尝试在 OpenMP 中为 Fortran 并行化一个循环，以解决圆柱坐标系中的拉普拉斯算子

Question

Apologies if the code is not great I am fairly new. I have the following code and I am unsure how to make it parallel via OpenMP, I have tried leaving the array (phi_test) as a shared variable which gives me a correct answer for a printed point however runs much slower ('this is point (60, 50) = 221.84875522778384') and the final 6-7 decimals vary each time, I have also tried setting the array as a reduction as I thought with prior to setting the boundary conditions, phi_test looks like it could be done via reduction. This again produces ('this is point (60, 50) = 221.84874961635666') where all of the decimals past dp 4 are now different but are consistent every time. I also plan to add a convergence condition to this loop and am wondering if this will be possible to do within the loop or whether I should do this outside of the loop.

I was just curious if anyone could point me or help me with the correct method to make this loop parallel however also appreciate any coding tips. If you need me to attach the rest of the code please just let me know, thank you.

! Making the loop parallel using OpenMP, I am currently struggling with this part 
!$omp parallel do default(none) shared(r_outer, l_outer, r_inner, l_inner, phi_initial, &
  !$omp seed, free_parameter, phi_test) private(rand_z, rand_r, outerloopiteration, U_potential) 
  !!$omp reduction(+: phi_test)
  bigloop : do outerloopiteration = 1, max_iters
   
    ! Creating two random numbers and converting them to integers to update grid points arbitrarily
    rand_z = nint(r_num(seed) * real(l_outer, kind=dp))
    rand_r = nint(r_num(seed) * real(r_outer, kind=dp))
   
    ! The code to solve the problem, split separately into two conditions for if rand_r = 0 or not      
    if  (rand_r == 0) then
      U_potential = 2.0_dp/3.0_dp * phi_test(1, rand_z) + (1.0_dp / 6.0_dp) * (phi_test(0, rand_z+1)&
       + phi_test(0, rand_z-1))
    else
      U_potential = 1.0_dp/4.0_dp * (phi_test(rand_r+1, rand_z) + phi_test(rand_r-1, rand_z) &
       + phi_test(rand_r, rand_z+1) + phi_test(rand_r, rand_z-1)) + &
      (1.0_dp/(8.0_dp*rand_r)) * (phi_test(rand_r+1, rand_z) - phi_test(rand_r-1, rand_z))
    end if
    
    ! Adjusting the new value at the random point using a free parameter to adjust convergence
    phi_test(rand_r, rand_z) = phi_test(rand_r, rand_z) + free_parameter * &
    (U_potential - phi_test(rand_r, rand_z))  
         
    ! Boundary conditions
    phi_test(0 , 0:l_inner) = phi_initial
    phi_test(0:r_inner, 0) = phi_initial
    phi_test(1:r_inner, 1:l_inner) = phi_initial
           
  end do bigloop
  !$omp end parallel do

I have tried moving the main section of the loop into its own subroutine however was then receiving a NaN for the point that I was trying to print. I have also tried using reduction and shared for the array however as previously described this produces different values for the answer. I am currently looking further into different ways of doing this and looking further into the OpenMP syntax.

Answer 1

Arbitrary updating of grid points is called "chaotic iteration" and 1. no one does that because it doesn't work terribly well (if at all) and 2. it's very hard to parallelize.

For an easy to code solution I suggest you look into a Jacobi iteration which is fully parallel, or SOR / Gauss-Seidel, which can be parallelized through multi-coloring or wavefronts. Even better would be a conjugate gradient method which converges much faster, but it's a little more coding.

For much background information, see my textbook https://theartofhpc.com/istc.html