+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system
+\begin{equation}
+A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
+\label{eq:03}
+\end{equation}
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. Algorithm~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold of GMRES respectively.
+
+\begin{algorithm}[t]
+\caption{Block Jacobi two-stage multisplitting method}
+\begin{algorithmic}[1]
+ \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+ \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+ \State Set the initial guess $x^0$
+ \For {$k=1,2,3,\ldots$ until convergence}
+ \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+ \State $x^k_\ell=Solve(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\label{solve}
+ \State Send $x_\ell^k$ to neighboring clusters\label{send}
+ \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters\label{recv}
+ \EndFor
+\end{algorithmic}
+\label{alg:01}
+\end{algorithm}
+
+The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{Szyld92,Bru95,Bai99,bahi07}. The multisplitting methods are convergent:
+\begin{itemize}
+\item if $A^{-1}>0$ and the splittings of matrix $A$ are weak regular when the iterations are synchronous, or
+\item if $A$ is M-matrix and its splittings are regular when the iterations are asynchronous.
+\end{itemize}
+
+In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on asynchronous model which allows the communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Algorithm~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged
+\begin{equation}
+k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
+\label{eq:04}
+\end{equation}
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold of the two-stage algorithm.
+
+
+
+
+
+
+
