The single instruction stream, multiple data stream Massively Parallel Processor (MPP) unit consists of 16,384 bit serial arithmetic processors configured as a 128 x 128 array whose speed can exceed that of current supercomputers (Cyber 205). The applicability of the MPP for solving reaction network problems is presented and discussed, including the mapping of the calculation to the architecture, and CPU timing comparisons

Carmichael, Gregory R.

Cho, Seog Y.

English

NASA Technical Reports Server

N87-26550
CHEMICAL NETWORK PROBLEMS SOLVED ON
NASA/GODDARD'S MASSIVELY PARALLEL PROCESSOR COMPUTER
Seog Y. Cho and Gregory R. Carmichael
Department of Chemical and Materials Engineering
University of Iowa
Iowa City, Iowa 52242
David M. Cohen and Mehmet H. Oguztuzun
Department of Computer Science
University of Iowa
Iowa City, Iowa 52242
ABSTRACT
The single instruction stream, multiple
data stream MPP processor consists of
16,384 bit serial arithmetic processors
configured as a 128 x 128 array whose
speed can exceed that of current
supercomputers (Cyber 205). This paper
presents and discusses the
applicability of the MPP for solving
reaction network problems including the
mapping of the calculation to the
architecture, and CPU timing
comparisons.
Keywords: Chemical Net Work Problems,
Eulerian Transport/Chemistry Model, Air
Pollution Model, Coupled ODE-IVP.
I_RODUCTIO_
A detailed model which describes the
transport and removal of photochemical
oxidants, and acidic species and
precursors in the troposphere has been
under development for the past nine
years. The present analysis consists
of about 30 coupled three-dimensional
time-dependent non-linear partial
differential equations and about 50-100
coupled non-linear ordinary
differential equations.
The model is representative of a number
of comprehensive Eulerian
transport/chemistry models being
developed for regional air pollution
problems. However, these models are
only feasible when run on
"supercomputers". Our model was
developed on the NASA-Langley CDC-STAR
computer and is currently being
exercised on the NASA-Langley Cyber
205, the NCAR CRAY-I and a FAC M240 in
Nagoya University, Japan. The
execution times are 0.025
CPU-sec
grids-time step
on the FAC _ _0,
CPU-sec
0.007 on the
grids-time step
CPU-sec
CRAY-I, and 0.70 on a
grids-time step
VAX 11/780. Thus, a 24-hour simulation
on the CRAY-I for the eastern United
States with 9500 grid points requires
100 CPU-minutes.
Our experience has shown that
transport/chemistry models can execute
about 70-100 times faster on the
"supercomputers". However, 100 CPU-
minutes/simulation-day is still too
large for most applications. Since
typical applications require
simulations of seven to ten days.
Therefore, to exercise these models
various simplifying assumptions are
used to decrease the CPU time.
However, these assumptions add
additional errors and uncertainties to
the model results. Faster computers
will enable the execution of the "best-
science" model version.
Currently about 90% of the CPU time is
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spent doing the chemistry
calculations. The chemistry introduces
the stiffness, the coupling, and the
non-linearity into the model. Thus,
the highest priority in continued model
development is to search for ways in
software and hardware to reduce the
chemistry calculation. The purpose of
this paper is to describe our attempts
to exploit massively parallel computer
architectures to accelerate the
chemistry calculations.
MODEL OVERVIEW
The regional-scale combined
transport/chemistry/deposition model is
Eulerian and treats 50 chemical
species. Thirty species are advected,
while the remaining species are short-
lived and are modeled using pseudo-
steady state methods. The mathematical
analysis consists of partial
differential equations for the advected
species and additional algebraic
equations for the steady-state
species. The advected species satisfy
_C.
1
-- + V (VC0) = V.K.VC
(i)
+ R. + S -G., i = i,...,30;
l 1 l
where C i is the gas-phase concentration
of the ith chemical species, V is the
wind velocity vector, K is the eddy
diffusivity tensor, R i denotes the
chemical reaction term, S i is the
source term, and G. is used to describe
the mass transfer _etween the gas and
condensed phases. The algebraic
equations for the gas-phase species
assumed to be at steady state are
written as
Ri(C I, C2,...,C50) = 0,
i = 31, ...,50.
(2)
These equations are representations of
general chemically reactive flow
problems.
166
Simulation of regional transport,
chemistry and deposition as described
by Equations (i) and (2) requires
numerical integration. The method
presently used is a combination of the
concept of fractional time steps and
one-dimensional finite elements. This
is referred to as Locally One-
Dimensional, Finite-Element Method
(LOD-FEM). The LOD procedures
(Mitchell, 1969) split the multi-
dimensional partial differential
equation into time dependent, one-
dimensional problems which are solved
sequentially. The transport equations
are solved using a Crank-Nicolson
Galerkin finite element technique.
Chemistry and mass transfer equations
are solved using an adaptation of the
semi-implicit Euler method proposed by
Preussner and Brand (1981).
SCOPING STUDIES
Test Problem I: Chemical Network
Problem
To evaluate the ability of the MPP to
calculate chemical network applications
a simple four species test problem was
selected. The four species (C I, C 2,
C 3, C 4) are involved in the following
chemical reactions:
k I
C I + C2 ----+ C 3 (a)
k
I
C 3 ----+ C I + C 2 (b)
(c)
(d)
The transport equations describing this
system is represented by Eq(1) with
i=1,2,3 and 4.
As mentioned in the model overview
section, one way of numerically solving
complex transport chemistry network
problems is to split the equation into
transport and chemistry parts. The
chemistry calculations using this
technique requires solving the set of
equations
3C.
1
m = R0
_t l
i=I, ...,
# of species
(5)
at each grid point in the discretized
space.
The use of the semi-implicit Euler
method to solve Eq(5) results in the
equations
dC
i djd--C-c [ H ck1 . 1
r 3 k= I
+ 7 p£. _ C
1 m
£ m
i=l, • • .4
(6)
This set of ODE-IVP's is solved within
each transport time step, i.e.,
to ( tr _ ttransport "
Now consider the case when we have
16,384 grid points in the discretized
spatial domain. Therefore each
chemical calculation within each
transport step requires the solution of
16,384 sets of Eq(6). To implement the
solution of these equations on the MPP
requires first the choice of how to map
the equations to the architecture. In
this case we have chosen to simply view
each processor as a grid point in the
discretized space, and to have each
processor solve its own set of Eq(6).
The algorithm for solution of Eq(6) is
written in Parallel Pascal and resides
on the VAX. The initial conditions and
constants are distributed to each
processor and the algorithm is executed
on the MPP and output is sent back to
the VAX.
The CPU time required for execution of
100 time steps on the MPP of this 4
species mechanism at 16,384 grid points
is 0.293 CPU-seconds. The same problem
was executed on the VAX-11/780 and
required 138 CPU-seconds. Thus for
this chemical network problem the MPP
executed a factor, of 470 times faster
than VAX 11/780!!
Test Problem 2: NO x Transport in
Eastern United States
To test combined transport/chemistry
network problems on the MPP a 3-
dimensional test problem describing the
transport and chemistry of NO, NO 2, 03 ,
and HNO 3 in the lower troposphere was
selected. The governing set of
equations is given by Eqn. (i). An
oversimplified chemical mechanism is
used in this test calculation, i.e.,
NO + 03 + NO 2 + 02 (e)
+
NO 2 + h_+o 2 NO + 0 3 (f)
NO 2 + OH ÷ I-INO 3 (g)
(The OH concentrations are given by an
empirical formula and the 02
concentrations are assumed constant.
Sample results for this test problem
calculated on a VAX 11/780 are shown in
Figures 1 and 2. Presented are the NO
x
emissions for the eastern United
States, and the 24-hour averaged
predicted surface concentrations of NO,
NO 2, and HNO 3. The meteorological
conditions simulated are those of July
4, 1974. The grid system used in the
simulation was 32 x 32 x 16, and a
transport time step of 15 minutes and a
chemistry time step of 1 second was
used.
This combined transport/chemistry
problem is currently being run on the
MPP. There are two choices for mapping
this problem to the MPP. One method is
to perform the chemistry calculations
on the MPP and the transport part of
the calculation on the VAX. This
167
AY ':' .2
Figure i. The emission of NO x at July 4, 1974 at surface.
0 100 _,
a) NOx b) HNO 3
Figure 2. Averaged concentration of July 4, 1974 at surface.
168
method is currently being tested. The
other method is to perform the entire
calculation on the MPP. We are
currently developing an algorithm to
solve the sets of tridiagonal matrices
on the MPP which arise from the
transport processes.
The above test calculation indicates
the MPP is well suited for chemical
network problems where each node can
hold the entire mechanism. Current
memory restrictions limit the size of
the chemical mechanism that can be
solved in this fashion. At present
each processor can hold 32 32-bit
variables. (It is planned to increase
the storage in the near future.)
However, it is possible to handle
larger chemical mechanisms. One way is
to group processors together. For
example if 128 words are required at
each node then four processors can work
together. This in turn would reduce
the maximum number of grids possible by
a factor of 4. Another way is to make
use of the staging memory.
SUMMARY
The suitablity of the MPP computer for
calculation of chemical network
problems is under evaluation. To date
the MPP has been used to calculate a
test problem which represents one
component of a sophisticated chemically
reactive flow problem. Specifically
the set of coupled ODE-IVP's describing
the chemical reactions occuring at
16,384 spatial grid points was
calculated. This problem is ideally
suited for the MPP because by using
operator splitting, the chemistry at
each grid point acts independently from
that of the other grids (within each
transport time step). This test
calculation showed that the MPP can
perform a 100-time-step calculation 470
times faster than the same calculation
on the VAX 11/780. Also since nearly
90% of CPU time of large chemically-
reactive flow problems is spent doing
the chemistry calculations, the MPP
architecture offers great potential for
CPU savings for model applications.
Coupled transport-chemistry problems
are now being tested on the MPP.
ACKNOWLEDGmentS
The authors wish to thank the staff of
NASA/Goddard, and specifically Jim
Abeles, for their assistance in helping
us use the MPP, the University of Iowa
Graduate College for providing funds to
carry out this project, and Chi-Chul
Yang, a graduate assistant in the
Department of Computer Science for
assisting with MPP programming.
REFEREMCES
lo Gabriel, R. P. "Massively Parallel
Computers: The Connection Machine
and NON-VON", Science, 231, 975-
978, 1986.
2. Hwang, K. and Briggs, F. A.
Computer Architecture and Parallel
3.
4.
Processing, McGraw-Hill, New York,
Table 6.3, 425, 1984.
Lerner, E. J. "Parallel Processing
Gets Down to Business",
Technolo_, July, 1985, pages 20-
28.
Mitchell A. R. Computational
Methods in Partial Differential
5.
Equations, John Wiley, New York,
1969.
Preussner P. R. and Brand K. P.
"Application of a Semi-Implicit
Euler Method to Mass Action
Kinetics", Chem Eng. Sci., 10,
1633-1641, 1981.
169


Chemical network problems solved on NASA/Goddard's massively parallel processor computer

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