/[escript]/trunk/doc/cookbook/example07.tex
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revision 3391 by caltinay, Thu Nov 25 05:05:50 2010 UTC revision 3392 by ahallam, Thu Dec 2 03:26:29 2010 UTC
# Line 77  object, it is also necessary to calculat Line 77  object, it is also necessary to calculat
77  diagonal in \autoref{eqn:gradrank1}.  diagonal in \autoref{eqn:gradrank1}.
78    
79  Thus when solving for equations containing the Laplacian one of two things must  Thus when solving for equations containing the Laplacian one of two things must
80  be completed. If the object \verb!p! is less then rank 1 the gradient is  be completed. If the object \verb!p! is less than rank 1 the gradient is
81  calculated via;  calculated via;
82  \begin{python}  \begin{python}
83  gradient=grad(p)  gradient=grad(p)
# Line 119  For example, a line 100m long is discret Line 119  For example, a line 100m long is discret
119  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
120  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
121  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.
122  This stability criterion is known as the Courant–Friedrichs–Lewy  This stability criterion is known as the Courant\textendash
123  condition given by  Friedrichs\textendash Lewy condition given by
124  \begin{equation}  \begin{equation}
125  dt=f\cdot \frac{dx}{v}  dt=f\cdot \frac{dx}{v}
126  \end{equation}  \end{equation}
# Line 128  where $dx$ is the mesh size and $f$ is a Line 128  where $dx$ is the mesh size and $f$ is a
128  $10E-4$, a safety factor of $f=0.1$ was used.  $10E-4$, a safety factor of $f=0.1$ was used.
129    
130  The wave frequency content also plays a part in numerical stability. The  The wave frequency content also plays a part in numerical stability. The
131  nyquist-sampling theorem states that a signals bandwidth content will be  Nyquist-sampling theorem states that a signals bandwidth content will be
132  accurately represented when an equispaced sampling rate $f _{n}$ is  accurately represented when an equispaced sampling rate $f _{n}$ is
133  equal to or greater then twice the maximum frequency of the signal  equal to or greater then twice the maximum frequency of the signal
134  $f_{s}$, or;  $f_{s}$, or;

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