poisson eq. with spectral method - fortran

Question

I wrote poisson eq. solver with spectral method. However, the obtained result does not coincide with the result of difference method with periodic boundary condition.
I think I am mistaken in the use of FFTW.
Could you tell me which part of the following code contains errors?
Thank you.
program main
implicit none
include 'fftw3.f'
integer(8) :: plan
integer, parameter :: j_max = 100, k_max = 100, m_max = j_max/2 + 1, n_max = k_max
integer :: j, k, m, n, mm, nn
real(8) :: v(1:j_max, 1:k_max), f(1:j_max, 1:k_max)
real(8) :: x_max, y_max, dx, dy, x, y, t_max, pi
complex(8), parameter :: im = (0.d0, 1.d0)
complex(8) :: vk(1:m_max, 1:n_max), fk(1:m_max, 1:n_max)
pi = 4.d0*atan(1.d0)
x_max = 2.d0*pi
y_max = 2.d0*pi
dx = x_max/j_max
dy = y_max/k_max
!*-- Initial Condition ---
do j = 1, j_max
x = dx*j
do k = 1, k_max
y = dy*k
f(j, k) = dexp(-(x - x_max/2)**2 -(y - y_max/2)**2)
enddo
enddo
!*-- FFT forward ---
call dfftw_plan_dft_r2c_2d(plan, j_max, k_max, v, vk, FFTW_ESTIMATE)
call dfftw_execute(plan)
call dfftw_plan_dft_r2c_2d(plan, j_max, k_max, f, fk, FFTW_ESTIMATE)
call dfftw_execute(plan)
do m = 1, m_max
do n = 1, n_max
if(m <= m_max/2 + 1) then
mm = m - 1
else
mm = m - 1 - m_max
endif
if(n <= n_max/2 + 1) then
nn = n - 1
else
nn = n - 1 - n_max
endif
if(mm == 0 .and. nn == 0) then
else
vk(m, n) = fk(m, n)/(mm**2 + nn**2)
endif
enddo
enddo
!*-- FFT backward ---
call dfftw_plan_dft_c2r_2d(plan, j_max, k_max, vk, v, FFTW_ESTIMATE)
call dfftw_execute(plan)
!*-- normalization ---
v = v/j_max/k_max
call dfftw_destroy_plan(plan)
end program main

Answer 1

Here you have a code that do what you ask for, take into account that the original data was in 'f1' and 'f2' in my case, the important comments are in english, some other in spanish, if you have problems to understand just tell me :)
// FFT CALCULATION
// Inicialización de elementos necesarios para el cálculo de la FFT
fftw_plan p1; // variable para almacenar la planificación de la FFT
fftw_plan p2; // variable para almacenar la planificación de la FFT
int N_fft= ancho*alto; //number of points of the image
fftw_complex *U1 =(fftw_complex*) fftw_malloc(sizeof(fftw_complex)*alto*((ancho/2)+1)); //puntero que apuntará al resultado de la FFT
fftw_complex *U2 =(fftw_complex*) fftw_malloc(sizeof(fftw_complex)*alto*((ancho/2)+1));
p1 = fftw_plan_dft_r2c_2d(alto,ancho, f1, U1, FFTW_ESTIMATE); // FFT planning
p2 = fftw_plan_dft_r2c_2d(alto,ancho, f2, U2, FFTW_ESTIMATE); // FFT planning
fftw_execute(p1); // FFT calculation
fftw_execute(p2); // FFT calculation
fftw_destroy_plan(p1);// Eliminación de la planificación de la FFT
fftw_destroy_plan(p2);// Eliminación de la planificación de la FFT
// Security saving of U1 and U2 in auxiliar variables because the ifft modifies the input data
for (int y = 0 ; y < alto ; y++){
for (int x = 0 ; x < (ancho/2)+1 ; x++){
U1_input_save[((ancho/2)+1)*y+x][0] = U1[((ancho/2)+1)*y+x][0];
U1_input_save[((ancho/2)+1)*y+x][1] = U1[((ancho/2)+1)*y+x][1];
U2_input_save[((ancho/2)+1)*y+x][0] = U2[((ancho/2)+1)*y+x][0];
U2_input_save[((ancho/2)+1)*y+x][1] = U2[((ancho/2)+1)*y+x][1];
}
}
// IFFT ( U1,U2 --> u1,u2)
//----IFFT-----
double *u1 = (double*) malloc(sizeof(double)*N_fft);//puntero que apuntará al resultado de la IFFT
double *u2 = (double*) malloc(sizeof(double)*N_fft);
fftw_plan p3;// variable para almacenar la planificación de la IFFT
fftw_plan p4;// variable para almacenar la planificación de la IFFT
p3 = fftw_plan_dft_c2r_2d(alto, ancho, U1, u1, FFTW_ESTIMATE);//planificación de la fft inversa
p4 = fftw_plan_dft_c2r_2d(alto, ancho, U2, u2, FFTW_ESTIMATE);//planificación de la fft inversa
fftw_execute(p3); // Calculo de la fft inversa
fftw_execute(p4); // Calculo de la fft inversa
fftw_destroy_plan(p3); // Eliminación de la planificación de la IFFT
fftw_destroy_plan(p4); // Eliminación de la planificación de la IFFT
// Normalización after IFFT important!
u1 = fftw_normalization(ancho,alto,N_fft,u1);
u2 = fftw_normalization(ancho,alto,N_fft,u2);
// Correction of U1 and U2, restoring the original data
for (int y = 0 ; y < alto ; y++){
for (int x = 0 ; x < (ancho/2)+1 ; x++){
U1[((ancho/2)+1)*y+x][0] = U1_input_save[((ancho/2)+1)*y+x][0];
U1[((ancho/2)+1)*y+x][1] = U1_input_save[((ancho/2)+1)*y+x][1];
U2[((ancho/2)+1)*y+x][0] = U2_input_save[((ancho/2)+1)*y+x][0];
U2[((ancho/2)+1)*y+x][1] = U2_input_save[((ancho/2)+1)*y+x][1];
}
}
// FIN CALCULATION FFT

Answer 2

In the FFT Forward process the '2d_c2r' function can modify the input values, then, if you use then later the results will not be correct, you can do a copy of the data before executing that function.