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การประชุมวิชาการเครือขายวิศวกรรมเครื่องกลแหงประเทศไทยครั้งที่ 17
15-17 ตุลาคม 2546 จังหวัดปราจีนบุรี
Parallel Computing on the Navier-Stokes Solver
with the Multigrid Method
Kiattisak Ngiamsoongnirn*, Ekachai Juntasaro*, Varangrat Juntasaro** and Putchong Uthayopas***
*School of Mechanical Engineering, Institute of Engineering, Suranaree University of Technology,
Nakhon Ratchasima 30000, Thailand,
Phone: (044)224410-2, Email: Kiatt2000@hotmail.com
**Department of Mechanical Engineering, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand,
Phone: (02)9428555 ext 1829, Email: ovrsk@ku.ac.th
***Department of Computer Engineering, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand,
Phone: (02)9428555 ext 1416, Email: pu@ku.ac.th
Abstract
This paper is aimed to present the combination of the parallel
computing and the multigrid method on the Navier-Stokes solver.
The combination is based on the concept of the object-oriented
programming (OOP), which consists of 4 independent modules:
Grid Generation, Navier-Stokes Solver, Multigrid Method and
Parallel Computing modules. The multigrid method is
implemented by employing the full approximation storage (FAS)
scheme for numerically solving the non-linear Navier-Stokes
equations. The overall computation is performed by using the
parallel computing in which a number of computers are
concurrently computed for the same task but on different sub-
data. The two-dimensional laminar flow in a cavity at Re=1,000 is
used as a test case. It is found that the computational time is
decreased significantly when employing the combination of the
multigrid method and the parallel computing.
1. Introduction
In recent years, Computational Fluid Dynamics (CFD) has
been a design tool in industries due to, for example, the lack of
instruments to measure some quantities in some dangerous
zones. In addition, the advantages of CFD are the low cost to
construct and the ability to immediately observe some phenomena
through a monitor of personal computer (PC). However, when the
flow is so complicated, the number of data points required to
capture the physics of flow has to be large enough. A single
computer is limited to its memory and speed. In other words, it
may compute one problem for a large number of iterations or it
cannot handle with a large number of data at all. The
convergence rate of iterative methods can be greatly improved by
using multigrid acceleration techniques. The multigrid methods
eliminate some errors on the coarser grid that cannot be
eliminated by the finer grid and then correct the solutions up to
the finest grid. The time consumption, moreover, can be
decreased by using a large number of computers to
simultaneously solve the same problem, which is the so-called
parallel computing. The data is partitioned to the smaller data
and assigned to each processor for performing the same function.
This kind of parallel computing is called the Single Instruction
Multiple Data (SIMD) architecture
In this paper the combination of the parallel computing and
the multigrid method on the Navier-Stoke solver is presented. The
solver code is carried out for the steady laminar flow in a two-
dimensional cavity. The numerical solutions are then compared
with Ghia, Ghia, and Shin [1]. To take into account the data
synchronization, interchanging and updating the boundary data at
the interface between a pair of adjacent processors are essential.
Therefore, the Message-Passing Interface (MPI) library is used in
the present work to perform such a task.
2. Governing Equations
The present work is focused on the incompressible laminar
flow which is governed by the continuity and Navier-Stokes
equations where all the fluid properties are treated to be constant.
2.1 Continuity Equation
The continuity equation is derived from one of the basic laws
of physics: if there is no mass generated in the control volume,
the amount of mass entering the control volume has to be equal
to the amount of mass passing out of the control volume.
()
0
j
j
u
xρ
∂=
∂ (1)
where is the fluid density and ρ
j
u are the velocity components.
2.2 Navier-Stokes Equations
The second law of Newton is all about the conservation law of
momentum. Applying the conservation law of momentum over a
fluid particle and treating it as the Newtonian fluid, the Navier-
Stokes equations for the steady incompressible laminar flow can
be derived in the form of tensor
() i
ji
j
jj
up
uu
i
x
xx
ρµ
∂∂∂
=−
∂∂∂
x
∂
∂
(2)
where is the viscosity and µ
p
is the pressure.
3. Numerical Method
In this work, the finite volume method is employed to
numerically solve the governing equations which consist of the
continuity and Navier-Stokes equations. The numbering system E,
W, N, and S are referred to East, West, North, and South
respectively.
Integrating the continuity equation over the control volume
gives
−+−=0
ew ns
FFFF (3)
where , , , and
.
( )
ee
Fuρ=
)
ss
( )
ww
Fuρ=( )
n
Fvρ=n
S
e
n
S
(
Fvρ=
Integrating the Navier-Stokes equations over the control
volume with the second-order central differencing scheme for the
diffusive terms and with the first-order upwind scheme for the
convective terms, the standard form of the finite volume equation
can be arranged as
PP E E WW NN SS
aaa aaφφφ φφ=+ +++ (4)
where aD ,
max(0, )
Ee F=+ −
aD ,
max( ,0)
Ww w
F=+
aD ,
max(0, )
Nn F=+ −
max( ,0)
Ss s
aD F=+
PEWN
aaaaa=+++
iP
p
Sd
x
∂
=−
∂
V
and (streamwise and cross-stream velocities
respectively).
uandvφ=
Once the continuity and Navier-Stokes equations have been
discretised, the Semi-Implicit Method for Pressure-Linked
Equation (SIMPLE) is employed to solve these discretised
equations in order to avoid the decoupling problem between
velocity and pressure fields. The collocated grid arrangement is
used in this work so that all variables are stored at the center of
each control volume. The Rhie-Chow interpolation is the process
of determining the mass flux that entering into and passing out of
the control volume to take into account for the nonlinear pressure
However, in the present work, the Rhie-Chow interpolation is not
used. The SIMPLE algorithm [2] for the simulation of steady
incompressible laminar flow can be summarized as follows:
(1) Initialize the velocity and pressure with the guessed
values.
(2) Solve the Navier-Stokes equations for the velocity
field.
(3) Solve the pressure correction equation for
′
p
.
(4) Correct the pressure and velocity by the pressure
correction
′
p
.
(5) Repeat steps (2)-(4) until the solution converges.
4. Multigrid Method
One drawback of conventional iterative methods is that they
cannot eliminate the low-frequency component of error effectively.
However, the low-frequency error on the fine grid appears as the
high-frequency error on the coarser grid [3]. Therefore, it is
worthwhile to eliminate the high-frequency error on the coarser
and coarser grid and then evaluate the corrections to correct all
the way up to the next finer and finer grid solutions. This strategy
is called the multigrid technique.
The multigrid method is the combination of the smoothing
process, in which the equations on each grid level are solved, and
the restriction and prolongation processes, which transfer the
current approximate solutions, equation residuals and correction
quantities between adjacent grid levels: the restriction transfers
the data down to the coarser grid and the prolongation transfers
the data up to the finer grid. The multigrid scheme specifies how
the coarse-grid problem is generated from the fine-grid problem
and what order the multiple grid levels are visited, i.e., the cycle
types, for example, the multigrid V-cycle is the one where a
recursive algorithm has the following steps: pre-smoothing
iterations, restriction to a coarser grid, solving the coarse grid
problem for coarse grid corrections, prolongation of corrections to
the fine grid, and post-smoothing iterations. The problems are
solved up and down in the same fashion which leads to the V
shape of the cycle, as shown in Fig. 1, the number of pre- and
post-smoothing iterations are specified inside the circles. The
number of iterations is fixed for any appropriate value and is the
same in the pre- and post-smoothing iterations but may be
different if the optimization is required.
G1
G2
G3
G4
Fig. 1 Multigrid V-Cycle
In the case of linear problems, a multigrid correction scheme
(CS) can be used effectively because of the solution errors are
directly proportional to the solution residuals. Therefore, only the
solution residuals are restricted to the next coarser grid to solve
the coarse grid problem for evaluating the corrections. For
nonlinear problems, however, the CS treatment cannot be
employed. For example, consider a system of nonlinear algebraic
equations, A(u) = f, where A is the matrix coefficient of vector u
and f is the vector of source term. Suppose that v is an
approximation to the exact solution u. If the solution error is
simply e = u – v and the residual is r = f – A(v). Substract the
original equation from the definition of the residual to give A(u) –
A(v) = r. Applying the nonlinear operator to the definition of the
error which can be written as A(e) = A(u - v). Since A is
nonlinear, the vector u and v cannot be easily split out of the
operator A, hence A(u - v) ≠ A(u) - A(v). Moreover, the relation
A(e) = A(u) – A(v) = r is no longer valid unlike linear problems.
Therefore, there is no simple linear residual equation and thus the
correction scheme cannot be used to solve the nonlinear
problems.
As stated earlier, the linear equation is possible to “transfer
the problem” from one grid to another by merely transferring the
residual. If the equation is nonlinear, the transfer of residual alone
is generally not possible [3]. Thus, in the case of nonlinear
equation, the solution must be transferred together with the
residual. This version of multigrid method is known as the full
approximation storage (FAS) scheme. The detailed derivation of
the FAS can be found in [4], this paper illustrates only three levels
of the FAS algorithm in which there are the following steps:
• Solve the finest-grid problem, , and then restrict
the current approximation and its residual to the intermediate
coarser grid:
( )
hh h
Av f=
())
hh
Av
22
(
hhh
h
rIf
=
−and v .
22hh
h
I=h
v
h
• Solve the intermediate-grid problem, 22 2
()
hh
Av f
=
, and
then restrict the current approximation and its residual to the
coarsest grid: 332 2
2(
hhh h
h
rIfA
2
())
h
v
=
−and 3
2
h
vI
3h
hv
2h
=
,
where 222 2
()
hhhhh
h
IvrfA
=
+.
• Solve the coarsest-grid problem, , and then
compute the coarsest-grid approximation to the error:
33 3
()
hh
Av f=h
2h333
2
hhh
h
evIv
=
−, where .
3332
2
()
hhhh
h
fAIv=+
3h
3h
r
Up to now, the restriction process has completed and the
further steps are the prolongation process:
• Interpolate the coarsest-grid error approximation up to the
intermediate grid and correct the current approximation of
such a grid: 222
3
hhh
h
vvIe
=
+.
• Solve the intermediate-grid problem, 22 2
()
hh
Av f
h
=
, with
the last updated solution as the initial guess and with the
same source term as in the restriction process and then
compute the intermediate-grid approximation to the error:
222hhh
h
evIv
h
=
−.
• Interpolate the intermediate-grid error approximation up to
the finest grid and correct the current approximation of such
a grid: 2
2
hhh
h
vvIe
h
=
+.
To incorporate the multigrid technique with the Navier-Stokes
equations, some special treatment must be taken carefully. The
velocity components are nonlinear but pressure is linear and both
the velocities and pressure appear in the system of equations.
Therefore, the pressure is solved through the pressure correction
equation by the multigrid correction scheme and the FAS is used
for the solution of the velocity components. The current
approximation of the velocity components, u and v, as well as the
residual of the momentum and pressure correction equations are
then restricted to the next coarser grid. Once a coarse grid has
been visited, the coarse-grid pressure is initialized with the
guessed value of zero every time, and all the coefficients together
with the mass flux must be recalculated on this grid with the
restricted solution, and then the process of the SIMPLE algorithm
is proceeded for a few iterations with the same steps as in a
single-grid problem. As the restriction process has gone down to
the coarsest grid and the coarsest-grid problem is solved, the
change in the velocity components and the current solution of the
pressure correction are then prolonged up to the next finer grid for
correcting the fine-grid current approximation of pressure and
velocity components: u and v with the change in velocity
components that prolonged from a coarse grid; p with the
prolonged pressure correction. Thereafter, the same operation as
the restriction process is repeated again, but the way is up
instead of down and the last updated solution is used as the initial
guess as well as the pressure which is initialized to zero on the
restriction process, and hence the change in velocity components
which is prolonged up to the next finer grid is the difference
between the current approximation and the restricted solution at
the restriction process. This is the multigrid V-cycle and one cycle
has been completed as the process goes up to the finest grid, the
bilinear interpolation is used to restrict the velocity components or
prolong the corrections and the residuals are restricted by simply
summing the residuals of the four fine grid control volumes that
make up each coarse grid control volume.
5. Parallelization Method
The basic idea of the parallel computing is that a number of
processors work in cooperation on a single task. For a distributed
memory architecture each processor possesses its own memory
and connects to another ones through the high-speed
interconnection network. This is called a cluster of computers.
Each processor can access only its own memory space.
Therefore, sending data into and receiving data from other
processors are carried out by means of a message-passing
interface. The message passing is done by using the MPI library,
which is portable to most computers [5]. The strategy of parallel
computing can be classified into two categories: data
parallelization and task, or function, parallelization. The data
parallelization is to partition the overall data into a number of sub-
data. The number of sub-data is equal to the number of
processors. Each sub-data is assigned to each processor where
each processor performs the same task or the same code on its
own sub-data. This strategy is called the Single Instruction
Multiple Data architecture (SIMD). The other is the functional
parallelization on the Multiple Instruction Multiple Data (MIMD)
architecture, which assigns different tasks or functions to different
data on different processors. In this case, each processor
possesses the same amount of data (or can be different) and this
data is performed with the function that is different from the
functions that are performed by other processors.
In the present work, the data parallel computing is chosen
because it is more appropriate for the present CFD algorithm than
the functional parallel computing. This is because the functions
used in the present algorithm are solved simultaneously. It should
be noted that if two computers possess the same name of
variable, and if the variable in the first computer is changed, the
second computer will not sense that changing as long as the
message is not sent to inform and exchange such a variable.
Thus, if the CFD code in the present work is operated on the
functional parallel computing, the message has to be sent every
time for every calculated data point. This gives rise to the time
overhead that increases the time consumption.
5.1 Parallel Algorithm
The parallel computing model should be discussed before
considering the parallel algorithm. In this paper the model of
parallel computing is the master-slave model [6]. The master is
assigned to initialize the necessary data such as boundary and
initial conditions and send this data to all slaves and the master
itself. The slaves receive the corresponding data from the master
and each slave carries out its own data with the same code as
the sequential algorithm does (see Fig. 2).
The algorithm of the parallel computing used in the present
work can be summarized as follows:
The master:
1) Determine the number of processors to be used in the
calculation and then evaluate the size of each sub-data
that is the size of the overall data divided by the
number of processors.
2) Allocate the memory space for the overall data and for
each sub-data.
3) Specify the physical boundary and initial conditions on
the overall data.
4) Send the initialized overall data to the corresponding
slave sub-data.
5) Perform the solver code on each sub-data until the
maximum error of all processors reaches the setting
criteria. In this step, the error of each processor has to
be interchanged among all processors to find the
maximum error. In addition, the data at the boundaries
of the adjacent processors have to be exchanged to
synchronize the data only on the finest grid.
6) Transfer each sub-data directly to its corresponding
location in the overall data.
7) Display the final results of the overall data (if
necessary).
The slave:
1) Determine the number of processors to be used in the
calculation and then evaluate the size of each sub-data
that is the size of the overall data divided by the
number of processors.
2) Allocate the memory space for the corresponding sub-
data or sub-domain.
3) Receive the corresponding location of the initialized
overall data from the master.
4) Perform the solver code on each sub-data until the
maximum error of all processors reaches the setting
criteria. In this step, the error of each processor has to
be interchanged among all processors to find the
maximum error. In addition, the data at the boundaries
of the adjacent processors have to be exchanged to
synchronize the only on the finest grid.
5) Send each sub-data back to the corresponding location
in the master overall data.
As all processors have executed the solver code, the
sequential steps of all processing nodes, as shown in Fig. 3, are
run concurrently. However, some processors may be idle if they
have been waiting for the messages from a pair of SEND-
RERCIEVE processors that have not sent the data. Since the
present work uses the standard blocking send/receive MPI
routine, the receiver will not be able to receive the data and has
to wait if the sender does not send any. This is the disadvantage
of poor design of parallel algorithm.
5.2 Communication Strategy
In step (5) of the master-processing node and step (4) of the
slave-processing node, the calculation has been carried out on
each sub-domain using the same code. In case of the sequential
code of two-dimensional problems, there are four physical
boundaries. It should be noted that the parallel code is the copy
of the sequential code; therefore, each sub-domain also has four
boundaries. The exchange of the data at the interface of the pair
node is necessary to avoid the fictitious internal boundary.
Master
Master Slave 1 Slave 2 Slave p-1
012345678910nx-1
Master send/receive data to/from itself and to/from all slave before /after calculation
012
nx/p
nx/p+1
nx/p
nx/p+1
nx/p
nx/p+1
nx/p
nx/p+1
012 012 012
nx = number of point in x- coordinate or the number of column in this figure
p = the number of processor include master and slave
changing data occurs after finish the calculation in each iteration
Fig.2 A model of parallel computing
P0
P0
P0
P0
P1
P1
P1
P1
P2
P2
P2
P2
P3
P3
P3
P3
P0 Master initialize the overall data
Master send data to slave
Exchange boundary data
Exchange boundary data
Calculation for 1 iteration
Calculation for 1 iteration
P1 P2 P3
P0
Slave send sub-data to the
master overall data
Fig.3 A sequence of each processor after executing a program
Data allocation for each sub-domain is depicted in Fig. 3. It
can be seen that the number of columns for each sub-domain
after partitioning is incremented by two columns (the gray
columns). These are called the ghost cells. It is served as the
boundary for each sub-domain in each node. The first location of
computing is the second column and now it is no longer the
boundary, likewise the last location of computing is not the last
column but it is the column before the last column (it is no longer
the boundary as well). Let us consider the exchange of the data
of the slaves 1 and 2. It occurs after the computing has done per
iteration. The first column of the slave 2 is treated as its left
boundary which requires the data from the fourth column of the
slave 1. Also the last column of the slave 1 is treated as its right
boundary which requires the data from the second column of the
slave 2. The others can be interpreted in a similar manner except
for the first column of the sub-domain of the master and the last
column of the last slave that are left empty. Therefore, it can be
seen similar to the sequential code, i.e., the second column in
Fig. 3 (number 1) of the sub-domain of the slave 1 which is the
fourth column (number 3) of the overall data of the master. It
seems as if the data of all slaves treated the same as the data of
the primary domain. On the other hand, the sixth and seventh
columns (number 5 and 6) of the overall domain of the master, if
performed by the sequential code, are updated immediately when
the computing is operated. However, the last column of the slave
1 (the right boundary of the slave 1) and the first column of the
slave 2 (the left boundary of the slave 2) are updated per
iteration. Therefore, they are still the semi-fictitious internal
boundary conditions. This causes the solution convergence to
slowdown.
7. Results and Discussion
A lid-driven cavity flow at Re=1000 is used as a test problem.
Fluid with u=1.0 m/s and density=1.0 kg/m3 flows past over the
top of the cavity as shown in Fig. 4 which depicts the velocity
contour. The computed u-velocity along the vertical centerline and
the computed v-velocity along the horizontal centerline of a
square cavity are compared with the results of Ghia et al [1]. The
results are agree very well with the reference data. Fig. 6 shows
the computing time used by the multigrid (MG) and the single grid
(SG) method, the single grid method uses about 5 times as many
as the multigrid method does.
6. Parallel Multigrid Implementation
There are two approaches to incorporate the parallel
computing with the multigrid method [7]: the domain
decomposition with multigrid and the multigrid with grid
partitioning. The domain decomposition is first applied to the finest
grid. Then, the multigrid method is used to solve problems inside
each block, that is, the multigrid V-cycle is applied in each local
finest grid. Since inter-domain connection is limited to the finest
level, thus communications are only required at this level. For the
multigrid with grid partitioning, however, the multigrid method is
used to solve the problem in the whole grid in which the grid is
partitioned among processors at each level. Therefore,
communications are required at each level
Fig.4 Velocity contour at Re=1000
Domain decomposition methods are often used with the finite
element method on parallel computers [8]. They are easier to
implement and require fewer communications only on the finest
grid. Additionally, they can be applied to general multiblock grids.
On the other hand, the domain decomposition methods lead to
algorithms which are numerically different from the sequential
version and have a negative impact on the convergence rate. Grid
partitioning retains the convergence rate of the sequential
algorithm. However, it requires more communication overhead
because the data exchange at each grid level. Several literatures
adopted the grid partitioning technique, for example, [8, 9], but in
the present work the domain decomposition is employed.
Fig. 5 u- and v-velocity at Re=1000 are compared with the
reference data
In order to test the parallel multigrid scheme, requested for
parallel machine, all cases are run and tested on the 8 single-
processing nodes of CAMETA cluster located at Computational
Fluid Dynamics Laboratory, Suranaree University of Technology,
Nakhon Ratchasima, Thailand. In Fig.7, the residuals are plotted
versus the computing time which are used by the multigrid without
parallel and the parallel multigrid method with 2, 4 and 8
processors respectively. It is found that the parallel computing
with the multigrid method does not improve the overall
computation performance. However, with 8 processors the time
consumed by parallel computing with the multigrid method is
slightly less than a single processor with the multigrid method.
This is because the number of grid points is large enough. The
multigrid technique is deteriorated in convergence rate when
using with the parallel computing because of the effect of the
internal boundary. It is observe in Fig. 7 that the line of the
parallel computing have strongly fluctuation with respect to time,
but the line of the multigrid method without parallel computing is
nearly smooth. It can be concluded that the oscillation occurred
is the effect of the internal boundary. Moreover, this work employs
the domain decomposition with the multigrid. Only the finest grid
is taken into account the synchronization of data, which
possesses the disadvantage of the rate of convergence. However,
the parallel version with 8 processors has overcome such a poor
rate of convergence, because the computing time is so fast that
the influence of internal boundary is diminished. It is interesting to
think further that if the larger the number of grid points and the
larger the number of processors are used, how much faster (or
slower) the parallel multigrid scheme is compared with the
multigrid method without parallel computing, this is left to study
further and since the present work is limited only with 8 number of
processors(nodes). Table 1 reports the time consumption of the
parallel computing with and without the multigrid method until the
solution converges. The parallel single grid method is about 7
times slower than the parallel multigrid method.
Table 1 Comparison of parallel with and without multigrid
Number of
Processors
SG
(time in s)
MG
(time in s)
2 2.349704e+04 3.338826e+03
4 1.189300e+04 1.613995e+03
8 6.338385e+03 9.328902e+02
Fig. 7 Multigrid with and without parallel computing
8. Conclusions
The multigrid method with and without the parallel computing
and the single grid method with and without the parallel
computing are studied. The program is firstly validated by
comparing the results at Re=1000 with [1] and very good
agreement is found. Without parallel computing, the efficiency of
the multigrid technique is estimated by comparing the computing
time of the single grid against multigrid, the multigrid technique
uses about 5 times less computing time than the single grid. The
performance of parallel multigrid is evaluated by comparing the
time consumption of the multigrid technique with a single
processor versus the single grid parallel computing. With parallel
computing, the multigrid method is about 7 times faster than the
single gird method. However, the parallel multigrid with 8
processors is slightly faster than the sequential multigrid
algorithm. For the parallel multigrid with 2 and 4 processors, the
sequential multigrid algorithm is faster than the parallel multigrid.
In brief, this present work with limited resource(hardware) and the
parallel computing in conjunction with the multigrid technique can
improve significantly an overall efficiency. The efficiency can be
improved by increasing the number of processing nodes and an
enhancement of the parallel multigrid scheme still has been
Fig. 6 Multigrid and single-grid result on a single processor
carrying on until to reach an extreme performance. In the future
work, the study will be focused on the 3D turbulence flow
simulation and the number of processing nodes is up to 16 nodes
and hence the parallel multigrid computation will be improved to
acquire a high efficiency as possible as.
9. Acknowledgement
The present work is financially supported by the National
Electronics and Computer Technology Center (NECTEC). This
support is greatly appreciated.
10. References
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incompressible Flow using the Navier-Stokes Equation and a
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