rr = dsqrt((xs-0.5d0)**2 + (ys-0.5d0)**2 + (zs-0.5d0)**2)
if(rr<=R0) then
   rho(icell_amr) = 1 + delta_in
else
   rho(icell_amr) = 1 + delta_out
end if

Top-Hat Nr. 1

delta_in = 190.9d0
delta_out = -0.1d0
R0 = 0.05d0

jobid_1697114: sf_src_mean set to zero, multigrid-solver switched on, convergence criteria set to 1e-12 and 1e-9, number of cells 128.

jobid_1702612: sf_src_mean calculated, multigrid-solver switched on, convergence criteria set to 1e-12 and 1e-9, number of cells 256. For analytical result the scipy.interate.trapz method is used.

The above rest compared to a different integration method to obtain the analytical result (scipy.interate.odeint).

jobid_1705277: placed centre of top-hat in centre of cells.

jobid_1705279: output sf field along x-y diagonal.

Top-Hat Nr. 2

delta_in = 23.77d0
delta_out = -0.1d0
R0 = 0.1d0

jobid_1699598: sf_src_mean set to zero, multigrid-solver switched on, convergence criteria set to 1e-12 and 1e-9, number of cells 256.

Complex chi field

The point in time when the chi field turns complex:

• jobid_1703821: param_b3=0.000001, negative eta at a= 0.843792366627832, 256^3 particles

• jobid_1707468: param_b3=0.000001, negative eta at a= 0.841407822646427, 512^3 particles

  • jobid_1708120: param_b3=0.0001, negative eta at a= 0.841407860117935, 512^3 particles (restart from snapshot 21 of jobid_1707468)

  • jobid_1708628: param_b3=0.00000001, negative eta at a= 0.841407820905272, 512^3 particles (restart from snapshot 21 of jobid_1707468)

• jobid_1708628: param_b3=0.00000001, negative eta at a= 0.841407763520329 , 512^3 particles

• jobid_1714440: param_b3=0.0001, negative eta at a= 0.841409089391650 , 512^3 particles

Amp = 0.000005d0
sigma = 0.09
alpha = 0.01
rho(icell_amr) = 1.0d0 + 2.0d0*beta_cvg/(3.0d0*omega_m*aexp) &
                       * 2.0d0*Amp*alpha/sigma**2 &
                       * (1.0d0 - 2.0d0*(xs-0.5d0)**2/sigma**2) &
                       * dexp(-(xs-0.5d0)**2/sigma**2)

Multigrid Off:

• jobid_1681479: initial guess of sf = 0.0d0, narrowing the convergence criteria to residual<1e-12, switched off multigrid-solver resulted in ecosmogs centered at zero, thus shifted down by an amount equal to Amp. The average value of sf_src_mean converges to 1e-8.
  
  repeating this test with initial guess of sf = Amp*(1.0d0 - alpha*dexp(-(xs-0.5d0)**2/sigma**2)) gives, jobid_1681810:
  
  The average value of sf_src_mean converges to 1e-10.
  (the same settings gave the correct result for the sine test too)

• jobid_1685023: repeating test 'jobid_1681479', but with abs(residual-residual_old)<1.0d-13 in multigrid_fine_commons_extradof.f90 at line [241,247], gives
  

Multigrid On:

• jobid_1681489: initial guess of sf = 0.0d0, narrowing the convergence criteria to residual<1e-12, switched on multigrid-solver resulted in ecosmogs centered at zero, thus shifted down by an amount equal to Amp. Furthermore there is unwanted evolution in the result
  
  repeating this test with initial guess of sf = Amp*(1.0d0 - alpha*dexp(-(xs-0.5d0)**2/sigma**2)) gives, jobid_1681796:
  

Code Changes:

• coarse and fine level solve the same equations.
• correct order in restriction and communication in fine-commons-extradof, and correct some double-precisions.

Multigrid On:

• jobid_1687566: correct analytical initial guess of sf, sf_src_mean calculated by code, uses non-linear eq. in multigrid_fine_fine_extradof.f90, narrow convergence criteria of residual<1e-12. After the code changes we get the following improvement:
  

• jobid_1687585: same as test 'jobid_1687566' but with a zero initial guess of sf gives:
  

Amp = 0.00000005d0
pii = 4.0d0*datan(1.0d0)
rho(icell_amr) = 1.0d0-8.0d0*beta_cvg*pii**2/(3.0d0*aexp*omega_m)
                     * Amp*dsin(2.0d0*pii*xs)

Findings:

• the sf_src_mean for chi is not close enough to zero (reaches 1e-8, but 1e-21 should be possible), whereas cbf's are (reaches 1e-31).
• it depends more on the initial guess than on the residual criteria

Tests:

• jobid_1680694: narrowing the convergence criteria from residual<1e-8 to residual<1e-10 in multigrid_fine_commons_extradof.f90 line 239 did not change the mismatch between the numerical and analytical result, but there is a slightly wider spread in the results of ecosmog
  

• jobid_1680760: we switched the multigrid-solver off by setting in multigrid_fine_commons.f90 line 161: levelmin_mg = levelmin. This resulted in numerical values of chi now being some factors smaller than analytic answer.

• jobid_1680763: fixed sf_src_mean to zero in multigrid_fine_commons_extradof.f90 at line 665. This resulted into a translations of different magnitude depending of aexp of chi-values. Thus the calculation of sf_src_mean and sf_src_mean2 seem to work correctly.

• jobid_1680768: changed the initial guess of sf by adding in amr_step.f90 at line 340: sf(icell_amr) = Amp*dsin(2.0d0*pii*xs). This improved the mean value of sf from 1e-7 to 1e-12 for the first iteration before in rises back to 1e-7 after V-cycle.

• jobid_1680770: changed the initial guess of sf by adding in amr_step.f90 at line 340: sf(icell_amr) = Amp*dsin(2.0d0*pii*xs) and set levelmin_mg = levelmin in multigrid_fine_commons.f90 line 161. This gives a perfect match between numerical and analytic results.

• jobid_1681120: using 0. as initial guess for sf and narrowing the convergence criteria to residual<1e-12 and levelmin_mg = levelmin, gives no improvent, but similar behaviour result to jobid_1680694.

• jobid_1681236: 0.0 initial guess for sf, setting op = op - 1.5d0*omega_m/beta_cvg*aexp*(rho(icell_amr(i))-rho_tot)*dx2 at appr. line [125,129,509,514] and sf_src(icell_amr(i)) = 1.5d0*omega_m/beta_cvg*aexp*(rho(icell_amr(i))-rho_tot) at appr. line [510,717] in multigrid_fine_fine_extradof.f90 gives average value of 1e-21. We switched multigrid-solver off. Thus, even if we did not enforce the correct solution on op, should we expect an average in the same ballpark as 1e-21. Ecosmog and analytical result match.

• jobid_1685033: same set-up as previous test, jobid_1681236, but with multigrid-solver switched on. Ecosmog and analytical result match.

Code Changes:

• coarse and fine level solve the same equations.

• jobid_1685040: 0.0 initial guess for sf, calculated sf_src_mean, code solves non-linear eq. in multigrid_fine_fine_extradof.f90, multigrid-solver switched on. The changes in the code do not change the result.
  

• jobid_1685406: 0.0 initial guess for sf, fixing sf_src_mean to zero, use non-linear eq. in multigrid_fine_fine_extradof.f90, and switching off multigrid-solver, gives same result as in test 'jobid_1680760'. Interestingly even though we fixed sf_src_mean to zero, the result does not move along the y-axis as in test 'jobid_1680763'.

Code Changes:

• correct order in restriction and communication in fine-commons-extradof, and correct some double-precisions.

• jobid_1687555: repeating test 'jobid_1685402' but with correct analytical initial guess for sf.
  

• jobid_1687577: repetition of test 'jobid_1685402', but this time fixing sf_src_mean to zero. This time we get the correct result, e.g. match between ecosmog and analytica answer.

• jobid_1687589: for clarification: zero initial guess for sf, sf_src_mean calculated by ecosmog, multigrid-solver on, use of non-linear eq. in multigrid_fine_fine_extradof.f90, gives the correct result: