaptativo:
\end_layout

\begin_layout Enumerate
Dar un paso con 
\begin_inset Formula $h$
\end_inset

 para obtener 
\begin_inset Formula $\omega_{i+1}$
\end_inset

 y dos con 
\begin_inset Formula $\frac{h}{2}$
\end_inset

 para obtener 
\begin_inset Formula $Y_{h}$
\end_inset

.
\end_layout

\begin_layout Enumerate
Obtener el error 
\begin_inset Formula $E\coloneqq \frac{2^{k}}{2^{k}-1}\Vert Y-\omega_{i+1}\Vert\approx\Vert z_{i}(t_{i+1})-\omega_{i+1}\Vert$
\end_inset

.
\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula $E>\varepsilon h$
\end_inset

, ajustar 
\begin_inset Formula $h$
\end_inset

 y volver a intentar desde el principio.
\end_layout

\begin_layout Enumerate
Aceptar el paso 
\begin_inset Formula $(t_{i}+h,\omega_{i+1})$
\end_inset

 y ajustar 
\begin_inset Formula $h$
\end_inset

 para el siguiente paso.
\end_layout

\begin_layout Standard
Para ajustar el paso:
\end_layout

\begin_layout Enumerate
Calcular 
\begin_inset Formula $q\coloneqq \left(\frac{\varepsilon h}{2E}\right)^{1/k}$
\end_inset

 y 
\begin_inset Formula $q'\coloneqq \min\{4,\max\{0.1,q\}\}$
\end_inset

, y hacer 
\begin_inset Formula $h\gets q'h$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Para cierta constante 
\begin_inset Formula $C$
\end_inset

, si 
\begin_inset Formula $y$
\end_inset

 es el resultado de aplicar un paso de tamaño 
\begin_inset Formula $qh$
\end_inset

, 
\begin_inset Formula $\Vert z_{i}(t_{i}+qh)-y\Vert\approx C(qh)^{k+1}=Cq^{k+1}h^{k+1}\approx q^{k+1}\Vert z_{i}(t_{i+1})-\omega_{i+1}\Vert\approx q^{k+1}E$
\end_inset

, pero 
\begin_inset Formula $q^{k+1}E\leq\varepsilon qh\iff q^{k}E\leq\varepsilon h\iff q\leq\left(\frac{\varepsilon h}{E}\right)^{1/k}$
\end_inset

.
 Entonces usamos 
\begin_inset Formula $2E$
\end_inset

 en vez de 
\begin_inset Formula $E$
\end_inset

 para tener cierto margen para evitar re-calcular y añadimos límites en
 
\begin_inset Formula $q'$
\end_inset

 por estabilidad.
\end_layout

\end_deeper
\begin_layout Enumerate
Usando umbrales 
\begin_inset Formula $h_{\min}$
\end_inset

 y 
\begin_inset Formula $h_{\max}$
\end_inset

 para el paso, si 
\begin_inset Formula $|h|<h_{\min}$
\end_inset

, informamos de un error, pues los errores de redondeo serían demasiado
 grandes, y si 
\begin_inset Formula $|h|>h_{\max}$
\end_inset

, hacemos 
\begin_inset Formula $h\gets h_{\max}\text{sgn}h$
\end_inset

.
\end_layout

\begin_layout Section
Método de Runge-Kutta-Fehlberg
\end_layout

\begin_layout Standard
Dados dos métodos de Runge-Kutta con los mismos pasos, 
\begin_inset Formula $(\omega_{i})$
\end_inset

 de orden 
\begin_inset Formula $p$
\end_inset

 y 
\begin_inset Formula $(\tilde{\omega}_{i})$
\end_inset

 de orden 
\begin_inset Formula $p+1$
\end_inset

, 
\begin_inset Formula $z_{i}(t_{i+1})-\omega_{i+1}\approx\tilde{\omega}_{i+1}-\omega_{i+1}$
\end_inset

.
 En efecto, para ciertos 
\begin_inset Formula $C_{1},C_{2}\in\mathbb{R}$
\end_inset

, 
\begin_inset Formula $z_{i}(t_{i}+h)-\omega_{i+1}\approx C_{1}h^{p+1}+O(h^{p+2})$
\end_inset

 y 
\begin_inset Formula $\tilde{z}_{i}(t_{i+1})-\tilde{\omega}_{i+1}\approx C_{2}h^{p+2}+O(h^{p+3})$
\end_inset

, luego 
\begin_inset Formula $z_{i}(t_{i}+h)-\omega_{i+1}\approx\tilde{z}_{i}(t_{i+1})-\tilde{\omega}_{i+1}+\tilde{\omega}_{i+1}-\omega_{i+1}\approx\tilde{\omega}_{i+1}-\omega_{i+1}+O(h^{p+2})$
\end_inset

.
\end_layout

\begin_layout Standard
El 
\series bold
método de Runge-Kutta-Fehlberg
\series default
 consiste en usar esta aproximación del error 
\begin_inset Formula $E$
\end_inset

 en el método anterior de paso adaptativo con los siguientes dos métodos
 de Runge-Kutta que tienen los mismos 
\begin_inset Formula $k_{i}$
\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\begin{array}{c|ccccc}
\frac{1}{4} & \frac{1}{4}\\
\frac{3}{8} & \frac{3}{32} & \frac{9}{32}\\
\frac{12}{13} & \frac{1932}{2197} & -\frac{7200}{2197} & \frac{7296}{2197}\\
1 & \frac{439}{216} & -8 & \frac{3680}{513} & -\frac{845}{4104}\\
\frac{1}{2} & -\frac{8}{27} & 2 & -\frac{443}{332} & \frac{1859}{4104} & -\frac{11}{40}\\
\hline \omega: & \frac{25}{216} & \frac{1408}{2565} & \frac{2197}{4104} & -\frac{1}{5}\\
\tilde{\omega}: & \frac{16}{135} & \frac{6656}{12825} & \frac{28561}{56430} & -\frac{9}{50} & \frac{2}{55}
\end{array}
\]

\end_inset


\end_layout

\begin_layout Standard
El método 
\begin_inset Formula $\omega$
\end_inset

 es de orden 4 y 
\begin_inset Formula $\tilde{\omega}$
\end_inset

 es de orden 5.
\end_layout

\end_body
\end_document