Gauss-Seidel迭代，OpenMP并行化本质上和串行不一样，但是是收敛的

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已于 2025-05-21 02:24:28 修改

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文章标签：自动性能优化 HPC Fortran

于 2025-05-20 22:52:43 首次发布

本文链接：https://blog.csdn.net/anlongstar/article/details/148100209

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1）代码

      PROGRAM OPENMP_GZ
      INCLUDE 'omp_lib.h'
      INTEGER, PARAMETER :: N = 101
      INTEGER I,J,L,MAXIT,T,S
      REAL(8), ALLOCATABLE, DIMENSION(:,:):: W,X,Y
      REAL(8) ERROR, EPS, T1, T2, dX, dY, C
      MAXIT = 100000 !最大迭代次数
      EPS = 1E-6 !收敛精度
      ERROR = 1E6 !初始误差
      L = 1
      ALLOCATE (W(N,N),X(N,N),Y(N,N))

      OPEN(1, FILE='101.msh')!从文件读取网格数据
      DO J=1,N
          DO I=1,N
              READ(1,*) X(I,J), Y(I,J)
          ENDDO
      ENDDO
      CLOSE(1)

      W(2:N-1,2:N-1) = 0.0! 设置边界条件
      DO J=1,N
         W(1,J) = EXP(-1.5*(X(1,J))**2 + 2*(Y(1,J))**2)
         W(N,J) = EXP(-1.5*(X(N,J))**2 + 2*(Y(N,J))**2)
      ENDDO
      DO I=1,N
         W(I,1) = EXP(-1.5*(X(I,1))**2 + 2*(Y(I,1))**2)
         W(I,N) = EXP(-1.5*(X(I,N))**2 + 2*(Y(I,N))**2)
      ENDDO

      T1 = OMP_GET_WTIME()
      DO S=1,100000000
      DO WHILE (L .LE. MAXIT .AND. ERROR .GT. EPS)
        ERROR = 0.0
!$OMP PARALLEL PRIVATE(I,J,dX,dY,C)  ! PARALLEL DO 指令允许指定哪些循环应由编译器并行化
!$OMP DO REDUCTION(MAX:ERROR)        ! DO 循环，通过 REDUCTION 计算最大误差
        DO J = 2, N-1
          DO I = 2, N-1
            dY = Y(I,J+1) - Y(I,J)
            dX = X(I+1,J) - X(I,J)
            C = W(I,J)
            W(I,J) = 0.5 * (dY*(W(I-1,J) + W(I+1,J)) + dX*(W(I,J-1) + W(I,J+1)) - 0.2*dX*dY)/(                                                         dX + dY)
            ERROR = MAX(ERROR, ABS(W(I,J) - C))
          END DO
        ENDDO
!$OMP END DO    !END PARALLEL DO 指令标记在 PARALLEL DO 指令中指定的 DO 循环的结束
!$OMP END PARALLEL
        L = L + 1
      END DO
      END DO
      T2 = OMP_GET_WTIME()
      PRINT *, 'computational time: ', T2-T1, 's'
      PRINT *, L

      OPEN(2,FILE='Res.dat')  ! 输出数据到文件
      DO J=1,N
        DO I=1,N
            WRITE(2,*) X(I,J), Y(I,J), W(I,J)
        END DO
      END DO
      CLOSE(2)

      DEALLOCATE (W,X,Y)
      END PROGRAM

2）编译命令

gfortran -fopenmp -o pa.exe Gauss–Seidel_parallel.f90

3）输入数据

见附件

3）运行结果

3000-3100行的数据：

串行：

0.40000000000000002 -0.41999999999999998 0.94283719429734447
0.41999999999999998 -0.41999999999999998 0.91517049547914420
0.44000000000000000 -0.41999999999999998 0.88692669347676489
0.46000000000000002 -0.41999999999999998 0.85820552287434693
0.47999999999999998 -0.41999999999999998 0.82910918519491994
0.50000000000000000 -0.41999999999999998 0.79974214641266483
0.52000000000000002 -0.41999999999999998 0.77021094585209326
0.54000000000000004 -0.41999999999999998 0.74062401904255837
0.56000000000000005 -0.41999999999999998 0.71109153718516127
0.57999999999999996 -0.41999999999999998 0.68172526596580540
0.59999999999999998 -0.41999999999999998 0.65263844650670655

并行：

0.40000000000000002 -0.41999999999999998 0.94283837147741767
0.41999999999999998 -0.41999999999999998 0.91517164464449596
0.44000000000000000 -0.41999999999999998 0.88692781352135375
0.46000000000000002 -0.41999999999999998 0.85820661272195631
0.47999999999999998 -0.41999999999999998 0.82911024380019038
0.50000000000000000 -0.41999999999999998 0.79974317276209350
0.52000000000000002 -0.41999999999999998 0.77021193896500240
0.54000000000000004 -0.41999999999999998 0.74062497797203064
0.56000000000000005 -0.41999999999999998 0.71109246101893997
0.57999999999999996 -0.41999999999999998 0.68172615382715873
0.59999999999999998 -0.41999999999999998 0.65263929755525862

串行：

10000-10010

-1.0000000000000000 0.97999999999999998 1.5231796120207264
-0.97999999999999998 0.97999999999999998 1.6350407390271877
-0.95999999999999996 0.97999999999999998 1.7454097502627000
-0.93999999999999995 0.97999999999999998 1.8566927732738419
-0.92000000000000004 0.97999999999999998 1.9699458586500573
-0.90000000000000002 0.97999999999999998 2.0856822347163213
-0.88000000000000000 0.97999999999999998 2.2041598308409540
-0.85999999999999999 0.97999999999999998 2.3254975194154395
-0.83999999999999997 0.97999999999999998 2.4497279948182107
-0.81999999999999995 0.97999999999999998 2.5768244045233648
-0.80000000000000004 0.97999999999999998 2.7067149417711622

并行：

-1.0000000000000000 0.97999999999999998 1.5231796120207264
-0.97999999999999998 0.97999999999999998 1.6350407359076693
-0.95999999999999996 0.97999999999999998 1.7454097440298204
-0.93999999999999995 0.97999999999999998 1.8566927639368240
-0.92000000000000004 0.97999999999999998 1.9699458462211767
-0.90000000000000002 0.97999999999999998 2.0856822192108919
-0.88000000000000000 0.97999999999999998 2.2041598122773096
-0.85999999999999999 0.97999999999999998 2.3254974978149128
-0.83999999999999997 0.97999999999999998 2.4497279702051098
-0.81999999999999995 0.97999999999999998 2.5768243769249448
-0.80000000000000004 0.97999999999999998 2.7067149112175963

5）分析

数据类型是双精度的，

REAL(8), ALLOCATABLE, DIMENSION(:,:):: W,X,Y

下面的代码存在强依赖，就是W(I,J)依赖于W(I-1,J)和W(I,J-1)

W(I,J) = 0.5 * (dY*(W(I-1,J) + W(I+1,J)) + dX*(W(I,J-1) + W(I,J+1)) - 0.2*dX*dY)/(dX + dY)

尽管迭代会逐步减少误差到满足收敛要求

DO WHILE (L .LE. MAXIT .AND. ERROR .GT. EPS)

6）

确实也是可以并行，但是只是结果不一样

根据EP判断收敛，确实是可以的。