KM2016 / Hyperbolic / testhyp

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\section*{Тестирование программы hyperbol}
При решении начальной задачи методом Рунге-Кутта число
обращений к функции для вычисления правой части равно 290.

Поточечные оценки точности аппроксимации
в узловых точках по пространству дается выражениями
\[\delta_i=\left|1-\frac{\tilde u_i}{u_i}\right|,\quad
\delta_i'=\left|1-\frac{\partial \tilde u_i/\partial t}
{\partial u_i/\partial t}\right|,\quad
\delta_i''=\left|1-\frac{\partial^2 \tilde u_i/\partial t^2}
{\partial^2 u_i/\partial t^2}\right|,\]
где $u_{i}=u(x_i,t)$, $\tilde u_{i}$ --
результаты численной аппроксимации решения исходной задачи в точке $(x_i,t)$.


Результаты представлены в таблице.
\begin{center}
\scriptsize
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|}\hline
\multicolumn{12}{|c|}{Момент времени $t_{0}=0$}\\
\hline
$x_i$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_i$&0&0&0&0&0&0&0&0&0&0&0\\
\hline
$\tilde u_i$&0&0&0&0&0&0&0&0&0&0&0\\
\hline
$\delta_i(\% )$&0&0&0&0&0&0&0&0&0&0&0\\
\hline
$\partial u_i/\partial t$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$\partial \tilde u_i/\partial t$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$\delta_i'(\%)$&0&0&0&0&0&0&0&0&0&0&0\\
\hline
\multicolumn{12}{|c|}{Момент времени $t_{1}=0.2$}\\
\hline
$x_i$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_i$&0.208&0.2538&0.302&0.3528&0.4067&0.464&0.5252&0.5906&0.6606&0.7356&0.816\\
\hline
$\tilde u_i$&0.2078&0.2536&0.3017&0.3525&0.4064&0.4637&0.5248&0.5902&0.6602&0.7352&0.8155\\
\hline
$\delta_i(\% )$&0.0799&0.075&0.0744&0.0726&0.0709&0.069&0.0671&0.0651&0.063&0.061&0.0586\\
\hline
$\partial u_i/\partial t$&1.12&1.407&1.729&2.092&2.5&2.96&3.478&4.059&4.709&5.434&6.24\\
\hline
$\partial \tilde u_i/\partial t$&1.118&1.404&1.726&2.088&2.496&2.955&3.472&4.053&4.703&5.428&6.233\\
\hline
$\delta_i'(\%)$&0.221&0.203&0.195&0.184&0.173&0.162&0.152&0.142&0.133&0.123&0.116\\
\hline
$\partial^2 u_i/\partial t^2$&1.2&2.074&3.293&4.915&6.998&9.6&12.78&16.59&21.09&26.34&32.4\\
\hline
$\partial^2 \tilde u_i/\partial t^2$&1.176&2.045&3.259&4.877&6.955&9.552&12.72&16.53&21.03&26.28&32.33\\
\hline
$\delta_i''(\%)$&2.01&1.39&1.02&0.783&0.618&0.501&0.414&0.347&0.297&0.253&0.23\\
\hline
\multicolumn{12}{|c|}{Момент времени $t_{2}=0.3$}\\
\hline
$x_i$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_i$&0.327&0.4067&0.4941&0.5906&0.6975&0.816&0.9475&1.093&1.255&1.433&1.629\\
\hline
$\tilde u_i$&0.3264&0.406&0.4933&0.5897&0.6965&0.8149&0.9463&1.092&1.253&1.431&1.627\\
\hline
$\delta_i(\% )$&0.168&0.159&0.153&0.146&0.14&0.133&0.126&0.119&0.112&0.105&0.101\\
\hline
$\partial u_i/\partial t$&1.27&1.667&2.141&2.706&3.375&4.16&5.075&6.132&7.346&8.727&10.29\\
\hline
$\partial \tilde u_i/\partial t$&1.265&1.66&2.133&2.697&3.365&4.149&5.063&6.119&7.331&8.712&10.27\\
\hline
$\delta_i'(\%)$&0.422&0.391&0.354&0.32&0.289&0.26&0.235&0.212&0.192&0.172&0.163\\
\hline
$\partial^2 u_i/\partial t^2$&1.8&3.11&4.939&7.373&10.5&14.4&19.17&24.88&31.64&39.51&48.6\\
\hline
$\partial^2 \tilde u_i/\partial t^2$&1.767&3.066&4.889&7.315&10.43&14.33&19.09&24.8&31.54&39.41&48.49\\
\hline
$\delta_i''(\%)$&1.82&1.43&1.02&0.788&0.622&0.504&0.416&0.35&0.297&0.255&0.236\\
\hline
\multicolumn{12}{|c|}{Момент времени $t_{3}=0.4$}\\
\hline
$x_i$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_i$&0.464&0.5906&0.7356&0.9021&1.093&1.312&1.561&1.845&2.165&2.525&2.928\\
\hline
$\tilde u_i$&0.4627&0.589&0.7338&0.9001&1.091&1.309&1.559&1.842&2.162&2.521&2.924\\
\hline
$\delta_i(\% )$&0.271&0.262&0.244&0.228&0.211&0.196&0.181&0.167&0.154&0.142&0.135\\
\hline
$\partial u_i/\partial t$&1.48&2.029&2.717&3.566&4.599&5.84&7.311&9.036&11.04&13.34&15.96\\
\hline
$\partial \tilde u_i/\partial t$&1.471&2.018&2.704&3.551&4.582&5.821&7.29&9.012&11.01&13.31&15.93\\
\hline
$\delta_i'(\%)$&0.6&0.583&0.497&0.434&0.378&0.331&0.291&0.257&0.227&0.204&0.186\\
\hline
$\partial^2 u_i/\partial t^2$&2.4&4.147&6.586&9.83&14&19.2&25.56&33.18&42.18&52.68&64.8\\
\hline
$\partial^2 \tilde u_i/\partial t^2$&2.363&4.085&6.518&9.752&13.91&19.1&25.45&33.06&42.06&52.54&64.66\\
\hline
$\delta_i''(\%)$&1.55&1.49&1.03&0.8&0.63&0.51&0.421&0.355&0.296&0.271&0.219\\
\hline
\end{tabular}
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