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*Serial Jacobi Iterative Scheme*

###

A single-process implementation of the

Jacobi Scheme as applied to the Laplace equation is included below. Note that:

A single-process implementation of the

Jacobi Scheme as applied to the Laplace equation is included below. Note that:

- F90 is used.
- System size, m, is determined at run time.
- Boundary conditions are handled by subroutine bc.
- Pointer arrays c, n, e, w, and s point to parts of the solution space, u.

They are used to avoid unnecessary memory usage

as well as to improve readability. - This scheme is very slow to converge and is not used in practice.
- This code is shown primarily to provide a starting point for

subsequent introduction of parallel concepts. It also serves as

a starting point for convergence rate improvement ideas.

```
! Solve Laplace equation using Jacobi iteration method
! Kadin Tseng, Boston University, November 1999
MODULE jacobi_module
IMPLICIT NONE
INTEGER, PARAMETER :: real4 = selected_real_kind(6,37)
INTEGER, PARAMETER :: real8 = selected_real_kind(15,307)
REAL(real8), DIMENSION(:,:), ALLOCATABLE :: unew
REAL(real8), DIMENSION(:,:), ALLOCATABLE, TARGET :: u ! solution array
REAL(real8) :: tol=1.d-4, gdel=1.0d0
REAL(real4) :: start_time, end_time
INTEGER :: m, iter = 0
PUBLIC
CONTAINS
SUBROUTINE bc(u, m)
! PDE: Laplacian u = 0; 0<=x<=1; 0<=y u(1:m ,1:m ) ! i ,j Current/Central for 1<=i<=m; 1<=j u(1:m ,2:m+1) ! i ,j+1 North (of Current)
e => u(2:m+1,1:m ) ! i+1,j East (of Current)
w => u(0:m-1,1:m ) ! i-1,j West (of Current)
s => u(1:m ,0:m-1) ! i ,j-1 South (of Current)
CALL bc(u, m) ! set up boundary values
DO WHILE (gdel > tol) ! iterate until error below threshold
iter = iter + 1 ! increment iteration counter
IF(iter > 5000) THEN
WRITE(*,*)'Iteration terminated (exceeds 5000)'
STOP ! nonconvergent solution
ENDIF
unew = ( n + e + w + s )*0.25 ! new solution, Eq. 3
gdel = MAXVAL(DABS(unew-c)) ! find local max error
IF(MOD(iter,10)==0) WRITE(*,"('iter,gdel:',i6,e12.4)")iter,gdel
c = unew ! update interior u
ENDDO
CALL CPU_TIME(end_time) ! stop timer
PRINT *,'Total cpu time =',end_time - start_time,' x 1'
PRINT *,'Stopped at iteration =',iter
PRINT *,'The maximum error =',gdel
DEALLOCATE (unew, u)
END PROGRAM Jacobi
```