The document provides an introduction to parallel computing in Fortran, focusing on solving a one-dimensional steady-state heat transfer problem through a discretized system of equations. It details the implementation of both direct and iterative methods, specifically using the Jacobi and Gauss-Seidel techniques, along with MPI for parallel computation. A sample Fortran program is included, which demonstrates the structure of the code for implementing the solution with MPI communications and iterative processing of data across multiple processors.

1. Hands on intro to
parallel computing in
Fortran
time < 3 minutes

2. Problem
Simple algebraic system
• One dimensional steady state heat transfer
or
• Discretize on 1 dimensional grid
• At point i,
∇ ⋅ k(∇T) = 0 k
∂2
T
∂x2
= 0
Ti+1 − 2Ti + Ti−1
δx2
= 0
Th Tc
i=0 i
i-1 i+1
x
δx δx

3. Single domain
System of Equations
• For i=1 to N
• The equivalent system
Ti−1 − 2Ti + Ti+1
δx2
−2 1 0 0 . . . 0
1 −2 1 0 . . . 0
0 ⋱ ⋱ ⋱ . . . 0
0 . . . 0 1 −2 1
0 . . . 0 0 1 −2 N×N
T1
T2
⋮
Tn−1
Tn
N×1
=
−Th
0
⋮
0
−Tc
i=0 i
i-1 i+1
δx
δx
Th Tc
i=N+1

4. Single domain
Solution of the System
• Solve directly!
• Or use Jacobi or Gauss Seidel method
• For a given system of linear equations
• The iterative solution is
a11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
xnew
1 = xold
1 +
1
a11
[b1 − (a11xold
1 + a12xold
2 + a13xold
3 )]
xnew
2 = xold
2 +
1
a21
[b2 − (a21xold
1 + a22xold
2 + a23xold
3 )]
xnew
3 = xold
3 +
1
a31
[b3 − (a31xold
1 + a32xold
2 + a33xold
3 )]
a11 a12 a13
a21 a22 a23
a31 a32 a33
x1
x2
x3
=
b1
b2
b3
x1 x2 x3

5. <
Parallel Computation
MPI
i=0 i
i-1 i+1
δx
Th Tc
i=N+1
i
i-1 i+1
i=0 i=n+1
Processor 1 Processor 2 Processor M
i
i-1 i+1
i=0 i=n+1
n =
N
Mprocs
MPI Send
MPI Recv
n+2 nodes
Aqeel Ahmed

6. Parallel Computation
Fortran and MPI
1 program laplacian_1d_parallel
2 use mpi_f08 ! call the MPI library
3 implicit none ! always declare variables (no auto type deduction)
4
5 ! declare variables
6 integer :: points_total, npoints, i, j, k, l, iter_max
7 real, dimension(:), allocatable :: T, T_full
8 real :: T_initial, Tr, Tl, omega, size, Temp, R
9 integer myrank, nranks, tag, ierror
10 type(MPI_Status) :: istatus
11
12 ! initialize MPI
13 call MPI_INIT(ierror)
14 call MPI_COMM_SIZE(MPI_COMM_WORLD, nranks, ierror) ! get no. of procs
15 call MPI_COMM_RANK(MPI_COMM_WORLD, myrank, ierror) ! get rank
16
17 ! total number of points
18 points_total = 2400
19 ! number of points per processor
20 npoints = int(points_total/nranks)
21 ! allocate T with 2 ghost points T(1) and T(npoints+2)
22 allocate(T(npoints+2))
23 if (myrank == 0) then
24 allocate(T_full(npoints*nranks)) ! this is for gather only
25 end if
26
27 ! initial conditions and parameters
28 Tl = 25;Tr = 18;T_initial = 20;omega = 1.5;Temp = 0.0;iter_max = 500000
29 T = T_initial
30 R = 0.0
31
32 ! set boundary conditions
33 if (myrank == 0) then
34 T(1) = Tl
35 end if
36 if (myrank == nranks-1) then
37 T(npoints+2) = Tr
38 end if
39
MPI
speci
fi
c
variables
Variables
declarations
Domain
discretisation
parameters
Problem
inputs
Boundary
conditions

7. Parallel Computation
Fortran and MPI
40 ! iterative solution of the system
41 do k = 1, iter_max
42 ! trasfer data at processors boundaries
43 do i = 0,nranks-2
44 if (myrank==i) then ! send the 2nd last point to next processor 1st point
45 Temp = T(npoints+1)
46 call MPI_SEND(Temp, 1, MPI_REAL, myrank+1, i, MPI_COMM_WORLD, ierror)
47 elseif (myrank==i+1) then ! receive the 2nd last point from the previous proc as 1st point
48 call MPI_RECV(Temp, 1, MPI_REAL, myrank-1, i, MPI_COMM_WORLD, istatus, ierror)
49 T(1) = Temp
50 end if
51 end do
52
53 do l = 1,nranks-1
54 ! send the second point T(2) of the current proc to previous proc last point T(npoints+2)
55 if (myrank==l) then ! send the second point to previous procs
56 Temp = T(2)
57 call MPI_SEND(Temp, 1, MPI_REAL, myrank-1, l, MPI_COMM_WORLD, ierror)
58 elseif (myrank==l-1) then ! receive the second point from the next procs as last point
59 call MPI_RECV(Temp, 1, MPI_REAL, myrank+1, l, MPI_COMM_WORLD, istatus, ierror)
60 T(npoints+2) = Temp
61 end if
62 end do
63
64 ! solution using Gauss-Seidel method with Successive Over Relaxation (SOR)
65 do i = 2,npoints+1
66 R = 0.5*omega*(T(i+1) - 2*T(i) + T(i-1))
67 T(i) = T(i) + R
68 end do
69
70 ! set boundary conditions
71 if (myrank == 0) then
72 T(1) = Tl
73 end if
74 if (myrank == nranks-1) then
75 T(npoints+2) = Tr
76 end if
77 end do
78
79 ! put T from all procs to root (master) in T_full
80 call MPI_Gather(T(2:npoints+1), npoints, MPI_REAL, T_full, npoints, &
81 MPI_REAL, 0, MPI_COMM_WORLD, ierror)
82
83 call MPI_FINALIZE(ierror)
MPI
communications
Iterative solution