$v=e^{-a}.u$ represents the general change of variables such that $a=\frac{b^{t}(x,y,z)}{2\eta}$.
Consequently, the numerical resolution of the diffusion problem (the self-adjoint
operator~(\ref{ch13:eq:04})) is done by optimization algorithms, in contrast to that
-of the convection-diffusion problem (non self-adjoint operator~(\ref{ch13:eq:03}))
+of the convection-diffusion problem (nonself-adjoint operator~(\ref{ch13:eq:03}))
which is done by relaxation algorithms. In the case of our studied algorithm, the convergence\index{convergence}
is ensured by M-matrix property; then, the performance is linked to the magnitude of
the spectral radius of the iteration matrix, which is independent of the condition
\begin{table}
\centering
+\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\bf Pb. size} & \multicolumn{3}{c|}{\bf Synchronous} & \multicolumn{3}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-7}
$800^{3}$ & $3,950.87$ & $899,088$ & $56.22$ & $3,636.57$ & $834,900$ & $51.91$ & $7.95$ \\ \hline
\end{tabular}
+\end{scriptsize}
\vspace{0.5cm}
\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 12 GPUs.}
\label{ch13:tab:02}
%%--------------------------%%
\section{Red-black ordering technique}
\label{ch13:sec:06}
-As is wellknown, the Jacobi method\index{iterative method!Jacobi} is characterized
+As is well known, the Jacobi method\index{iterative method!Jacobi} is characterized
by a slow convergence\index{convergence} rate compared to some iterative methods\index{iterative method}
(for example, Gauss-Seidel method\index{iterative method!Gauss-Seidel}). So, in this
section, we present some solutions to reduce the execution time and the number of
we apply the point red-black ordering\index{iterative method!red-black ordering}
accordingly to the $y$-coordinate, as is shown in Figure~\ref{ch13:fig:06.02}. In
this case, the vector elements having even $y$-coordinate are computed in parallel
-using the values of those having odd $y$-coordinate and then viceversa. Moreover,
+using the values of those having odd $y$-coordinate and then vice-versa. Moreover,
in the GPU implementation of the parallel projected Richardson method (Section~\ref{ch13:sec:04}),
we have shown that a subproblem of size $(NX\times ny\times nz)$ is decomposed into
$nz$ grids of size $(NX\times ny)$. Then, each kernel is executed in parallel by
