/[escript]/trunk/doc/user/lumping.tex
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revision 4285 by jfenwick, Tue Jan 22 09:30:23 2013 UTC revision 4286 by caltinay, Thu Mar 7 04:28:11 2013 UTC
# Line 57  This equation can be interpreted as a PD Line 57  This equation can be interpreted as a PD
57  which must be solved at each time-step.  which must be solved at each time-step.
58  In the notation of equation~\ref{LINEARPDE.SINGLE.1} we thus set $D=1$ and  In the notation of equation~\ref{LINEARPDE.SINGLE.1} we thus set $D=1$ and
59  $X=-c^2 u^{(n-1)}_{,i}$. Furthermore, in order to maintain stability,  $X=-c^2 u^{(n-1)}_{,i}$. Furthermore, in order to maintain stability,
60  the time step size needs to fullfill the Courant–Friedrichs–Lewy condition  the time step size needs to fulfill the Courant–Friedrichs–Lewy condition
61  (CFL condition).  (CFL condition).
62  \index{Courant condition}  \index{Courant condition}
63  \index{explicit scheme!Courant condition}  \index{explicit scheme!Courant condition}
# Line 70  we use $f=\frac{1}{6}$. Line 70  we use $f=\frac{1}{6}$.
70    
71  Figure~\ref{FIG LUMPING VALET A} depicts a temporal comparison between four  Figure~\ref{FIG LUMPING VALET A} depicts a temporal comparison between four
72  alternative solution algorithms: the exact solution; using a full mass matrix;  alternative solution algorithms: the exact solution; using a full mass matrix;
73  using HRZ lumping; and row sum lumping. The domain utilsed rectangular order 1  using HRZ lumping; and row sum lumping. The domain utilised rectangular order 1
74  elements (element size is $0.01$) with observations taken at the point  elements (element size is $0.01$) with observations taken at the point
75  $(\frac{1}{2},\frac{1}{2})$.  $(\frac{1}{2},\frac{1}{2})$.
76  All four solutions appear to be identical for this example. This is not the case  All four solutions appear to be identical for this example. This is not the case
# Line 84  when compared with the analytical soluti Line 84  when compared with the analytical soluti
84    
85  \begin{figure}[ht]  \begin{figure}[ht]
86  \centerline{\includegraphics[width=7cm]{lumping_valet_a_1}}  \centerline{\includegraphics[width=7cm]{lumping_valet_a_1}}
87  \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the accelaraton formulation~\ref{LUMPING WAVE VALET 2} of the  \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the acceleration formulation~\ref{LUMPING WAVE VALET 2} of the
88  Velet scheme with order $1$ elements, element size $dx=0.01$, and $c=1$.}  Velet scheme with order $1$ elements, element size $dx=0.01$, and $c=1$.}
89  \label{FIG LUMPING VALET A}  \label{FIG LUMPING VALET A}
90  \end{figure}  \end{figure}
91    
92  \begin{figure}[ht]  \begin{figure}[ht]
93  \centerline{\includegraphics[width=7cm]{lumping_valet_a_2}}  \centerline{\includegraphics[width=7cm]{lumping_valet_a_2}}
94  \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the accelaraton formulation~\ref{LUMPING WAVE VALET 2} of the  \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the acceleration formulation~\ref{LUMPING WAVE VALET 2} of the
95  Velet scheme with order $2$ elements, element size $0.01$, and $c=1$.}  Velet scheme with order $2$ elements, element size $0.01$, and $c=1$.}
96  \label{FIG LUMPING VALET B}  \label{FIG LUMPING VALET B}
97  \end{figure}  \end{figure}
# Line 119  The displacement solution is depicted in Line 119  The displacement solution is depicted in
119  domain utilised order $1$ elements (for order $2$, both  domain utilised order $1$ elements (for order $2$, both
120  lumping methods are unstable). The solutions for the exact and the full mass  lumping methods are unstable). The solutions for the exact and the full mass
121  matrix approximation are almost identical while the lumping solutions, whilst  matrix approximation are almost identical while the lumping solutions, whilst
122  identical to each other, exhibit a considerably faster wave-front propagration  identical to each other, exhibit a considerably faster wave-front propagation
123  and a decaying amplitude.  and a decaying amplitude.
124    
125  \subsection{Advection equation}  \subsection{Advection equation}
# Line 231  The examples in this section have demons Line 231  The examples in this section have demons
231  of both HRZ and row sum lumping with comparisons to the exact and full mass  of both HRZ and row sum lumping with comparisons to the exact and full mass
232  matrix solutions. Wave propagation type problems that utilise lumping, produce  matrix solutions. Wave propagation type problems that utilise lumping, produce
233  results simular the full mass matrix at significantly  results simular the full mass matrix at significantly
234  lower computation cost. An accelleration based formulation, with HRZ lumping  lower computation cost. An acceleration based formulation, with HRZ lumping
235  should be implemented for such problems, and can be appied to both order $1$ and  should be implemented for such problems, and can be applied to both order $1$ and
236   order $2$ elements.   order $2$ elements.
237    
238  In transport type problems, it is essential that row sum lumping is used to  In transport type problems, it is essential that row sum lumping is used to

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