#LyX 2.3 created this file. For more info see http://www.lyx.org/ \lyxformat 544 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass book \use_default_options true \maintain_unincluded_children false \language spanish \language_package default \inputencoding auto \fontencoding global \font_roman "default" "default" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \paperfontsize default \spacing single \use_hyperref false \papersize default \use_geometry false \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 1 \use_minted 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style french \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \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 $|h|h_{\max}$ \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