57 
which must be solved at each timestep. 
which must be solved at each timestep. 
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 
$X=c^2 u^{(n1)}_{,i}$. Furthermore, in order to maintain stability, 
$X=c^2 u^{(n1)}_{,i}$. 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} 
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 
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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 
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 
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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} 
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 
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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 wavefront propagration 
identical to each other, exhibit a considerably faster wavefront propagation 
123 
and a decaying amplitude. 
and a decaying amplitude. 
124 


125 
\subsection{Advection equation} 
\subsection{Advection equation} 
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 