 # Diff of /trunk/doc/cookbook/example07.tex

revision 3383 by ahallam, Tue Nov 23 00:29:07 2010 UTC revision 3384 by ahallam, Thu Nov 25 03:28:01 2010 UTC
# Line 121  For example, a line 100m long is discret Line 121  For example, a line 100m long is discret
121  a wave enters with a propagation velocity of 100m/s then the travel time for  a wave enters with a propagation velocity of 100m/s then the travel time for
122  the wave between each node will be 0.01 seconds. The time step, must therefore  the wave between each node will be 0.01 seconds. The time step, must therefore
123  be significantly less then this. Of the order $10E-4$ would be appropriate.  be significantly less then this. Of the order $10E-4$ would be appropriate.
124    This stability criterion is known as the Courant–Friedrichs–Lewy
125    condition given by
126    \begin{equation}
127    dt=f\cdot \frac{dx}{v}
128    \end{equation}
129    where $dx$ is the mesh size and $f$ is a safety factor. To obtain a time step of
130    $10E-4$, a safety factor of $f=0.1$ was used.
131
132  The wave frequency content also plays a part in numerical stability. The  The wave frequency content also plays a part in numerical stability. The
133  nyquist-sampling theorem states that a signals bandwidth content will be  nyquist-sampling theorem states that a signals bandwidth content will be
# Line 130  $f_{s}$, or; Line 137  $f_{s}$, or;
137  \begin{equation} \label{eqn:samptheorem}  \begin{equation} \label{eqn:samptheorem}
138   f_{n} \geqslant f_{s}   f_{n} \geqslant f_{s}
139  \end{equation}  \end{equation}
140  For example a 50Hz signal will require a sampling rate greater then 100Hz or  For example, a 50Hz signal will require a sampling rate greater then 100Hz or
141  one sample every 0.01 seconds. The wave equation relies on a spatial frequency,  one sample every 0.01 seconds. The wave equation relies on a spatial frequency,
142  thus the sampling theorem in this case applies to the solution mesh spacing.  thus the sampling theorem in this case applies to the solution mesh spacing.
143  This relationship confirms that the frequency content of the input signal  This relationship confirms that the frequency content of the input signal
# Line 138  directly affects the time discretisation Line 145  directly affects the time discretisation
145
146  To accurately model the wave equation with high resolutions and velocities  To accurately model the wave equation with high resolutions and velocities
147  means that very fine spatial and time discretisation is necessary for most  means that very fine spatial and time discretisation is necessary for most
148  problems.  problems.  This requirement makes the wave equation arduous to
This requirement makes the wave equation arduous to
149  solve numerically due to the large number of time iterations required in each  solve numerically due to the large number of time iterations required in each
150  solution. Models with very high velocities and frequencies will be the worst  solution. Models with very high velocities and frequencies will be the worst
151  affected by this problem.  affected by this problem.
# Line 296  requirements of a problem. Care should b Line 302  requirements of a problem. Care should b
302  function can only be used when the $A$, $B$ and $C$ coefficients of the  function can only be used when the $A$, $B$ and $C$ coefficients of the
303  general form are zero.  general form are zero.
304