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vlaplex/plasmaExpansion.f90
JGonzalez 8eab3b5610 I'm stupid and I deleted the previous repository... 
So, code is working, this case reproduce a Diko's peak with Poisson
equation by changing the boundary conditions over time.
2024-09-26 17:58:45 +02:00

663 lines
20 KiB
Fortran
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!Physical and mathematical constants
module constantParameters
implicit none
public
integer, parameter:: dp = kind(0.d0) ! Precision
real(dp), parameter:: PI = 4.0_dp*ATAN(1.0_dp) ! Number pi
real(dp), parameter:: qe = 1.60217662e-19_dp ! Elementary charge
real(dp), parameter:: kb = 1.38064852e-23_dp ! Boltzmann constants SI
real(dp), parameter:: eV2J = qe ! Electron volt to Joule conversion
real(dp), parameter:: eps_0 = 8.8542e-12_dp ! Epsilon_0
real(dp), parameter:: eV_to_K = 11604.5_dp ! Convert eV to K
real(dp), parameter:: cm3_to_m3 = 1.0e6_dp ! Convert cm^-3 to m^-3
end module constantParameters
module referenceValues
use constantParameters, only: dp
implicit none
real(dp):: L_ref, t_ref, n_ref, u_ref, Temp_ref ! Reference values
real(dp):: phi_ref, p_ref ! Reference values
end module referenceValues
module output
use constantParameters, only: dp
implicit none
public
private:: dataRef_id, dataBC_id, dataF_id, dataPhi_id, dataCum_id
private:: formatFloat, formatSep, formatTime
integer:: everyOutput, everyWrite
character(:), allocatable:: pathOutput
integer, parameter:: dataRef_id = 10
integer, parameter:: dataBC_id = 20
integer, parameter:: dataF_id = 30
integer, parameter:: dataPhi_id = 40
integer, parameter:: dataCum_id = 50
character(len=7), parameter:: formatFloat = 'ES0.4e3'
character(len=3), parameter:: formatSep = '","'
character(len=7):: formatTime
contains
subroutine setTimeFormat(nt)
integer, intent(in):: nt
integer:: l
l=max(1,ceiling(log10(real(nt))))
write(formatTime, '(A2,I0,".",I0,A1)') '(I',l,l,')'
end subroutine setTimeFormat
subroutine createPath()
character(8) :: date_now
character(10) :: time_now
call date_and_time(date_now, time_now)
!Compose the folder name
pathOutput = date_now(1:4) // '-' // date_now(5:6) // '-' // date_now(7:8) // '_' // &
time_now(1:2) // '.' // time_now(3:4) // '.' // time_now(5:6) // '/'
call system('mkdir ' // pathOutput)
end subroutine createPath
subroutine writeOutputF(t, dt, nr, r, nv, v, f)
use referenceValues, only: L_ref, n_ref, u_ref, t_ref
integer, intent(in):: t
integer, intent(in):: nr, nv
real(dp), intent(in):: dt
real(dp), intent(in):: r(1:nr)
real(dp), intent(in):: v(1:nv)
real(dp), intent(in):: f(1:nr,1:nv)
character(len=30) :: myfmt
character(:), allocatable:: filename
integer:: i
character(len=10):: timeString
write (timeString, formatTime) t
filename = 'time_' // trim(timeString) // '_f_i.csv'
write (*, '(A, A)') 'Writing: ', filename
open(unit=dataF_id, file=pathOutput // filename)
write(dataF_id, '(A)') "t (s)"
write(dataF_id, '('//formatFloat//')') t*dt*t_ref
write(myfmt, "(I0)") nr
myfmt = '(A,' // trim(myfmt) // '(' // formatSep // ',' // formatFloat // '))'
write(dataF_id, myfmt) "v (m/s) / r (m)", r*L_ref
write(myfmt, "(I0)") nr
myfmt = '(' // formatFloat // ',' // trim(myfmt) // '(' // formatSep // ',' // formatFloat // '))'
do i = 1, nv
write(dataF_id, myfmt) v(i)*u_ref, f(:,i)*n_ref/u_ref
end do
close(unit=dataF_id)
end subroutine writeOutputF
subroutine writeOutputPhi(t, dt, nr, r, phi, n_e)
use constantParameters, only: eV_to_K
use referenceValues, only: L_ref, phi_ref, t_ref, n_ref
integer, intent(in):: t
integer, intent(in):: nr
real(dp), intent(in):: dt
real(dp), intent(in):: r(1:nr)
real(dp), intent(in):: phi(1:nr)
real(dp), intent(in):: n_e(1:nr)
character(:), allocatable:: filename
integer:: i
character(len=10):: timeString
write (timeString, formatTime) t
filename = 'time_' // trim(timeString)//'_phi.csv'
write (*, '(A, A)') 'Writing: ', filename
open(unit=dataPhi_id, file=pathOutput//filename)
write(dataPhi_id, '(A)') "t (s)"
write(dataPhi_id, '('//formatFloat//')') t*dt*t_ref
write(dataPhi_id, '(A,2('//formatSep//',A))') "r (m)","phi (V)","n_e (m^-3)"
do i = 1, nr
write(dataPhi_id, '('//formatFloat//',2('//formatSep //','//formatFloat//'))') &
r(i)*L_ref, &
phi(i)*phi_ref, &
n_e(i)*n_ref
end do
close(unit=dataPhi_id)
end subroutine writeOutputPhi
subroutine writeOutputMom(t, dt, nr, r, n_i, u_i, T_i, Z)
use constantParameters, only: eV_to_K
use referenceValues, only: L_ref, t_ref, n_ref, u_ref, Temp_ref
integer, intent(in):: t
integer, intent(in):: nr
real(dp), intent(in):: dt
real(dp), intent(in):: r(1:nr)
real(dp), intent(in):: n_i(1:nr)
real(dp), intent(in):: u_i(1:nr)
real(dp), intent(in):: T_i(1:nr)
real(dp), intent(in):: Z(1:nr)
character(:), allocatable:: filename
integer:: i
character(len=10):: timeString
write (timeString, formatTime) t
filename = 'time_' // trim(timeString)//'_mom_i.csv'
write (*, '(A, A)') 'Writing: ', filename
open(unit=dataPhi_id, file=pathOutput//filename)
write(dataPhi_id, '(A)') "t (s)"
write(dataPhi_id, '('//formatFloat//')') t*dt*t_ref
write(dataPhi_id, '(A,4('//formatSep//',A))') "r (m)","n_i (m^-3)","u_i (m s^-1)", "T_i (eV)","Zave"
do i = 1, nr
write(dataPhi_id, '('//formatFloat//',4('//formatSep //','//formatFloat//'))') &
r(i)*L_ref, &
n_i(i)*n_ref, &
u_i(i)*u_ref, &
T_i(i)*Temp_ref/ev_to_K, &
Z(i)
end do
close(unit=dataPhi_id)
end subroutine writeOutputMom
subroutine writeOutputBoundary(t, dt, n, u, Temp, Z)
use constantParameters, only: eV_to_K
use referenceValues, only: t_ref, n_ref, u_ref, Temp_ref
integer, intent(in):: t
real(dp), intent(in):: dt
real(dp), intent(in):: n, u, Temp, Z
character(len=6), parameter:: filename = 'bc.csv'
logical:: res
inquire(file=pathOutput // filename, exist=res)
if (.not. res) then
write (*, '(A, A)') 'Writing: ', filename
open(unit=dataBC_id, file=pathOutput // filename, action='write', position='append')
write(dataBC_id, '(A,4(' // formatSep // ',A))') 't (s)', 'n (m^-3)', 'u (m s^-2)', 'T (eV)', 'Z'
close(dataBC_id)
end if
open(unit=dataBC_id, file=pathOutput // filename, action='write', position='append')
write(dataBC_id, '(' // formatFloat // ',4('// formatSep // ',' // formatFloat // '))') &
t*dt*t_ref, n*n_ref, u*u_ref, Temp*Temp_ref/eV_to_K, Z
close(dataBC_id)
end subroutine writeOutputBoundary
subroutine writeOutputRef()
use referenceValues, only: t_ref, L_ref, n_ref, u_ref, Temp_ref, phi_ref
character(len=7), parameter:: filename = 'ref.csv'
write (*, '(A, A)') 'Writing: ', filename
open(unit=dataRef_id, file=pathOutput // filename)
write(dataRef_id, '(A,5(' // formatSep // ',A))') 't_ref (s)', 'L_ref (m)', 'n_ref (m^-3)', &
'u_ref (m s^-1)', 'T_ref (K)', 'phi_ref (V)'
write(dataRef_id, '(' // formatFloat // ',5('// formatSep // ',' // formatFloat // '))') &
t_ref, L_ref, n_ref, &
u_ref, Temp_ref, phi_ref
close(dataRef_id)
end subroutine writeOutputRef
subroutine writeOutputFCum(t, dt, r, nv, v, f)
use referenceValues, only: L_ref, n_ref, u_ref, t_ref
integer, intent(in):: t
real(dp), intent(in):: dt
integer, intent(in):: nv
real(dp), intent(in):: r
real(dp), intent(in):: v(1:nv)
real(dp), intent(in):: f(1:nv)
character(len=30) :: myfmt
character(:), allocatable:: filename
integer:: i
character(len=10):: timeString
write (timeString, formatTime) t
filename = 'time_' // trim(timeString) // '_fCum_i.csv'
write (*, '(A, A)') 'Writing: ', filename
open(unit=dataCum_id, file=pathOutput // filename)
write(dataCum_id, '(A)') "t (s)"
write(dataCum_id, '('//formatFloat//')') t*dt*t_ref
write(myfmt, "(I0)") 1
myfmt = '(A,' // trim(myfmt) // '(' // formatSep // ',' // formatFloat // '))'
write(dataCum_id, myfmt) "v (m/s) / r (m)", r*L_ref
write(myfmt, "(I0)") 1
myfmt = '(' // formatFloat // ',' // trim(myfmt) // '(' // formatSep // ',' // formatFloat // '))'
do i = 1, nv
write(dataCum_id, myfmt) v(i)*u_ref, f(i)*n_ref/u_ref
end do
close(unit=dataCum_id)
end subroutine writeOutputFCum
end module output
module eos
use constantParameters, only: dp
implicit none
private
public:: T_to_Z
contains
pure function T_to_Z(T) result(Z)
use constantParameters, only: eV_to_K
use referenceValues, only: Temp_ref
implicit none
real(dp), intent(in):: T
real(dp):: Z
! Z = (Temp_ref * T / eV_to_K)**0.6
! Z = max(Z, 1.0_dp)
! Z = min(Z, 22.0_dp)
Z = 12.0_dp
end function T_to_Z
end module eos
program plasmaExpansion
use constantParameters, only: dp, kb, qe, eps_0, ev_to_K, cm3_to_m3, PI
use output
use referenceValues
use eos, only: T_to_Z
use omp_lib
implicit none
real(dp), parameter:: m_i = 1.9712258e-25_dp ! Tin aton mass in kg
real(dp), parameter:: gam = 1.0_dp ! Adiabatic coefficient
real(dp):: r0, rf
real(dp), allocatable, dimension(:):: r
real(dp):: v0, vf
real(dp), allocatable, dimension(:):: v
real(dp):: t0, tf
real(dp):: dr, dv, dt
integer:: nr, nv, nt
integer:: i, j, t
integer:: j0 ! First integer of positive velocity
real(dp):: Temp_bc ! Temperature
real(dp):: Temp0, TempF
real(dp):: n_ecr ! Electron critical density for the laser
real(dp):: c_s ! Ion sound speed
real(dp):: u_bc ! Injection velocity
real(dp):: u_bc0, u_bcF
real(dp):: n_bc ! Injection density
real(dp):: n_bc0, n_bcF
integer:: t_bc0, t_bcF
real(dp), allocatable, dimension(:,:):: f_i, f_i_old
real(dp), allocatable, dimension(:):: f0 ! Boundary at r = x_0
real(dp), allocatable, dimension(:):: n_i
real(dp), allocatable, dimension(:):: u_i
real(dp), allocatable, dimension(:):: E_i
real(dp), allocatable, dimension(:):: T_i
real(dp), allocatable, dimension(:):: n_e
real(dp), allocatable, dimension(:):: Zave
real(dp), allocatable, dimension(:):: diag, diag_low, diag_high
real(dp), allocatable, dimension(:,:):: A
real(dp), allocatable, dimension(:):: Res
real(dp), allocatable, dimension(:):: b
integer:: info
real(dp), allocatable, dimension(:):: phi, phi_old, E
real(dp):: phiConv
real(dp):: phi0
integer:: k
real(dp), allocatable, dimension(:):: fCum_i
real(dp):: rCum
integer:: rCum_index
! Set number of threads
call omp_set_num_threads(8)
! Set reference numbers (in SI units)
Temp_ref = 60.0_dp * eV_to_K
n_ref = 1.0e19_dp * cm3_to_m3
t_ref = sqrt(eps_0 * m_i / (n_ref * 1.0_dp * qe**2)) ! 1.0_dp represents Z = 1 for reference values
u_ref = sqrt(kb * Temp_ref / m_i)
L_ref = u_ref * t_ref
phi_ref = kb * Temp_ref / qe
p_ref = n_ref * kb * Temp_ref
! Set position to calculate cumulative sum of f (non-dimensional units)
rCum = 5.0e-3 / L_ref
! Set input parameters (remember these have to be in non-dimensional units)
Temp0 = 60.0_dp * eV_to_K / Temp_ref
TempF = 10.0_dp * eV_to_K / Temp_ref
n_ecr = 1.0e19_dp * cm3_to_m3 / n_ref
c_s = sqrt(T_to_Z(Temp0) * gam * Temp0)
u_bc0 = sqrt(Temp0)
u_bcF = sqrt(TempF)
n_bc0 = n_ecr / T_to_Z(Temp0)
n_bcF = n_ecr*1.0e-1 / T_to_Z(Temp0)
! Set domain boundaries (non-dimensional units)
r0 = 200.0e-6_dp / L_ref
rf = 1.0e-2_dp / L_ref
dr = 1.0e3_dp
nr = nint((rf - r0) / dr) + 1
dr = (rf - r0) / float(nr-1)
allocate(r(1:nr))
do i = 1, nr
r(i) = dr * float(i-1) + r0
end do
! Index for cumulative sum
rCum_index = minloc(abs(r - rCum), 1)
v0 =-1.0e1_dp*c_s
vf = 2.0e1_dp*c_s
dv = 1.0e-1_dp
nv = nint((vf - v0) / dv) + 1
dv = (vf - v0) / float(nv-1)
allocate(v(1:nv))
do j = 1, nv
v(j) = dv * float(j-1) + v0
end do
! Shift v mesh so it passes by 0
v = v - (minval(abs(v)))
j0 = minloc(abs(v), 1)
if (v(j0) < 0.0_dp) then
j0 = j0 + 1
end if
t0 = 0.0_dp
tf = 1.0e-6_dp / t_ref
! tf = 1.0e1_dp * (rf - r0) / c_s
dt = 1.0e-2_dp*dr/c_s
nt = nint((tf - t0) / dt)
dt = (tf - t0) / float(nt)
t_bc0 = nint( 20.0e-9_dp / t_ref / dt)
t_bcF = nint( 25.0e-9_dp / t_ref / dt)
everyOutput = nint(1.0e-9_dp/t_ref/dt)
if (everyOutput == 0) then
everyOutput = 1
end if
everyWrite = everyOutput/10
if (everyWrite == 0) then
everyWrite = 1
end if
write(*, '(A,ES0.4e3)') 'CFL: ', dt*vf/dr
! Allocate vectors
allocate(f_i(1:nr,1:nv), f_i_old(1:nr,1:nv))
allocate(n_i(1:nr))
allocate(u_i(1:nr), E_i(1:nr), T_i(1:nr))
allocate(Zave(1:nr))
allocate(n_e(1:nr))
allocate(phi(1:nr), phi_old(1:nr), E(1:nr))
allocate(fCum_i(1:nv))
f_i = 0.0_dp
f_i_old = 0.0_dp
n_i = 0.0_dp
u_i = 0.0_dp
E_i = 0.0_dp
T_i = 0.0_dp
n_e = 0.0_dp
Zave = 0.0_dp
phi = 0.0_dp
phi_old = 0.0_dp
E = 0.0_dp
fCum_i = 0.0_dp
! Allocate matrix for Poisson equation
allocate(diag(1:nr), diag_low(1:nr-1), diag_high(1:nr-1))
allocate(b(1:nr))
diag = 0.0_dp
diag_low = 0.0_dp
diag_high = 0.0_dp
b = 0.0_dp
diag = -2.0_dp / dr**2
diag_low = 1.0_dp / dr**2 * r(1:nr-1) / r(2:nr)
diag_high = 1.0_dp / dr**2 * r(2:nr) / r(1:nr-1)
! diag(1) = 1.0_dp ! Dirichlet
! diag_high(1) = 0.0_dp ! Dirichlet
diag_high(1) = 2.0_dp / dr**2 ! Neumann
diag(nr) = 1.0_dp ! Dirichlet
diag_low(nr-1) = 0.0_dp ! Dirichlet
! diag_low(nr-1) = 2.0_dp / dr**2 ! Neumann
allocate(A(1:nr,1:nr))
A = 0.0_dp
A(1,1) = diag(1)
A(1,2) = diag_high(1)
do i = 2, nr - 1
A(i, i-1) = diag_low(i-1)
A(i, i) = diag(i)
A(i, i+1) = diag_high(i)
end do
A(nr,nr-1) = diag_low(nr-1)
A(nr,nr) = diag(nr)
allocate(Res(1:nr))
Res = 0.0_dp
! Set boundary values
! phi0 = 0.0_dp / phi_ref ! Dirichlet
! phi(1) = phi0 ! Dirichlet
phi0 = phi(1) ! Neumann
phi(nr) = 0.0_dp ! Dirichlet
allocate(f0(j0:nv))
f0 = 0.0_dp
! Output initial values
call createPath()
call setTimeFormat(nt)
t = 0
call writeOutputRef()
call writeOutputF(t, dt, nr, r, nv, v, f_i_old)
call writeOutputPhi(t, dt, nr, r, phi, n_e)
call writeOutputMom(t, dt, nr, r, n_i, u_i, T_i, Zave)
! Main loop
Temp_bc = Temp0
u_bc = u_bc0
n_bc = n_bc0
do t = 1, nt
if (t > t_bc0 .and. t <= t_bcF) then
Temp_bc = (TempF - Temp0) / float(t_bcF - t_bc0)*(t - t_bc0) + Temp0
u_bc = (u_bcF - u_bc0) / float(t_bcF - t_bc0)*(t - t_bc0) + u_bc0
n_bc = (n_bcF - n_bc0) / float(t_bcF - t_bc0)*(t - t_bc0) + n_bc0
else if (t > t_bcF) then
Temp_bc = TempF
u_bc = u_bcF
n_bc = n_bcF
end if
call writeOutputBoundary(t, dt, n_bc, u_bc, Temp_bc, T_to_Z(Temp_bc))
f0(j0:nv) = n_bc / sqrt(PI*Temp_bc) * exp(-(v(j0:nv) - u_bc)**2 / Temp_bc)
! Boundary conditions
! r = r0, v>0
f_i_old(1,j0:nv) = f0
f_i(1,j0:nv) = f_i_old(1,j0:nv)
T_i(1) = Temp_bc
Zave(1) = T_to_Z(Temp_bc)
! r = rf, v<0
f_i_old(nr,1:j0-1) = 0.0_dp
f_i(nr,1:j0-1) = f_i_old(nr,1:j0-1)
! set edge velocities to 0
f_i_old(:,1) = 0.0_dp
f_i_old(:,nv) = 0.0_dp
! Advect in the r direction
!$omp parallel do
do i = 1, nr
! Advect negative velocity
if (i < nr) then
f_i(i,1:j0-1) = f_i_old(i,1:j0-1) - v(1:j0-1)*dt/dr/r(i)**2*(r(i+1)**2*f_i_old(i+1,1:j0-1) - &
r(i )**2*f_i_old(i ,1:j0-1))
end if
! Advect positive velocity
if (i > 1) then
f_i(i,j0:nv) = f_i_old(i, j0:nv) - v( j0:nv)*dt/dr/r(i)**2*(r(i )**2*f_i_old(i , j0:nv) - &
r(i-1)**2*f_i_old(i-1, j0:nv))
end if
n_i(i) = sum(f_i(i,:)*dv)
if (n_i(i) > 0.0_dp) then
u_i(i) = sum(v(:) *f_i(i,:)*dv) / n_i(i)
E_i(i) = sum(v(:)**2*f_i(i,:)*dv) / n_i(i)
T_i(i) = 2.0_dp*E_i(i) - 2.0_dp*u_i(i)**2
Zave(i) = T_to_Z(T_i(i))
else
u_i(i) = 0.0_dp
T_i(i) = 0.0_dp
Zave(i) = 0.0_dp
end if
end do
!$omp end parallel do
! Solve Poission (maximum number of iterations, break if convergence is reached before)
do k = 1, 200
! Store previous value
phi_old = phi
! Calculate distribution of electrons
n_e = Zave(1) * n_i(1) * exp((phi_old - phi0) / T_i(1))
! Diagonal matrix for Newton integration scheme
diag = -2.0_dp / dr**2 - n_e
diag_low = 1.0_dp / dr**2 * r(1:nr-1) / r(2:nr)
diag_high = 1.0_dp / dr**2 * r(2:nr) / r(1:nr-1)
! diag(1) = 1.0_dp ! Dirichlet
! diag_high(1) = 0.0_dp ! Dirichlet
diag_high(1) = 2.0_dp / dr**2 - n_e(1) ! Neumann
diag(nr) = 1.0_dp ! Dirichlet
diag_low(nr-1) = 0.0_dp ! Dirichlet
! diag_low(nr-1) = 2.0_dp / dr**2 - n_e(nr) ! Neumann
! Calculate charge density
b = -(Zave*n_i - n_e)
! Apply boundary conditions
! b(1) = phi0 ! Dirichlet
b(nr) = 0.0_dp ! Dirichlet
! Calculate residual
!$omp parallel workshare
Res = -(MATMUL(A, phi_old) - b)
!$omp end parallel workshare
! Res(1) = 0.0_dp ! Dirichlet
Res(nr) = 0.0_dp ! Dirichlet
! Iterate system
call dgtsv(nr, 1, diag_low, diag, diag_high, Res, nr, info)
phi = phi_old + Res
phi0 = phi(1) ! Neumann
phi(nr-10:nr) = phi(nr-11)
! Check if the solution has converged
phiConv = maxval(abs(Res),1)
if (phiConv < 1.0e-2_dp) then
exit
end if
end do
! Calculate electric field
! E(1) = - (phi(2) - phi(1)) / dr ! Dirichlet
E(1) = 0.0_dp ! Neumann
!$omp parallel do
do i = 2, nr-1
E(i) = - 0.5_dp*(phi(i+1) - phi(i-1)) / dr
end do
!$omp end parallel do
E(nr) = - (phi(nr) - phi(nr-1)) / dr ! Dirichlet
! E(nr) = 0.0_dp ! Neumann
! Trick to avoid problems at the sheath
E(nr-5:nr) = 0.0_dp
! Update intermediate f
f_i_old = f_i
! Advect in the v direction
! i = 1, v<0
i = 1
if (E(i) >= 0.0_dp) then
f_i(i,2:j0-2) = f_i_old(i,2:j0-2) - Zave(i)*E(i)*dt/dv*(f_i_old(i,2:j0-2) - f_i_old(i,1:j0-3))
else
f_i(i,2:j0-2) = f_i_old(i,2:j0-2) - Zave(i)*E(i)*dt/dv*(f_i_old(i,3:j0-1) - f_i_old(i,2:j0-2))
end if
! i = 2, nr-1; all v
!$omp parallel do
do i = 2, nr-1
if (E(i) >= 0.0_dp) then
f_i(i,2:nv-1) = f_i_old(i,2:nv-1) - Zave(i)*E(i)*dt/dv*(f_i_old(i,2:nv-1) - f_i_old(i,1:nv-2))
else
f_i(i,2:nv-1) = f_i_old(i,2:nv-1) - Zave(i)*E(i)*dt/dv*(f_i_old(i,3:nv) - f_i_old(i,2:nv-1))
end if
end do
!$omp end parallel do
! i = nr, v>=0
i = nr
if (E(i) >= 0.0_dp) then
f_i(i,j0+1:nv-1) = f_i_old(i,j0+1:nv-1) - Zave(i)*E(i)*dt/dv*(f_i_old(i,j0+1:nv-1) - f_i_old(i,j0:nv-2))
else
f_i(i,j0+1:nv-1) = f_i_old(i,j0+1:nv-1) - Zave(i)*E(i)*dt/dv*(f_i_old(i,j0+2:nv) - f_i_old(i,j0+1:nv-1))
end if
! Reset values for next iteration
f_i_old = f_i
fCum_i = fCum_i + f_i_old(rCum_index,:)
! Write output
if (mod(t,everyOutput) == 0 .or. t == nt) then
call writeOutputF(t, dt, nr, r, nv, v, f_i_old)
call writeOutputPhi(t, dt, nr, r, phi, n_e)
call writeOutputMom(t, dt, nr, r, n_i, u_i, T_i, Zave)
call writeOutputFCum(t, dt, r(rCum_index), nv, v, fCum_i)
end if
! Write progress
if (mod(t,everyWrite) == 0) then
write (*, '(I10, A, I10)' ) t, '/', nt
write (*, '(A, ES0.4e3,","ES0.4e3)') 'phi max,min: ', maxval(phi)*phi_ref, minval(phi)*phi_ref
end if
end do
end program plasmaExpansion
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