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mod_lorenz63_fwd.f90
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mod_lorenz63_fwd.f90
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module mod_lorenz63_fwd
!$$$ program documentation block
! . . .
! program name: mod_lorenz63_fwd
! programmer: da,cheng org: umd date: 2015-Jan-05
!
! purpose:
! forward model for 3-variable lorenz model.
! The module structure follows Miyoshi's lorenz96 module
!
! revision history:
! 2015-Jan-05 da - creator
! 2018-oct-12 Da,Cheng - merge all schemes together
!
! file dependencies:
!
! attributes:
! language: fortran 90
! machine :
!
!
!$$$ end documentation block
use mod_type, only: r_kind => rdef
implicit none
private
!------public vars & subs
integer, parameter,public :: nx = 3
real(r_kind),parameter,public :: para_default(3) = (/ 10.d0, 28.d0, 8.d0/3 /)
public :: t_lorenz63
public :: step, tl_step, adj_step
public :: lorenz63_rk4, tl_lorenz63_rk4, adj_lorenz63_rk4
!public :: step_rk4, tl_step_rk4, adj_step_rk4 ! RK-4 scheme
!public :: step_ft, tl_step_ft, adj_step_ft ! Forward in time scheme
public :: get_tlm
!public :: get_tlm_1 ! exact TLM
!public :: get_tlm_2 ! approximated TLM with perturbation method
!public :: get_tlm_3 ! appromxiated TLM directly from analytical jacobian
!public :: jacobian
!public :: tendency, tl_tendency, adj_tendency
!------private vars & subs
real(r_kind),private :: dx = 1.D-10
!------solver selection
interface get_tlm
module procedure get_tlm_1
!module procedure get_tlm_2
!module procedure get_tlm_3
endinterface
interface step
module procedure step_rk4
!module procedure step_ft
endinterface
interface tl_step
module procedure tl_step_rk4
!module procedure tl_step_ft
endinterface
interface adj_step
module procedure adj_step_rk4
!module procedure adj_step_ft
endinterface
type t_lorenz63
integer :: nt
real(r_kind) :: dt
real(r_kind) :: para(nx)
real(r_kind) :: x0(nx)
real(r_kind) :: xn(nx)
endtype
!subroutine lorenz63_rk4( xin, xout, para, nt, dt )
contains
!-----------------------------------------------------------------------------------------
! output explicit TLM matrix
!-----------------------------------------------------------------------------------------
!
! exact TLM
!
subroutine get_tlm_1(xin,para,dt,M)
implicit none
real(r_kind),intent(in) :: xin(nx)
real(r_kind),intent(in) :: para(nx)
real(r_kind),intent(in) :: dt
real(r_kind),intent(out) :: M(nx,nx)
real(r_kind) :: xout(nx), xinp(nx), xoutp(nx)
real(r_kind) :: buft
integer :: n
call step(xin, para, dt, buft, xout)
do n = 1, nx
xinp=0.d0
xinp(n) = 1.d0
call tl_step( xin, xinp, para, dt, buft, xout, xoutp )
M(:,n) = xoutp
enddo
endsubroutine
!
! use small perturbed run to approximate TLM
!
subroutine get_tlm_2(xin,para,dt,M)
implicit none
real(r_kind),intent(in) :: xin(nx)
real(r_kind),intent(in) :: para(nx)
real(r_kind),intent(in) :: dt
real(r_kind),intent(out) :: M(nx,nx)
real(r_kind) :: xout(nx), xinp(nx), xoutp(nx)
real(r_kind) :: buft
integer :: n
call step(xin, para, dt, buft, xout)
do n = 1, nx
xinp = xin
xinp(n) = xinp(n) + dx
call step(xinp, para, dt, buft, xoutp)
M(:,n) = (xoutp-xout)/dx
enddo
endsubroutine
!
! use jacobian & fwd scheme to approximate TLM
!
subroutine get_tlm_3(xin,para,dt,M)
implicit none
real(r_kind),intent(in) :: xin(nx)
real(r_kind),intent(in) :: para(nx)
real(r_kind),intent(in) :: dt
real(r_kind),intent(out) :: M(nx,nx)
real(r_kind) :: DJ(nx,nx), I(nx,nx)
integer :: n
I=0.d0
do n = 1, nx
I(n,n) = 1.d0
enddo
call jacobian( xin, para, DJ)
M(:,:) = I+ DJ*dt
endsubroutine
subroutine jacobian(xin, para, Jac)
implicit none
real(r_kind),intent(in) :: xin(nx)
real(r_kind),intent(in) :: para(nx)
real(r_kind),intent(out) :: Jac(nx,nx)
real(r_kind) :: sigma, rho, beta
sigma = para(1)
rho = para(2)
beta = para(3)
Jac(1,:) = (/-sigma, sigma, 0.d0/)
Jac(2,:) = (/rho-xin(3), -1.d0, -xin(1)/)
Jac(3,:) = (/xin(2), xin(1), -beta/)
endsubroutine
!-----------------------------------------------------------------------------------------
! FWD, TL, ADJ of lorenz63 with Runge-Kutta 4th order method
!-----------------------------------------------------------------------------------------
! multi-step RK-4 FWD
!subroutine lorenz63_rk4( xin, para, nt, dt, t, xout )
subroutine lorenz63_rk4( xin, xout, para, nt, dt )
implicit none
! passed args
real(r_kind),intent(in ) :: xin(nx)
real(r_kind),intent(in ) :: para(nx)
integer, intent(in ) :: nt
real(r_kind),intent(in ) :: dt
!real(r_kind),intent(inout) :: t
real(r_kind),intent( out) :: xout(nx)
! local vars
real(r_kind) :: t
real(r_kind) :: cp_xin(nx)
real(r_kind) :: x2(nx), x3(nx), x4(nx)
real(r_kind) :: tend1(nx), tend2(nx), tend3(nx), tend4(nx)
integer :: k
cp_xin(:) = xin(:)
do k = 1, nt
! RK4 1st stage
Call tendency( cp_xin, tend1, para )
x2(:) = cp_xin(:) + 0.5d0 * dt * tend1(:)
! RK4 2nd stage
Call tendency( x2, tend2, para )
x3(:) = cp_xin(:) + 0.5d0 * dt * tend2(:)
! RK4 3rd stage
Call tendency( x3, tend3, para )
x4(:) = cp_xin(:) + dt * tend3(:)
! RK4 4th stage
Call tendency( x4, tend4, para )
xout(:) = cp_xin(:) + &
dt * ( tend1(:) + 2.0d0*tend2(:) + 2.0d0*tend3(:) + tend4(:) )/6.0d0
t = t+dt
cp_xin(:) = xout(:)
enddo
endsubroutine
! multi-step RK-4 TL
!subroutine tl_lorenz63_rk4( xin9, xin, para, nt, dt, t, xout9, xout )
subroutine tl_lorenz63_rk4( xin9, xin, xout9, xout, para, nt, dt)
implicit none
! passed args
real(r_kind),intent(in ) :: xin9(nx)
real(r_kind),intent(in ) :: xin(nx)
real(r_kind),intent(in ) :: para(nx)
integer, intent(in ) :: nt
real(r_kind),intent(in ) :: dt
!real(r_kind),intent(inout) :: t
real(r_kind),intent( out) :: xout9(nx)
real(r_kind),intent( out) :: xout(nx)
! local vars
real(r_kind) :: t
real(r_kind) :: cp_xin(nx)
real(r_kind) :: cp_xin9(nx)
real(r_kind) :: x2(nx), x3(nx), x4(nx)
real(r_kind) :: x29(nx), x39(nx), x49(nx)
real(r_kind) :: tend1(nx), tend2(nx), tend3(nx), tend4(nx)
real(r_kind) :: tend19(nx), tend29(nx), tend39(nx), tend49(nx)
integer :: k
cp_xin(:) = xin(:)
cp_xin9(:) = xin9(:)
do k = 1, nt
! RK4 1st stage
call tl_tendency( cp_xin9, cp_xin, tend19, tend1, para )
x2(:) = cp_xin(:) + 0.5d0 * dt * tend1(:)
x29(:) = cp_xin9(:) + 0.5d0 * dt * tend19(:)
! RK4 2nd stage
call tl_tendency( x29, x2, tend29, tend2, para )
x3(:) = cp_xin(:) + 0.5d0 * dt * tend2(:)
x39(:) = cp_xin9(:) + 0.5d0 * dt * tend29(:)
! RK4 3rd stage
call tl_tendency( x39, x3, tend39, tend3, para )
x4(:) = cp_xin(:) + dt * tend3(:)
x49(:) = cp_xin9(:) + dt * tend39(:)
! RK4 4th stage
call tl_tendency( x49, x4, tend49, tend4, para )
xout(:) = cp_xin(:) + &
dt * ( tend1(:) + 2.0d0*tend2(:) + 2.0d0*tend3(:) + tend4(:) )/6.0d0
xout9(:) = cp_xin9(:) + &
dt * ( tend19(:) + 2.0d0*tend29(:) + 2.0d0*tend39(:) + tend49(:) )/6.0d0
t = t+dt
if (k<nt) then
cp_xin(:) = xout(:)
cp_xin9(:) = xout9(:)
endif
enddo
endsubroutine
! multi-step RK-4 ADJ
!subroutine adj_lorenz63_rk4( xin9, xin, para, nt, dt, t, xout9, xout )
subroutine adj_lorenz63_rk4( xin9, xin, xout9, xout, para, nt, dt )
implicit none
! passed args
real(r_kind),intent(in ) :: xin9(nx)
real(r_kind),intent(inout) :: xin(nx)
real(r_kind),intent(in ) :: para(nx)
integer, intent(in ) :: nt
real(r_kind),intent(in ) :: dt
!real(r_kind),intent(inout) :: t
real(r_kind),intent(inout) :: xout9(nx)
real(r_kind),intent(inout) :: xout(nx)
! local vars
real(r_kind) :: t
real(r_kind) :: cp_xin(nx)
real(r_kind) :: cp_xin9(nx,nt)
real(r_kind) :: x2(nx), x3(nx), x4(nx)
real(r_kind) :: x29(nx,nt), x39(nx,nt), x49(nx,nt)
real(r_kind) :: tend1(nx), tend2(nx), tend3(nx), tend4(nx)
real(r_kind) :: tend19(nx,nt), tend29(nx,nt), tend39(nx,nt), tend49(nx,nt)
integer :: k
! base trajectory calculation
cp_xin9(:,1) = xin9(:)
do k = 1, nt
Call tendency( cp_xin9(:,k), tend19(:,k), para )
x29(:,k) = cp_xin9(:,k) + 0.5d0 * dt * tend19(:,k)
Call tendency( x29(:,k), tend29(:,k), para )
x39(:,k) = cp_xin9(:,k) + 0.5d0 * dt * tend29(:,k)
Call tendency( x39(:,k), tend39(:,k), para )
x49(:,k) = cp_xin9(:,k) + dt * tend39(:,k)
Call tendency( x49(:,k), tend49(:,k), para )
xout9(:) = cp_xin9(:,k) + &
dt * ( tend19(:,k) + 2.0d0*tend29(:,k) + 2.0d0*tend39(:,k) + tend49(:,k) )/6.0d0
if (k<nt) cp_xin9(:,k+1) = xout9(:)
enddo
do k = nt, 1, -1
t = t - dt
xin = 0.d0; x2 = 0.d0; x3 = 0.d0; x4 = 0.d0
tend1 = 0.d0; tend2=0.d0; tend3=0.d0; tend4=0.d0
! ADJ: RK4 4th stage
!xout(:) = xin(:) + &
! dt * ( tend1(:) + 2.0d0*tend2(:) + 2.0d0*tend3(:) + tend4(:) )/6.0d0
xin(:) = xout(:)
tend1(:) = dt*xout(:)/6.d0
tend2(:) = dt*2.0d0*xout(:)/6.d0
tend3(:) = dt*2.0d0*xout(:)/6.d0
tend4(:) = dt*xout(:)/6.d0
!call tl_tendency( x49, x4, tend49, tend4, para )
call adj_tendency( x49(:,k), x4, tend49(:,k), tend4, para )
! ADJ: RK4 3rd stage
!x4(:) = xin(:) + dt * tend3(:)
xin(:) = x4(:) + xin(:)
tend3(:) = dt * x4(:) + tend3(:)
!call tl_tendency( x39, x3, tend39, tend3, para )
call adj_tendency( x39(:,k), x3, tend39(:,k), tend3, para )
! ADJ: RK4 2nd stage
!x3(:) = xin(:) + 0.5d0 * dt * tend2(:)
xin(:) = x3(:) + xin(:)
tend2(:) = 0.5d0 * dt * x3(:) + tend2(:)
!call tl_tendency( x29, x2, tend29, tend2, para )
call adj_tendency( x29(:,k), x2, tend29(:,k), tend2, para )
! ADJ: RK4 1st stage
!x2(:) = xin(:) + 0.5d0 * dt * tend1(:)
xin(:) = x2(:) + xin(:)
tend1(:) = 0.5d0 * dt * x2(:) + tend1(:)
!call tl_tendency( xin9, xin, tend19, tend1, para )
call adj_tendency( cp_xin9(:,k), xin, tend19(:,k), tend1, para )
! go back to the previous step
if (k>1) xout(:) = xin(:)
enddo
endsubroutine
! RK-4 FWD
subroutine step_rk4( xin, para, dt, t, xout )
implicit none
! passed args
real(r_kind),intent(in ) :: xin(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent(in ) :: dt
real(r_kind),intent(inout) :: t
real(r_kind),intent( out) :: xout(nx)
! local vars
real(r_kind) :: x2(nx), x3(nx), x4(nx)
real(r_kind) :: tend1(nx), tend2(nx), tend3(nx), tend4(nx)
! RK4 1st stage
Call tendency( xin, tend1, para )
x2(:) = xin(:) + 0.5d0 * dt * tend1(:)
! RK4 2nd stage
Call tendency( x2, tend2, para )
x3(:) = xin(:) + 0.5d0 * dt * tend2(:)
! RK4 3rd stage
Call tendency( x3, tend3, para )
x4(:) = xin(:) + dt * tend3(:)
! RK4 4th stage
Call tendency( x4, tend4, para )
xout(:) = xin(:) + &
dt * ( tend1(:) + 2.0d0*tend2(:) + 2.0d0*tend3(:) + tend4(:) )/6.0d0
t = t+dt
endsubroutine
! RK-4 TL
subroutine tl_step_rk4( xin9, xin, para, dt, t, xout9, xout )
implicit none
! passed args
real(r_kind),intent(in ) :: xin9(nx)
real(r_kind),intent(in ) :: xin(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent(in ) :: dt
real(r_kind),intent(inout) :: t
real(r_kind),intent( out) :: xout9(nx)
real(r_kind),intent( out) :: xout(nx)
! local vars
real(r_kind) :: x2(nx), x3(nx), x4(nx)
real(r_kind) :: x29(nx), x39(nx), x49(nx)
real(r_kind) :: tend1(nx), tend2(nx), tend3(nx), tend4(nx)
real(r_kind) :: tend19(nx), tend29(nx), tend39(nx), tend49(nx)
! RK4 1st stage
call tl_tendency( xin9, xin, tend19, tend1, para )
x2(:) = xin(:) + 0.5d0 * dt * tend1(:)
x29(:) = xin9(:) + 0.5d0 * dt * tend19(:)
! RK4 2nd stage
call tl_tendency( x29, x2, tend29, tend2, para )
x3(:) = xin(:) + 0.5d0 * dt * tend2(:)
x39(:) = xin9(:) + 0.5d0 * dt * tend29(:)
! RK4 3rd stage
call tl_tendency( x39, x3, tend39, tend3, para )
x4(:) = xin(:) + dt * tend3(:)
x49(:) = xin9(:) + dt * tend39(:)
! RK4 4th stage
call tl_tendency( x49, x4, tend49, tend4, para )
xout(:) = xin(:) + &
dt * ( tend1(:) + 2.0d0*tend2(:) + 2.0d0*tend3(:) + tend4(:) )/6.0d0
xout9(:) = xin9(:) + &
dt * ( tend19(:) + 2.0d0*tend29(:) + 2.0d0*tend39(:) + tend49(:) )/6.0d0
t = t+dt
endsubroutine
! RK-4 ADJ
subroutine adj_step_rk4( xin9, xin, para, dt, t, xout9, xout )
implicit none
! passed args
real(r_kind),intent(in ) :: xin9(nx)
real(r_kind),intent(inout) :: xin(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent(in ) :: dt
real(r_kind),intent(inout) :: t
real(r_kind),intent(inout) :: xout9(nx)
real(r_kind),intent(inout) :: xout(nx)
! local vars
real(r_kind) :: x2(nx), x3(nx), x4(nx)
real(r_kind) :: x29(nx), x39(nx), x49(nx)
real(r_kind) :: tend1(nx), tend2(nx), tend3(nx), tend4(nx)
real(r_kind) :: tend19(nx), tend29(nx), tend39(nx), tend49(nx)
! base trajectory calculation
t = t - dt
Call tendency( xin9, tend19, para )
x29(:) = xin9(:) + 0.5d0 * dt * tend19(:)
Call tendency( x29, tend29, para )
x39(:) = xin9(:) + 0.5d0 * dt * tend29(:)
Call tendency( x39, tend39, para )
x49(:) = xin9(:) + dt * tend39(:)
Call tendency( x49, tend49, para )
xout9(:) = xin9(:) + &
dt * ( tend19(:) + 2.0d0*tend29(:) + 2.0d0*tend39(:) + tend49(:) )/6.0d0
xin = 0.d0; x2 = 0.d0; x3 = 0.d0; x4 = 0.d0
tend1 = 0.d0; tend2=0.d0; tend3=0.d0; tend4=0.d0
! ADJ: RK4 4th stage
!xout(:) = xin(:) + &
! dt * ( tend1(:) + 2.0d0*tend2(:) + 2.0d0*tend3(:) + tend4(:) )/6.0d0
xin(:) = xout(:)
tend1(:) = dt*xout(:)/6.d0
tend2(:) = dt*2.0d0*xout(:)/6.d0
tend3(:) = dt*2.0d0*xout(:)/6.d0
tend4(:) = dt*xout(:)/6.d0
!call tl_tendency( x49, x4, tend49, tend4, para )
call adj_tendency( x49, x4, tend49, tend4, para )
! ADJ: RK4 3rd stage
!x4(:) = xin(:) + dt * tend3(:)
xin(:) = x4(:) + xin(:)
tend3(:) = dt * x4(:) + tend3(:)
!call tl_tendency( x39, x3, tend39, tend3, para )
call adj_tendency( x39, x3, tend39, tend3, para )
! ADJ: RK4 2nd stage
!x3(:) = xin(:) + 0.5d0 * dt * tend2(:)
xin(:) = x3(:) + xin(:)
tend2(:) = 0.5d0 * dt * x3(:) + tend2(:)
!call tl_tendency( x29, x2, tend29, tend2, para )
call adj_tendency( x29, x2, tend29, tend2, para )
! ADJ: RK4 1st stage
!x2(:) = xin(:) + 0.5d0 * dt * tend1(:)
xin(:) = x2(:) + xin(:)
tend1(:) = 0.5d0 * dt * x2(:) + tend1(:)
!call tl_tendency( xin9, xin, tend19, tend1, para )
call adj_tendency( xin9, xin, tend19, tend1, para )
endsubroutine
!-----------------------------------------------------------------------------------------
! FWD, TL, ADJ of lorenz63 with forward in time method
!-----------------------------------------------------------------------------------------
subroutine step_ft( xin, para, dt, t, xout )
implicit none
! passed args
real(r_kind),intent(in ) :: xin(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent(in ) :: dt
real(r_kind),intent(inout) :: t
real(r_kind),intent( out) :: xout(nx)
! local vars
real(r_kind) :: tend1(nx)
Call tendency( xin, tend1, para )
xout(:) = xin(:) + dt * tend1(:)
t = t+dt
endsubroutine
subroutine tl_step_ft( xin9, xin, para, dt, t, xout9, xout )
implicit none
! passed args
real(r_kind),intent(in ) :: xin(nx)
real(r_kind),intent(in ) :: xin9(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent(in ) :: dt
real(r_kind),intent(inout) :: t
real(r_kind),intent( out) :: xout(nx)
real(r_kind),intent( out) :: xout9(nx)
! local vars
real(r_kind) :: tend1(nx)
real(r_kind) :: tend19(nx)
call tl_tendency( xin9, xin, tend19, tend1, para )
xout(:) = xin(:) + dt * tend1(:)
xout9(:) = xin9(:) + dt * tend19(:)
t = t+dt
endsubroutine
subroutine adj_step_ft( xin9, xin, para, dt, t, xout9, xout )
implicit none
! passed args
real(r_kind),intent( out) :: xin(nx)
real(r_kind),intent(in ) :: xin9(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent(in ) :: dt
real(r_kind),intent(inout) :: t
real(r_kind),intent(inout) :: xout(nx)
real(r_kind),intent(inout) :: xout9(nx)
! local vars
real(r_kind) :: tend1(nx)
real(r_kind) :: tend19(nx)
! base trajectory
call step_ft( xin9, para, dt, t, xout9 )
t = t-dt
!TL: xout(:) = xin(:) + dt * tend1(:)
xin(:) = xout(:)
tend1(:) = dt*xout(:)
!TL: call tl_tendency( xin9, xin, tend1, tend19, tend1, para )
call adj_tendency( xin9, xin, tend19, tend1, para )
endsubroutine
!-----------------------------------------------------------------------------------------
! FWD, TL, ADJ of tendency term
!-----------------------------------------------------------------------------------------
subroutine tendency( xin, tend, para )
implicit none
real(r_kind),intent(in ) :: xin(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent( out) :: tend(nx)
tend(1) = -para(1)*( xin(1) - xin(2) )
tend(2) = para(2)*xin(1) - xin(2) - xin(1)*xin(3)
tend(3) = xin(1)*xin(2) - para(3)*xin(3)
endsubroutine
subroutine tl_tendency( xin9, xin, tend9, tend, para )
implicit none
real(r_kind),intent(in ) :: xin(nx)
real(r_kind),intent(in ) :: xin9(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent( out) :: tend(nx)
real(r_kind),intent( out) :: tend9(nx)
! first copy the original code
! then based on the original code, insert the trajectory var directly
! modify fwd codes by adding 9
tend(1) = -para(1)*( xin(1) - xin(2) )
tend9(1) = -para(1)*( xin9(1) - xin9(2) )
tend(2) = para(2)*xin(1) - xin(2) - xin(1)*xin9(3) - xin9(1)*xin(3)
tend9(2) = para(2)*xin9(1) - xin9(2) - xin9(1)*xin9(3)
tend(3) = xin9(1)*xin(2) + xin(1)*xin9(2) - para(3)*xin(3)
tend9(3) = xin9(1)*xin9(2) - para(3)*xin9(3)
endsubroutine
subroutine adj_tendency( xin9, xin, tend9, tend, para )
implicit none
real(r_kind),intent(inout) :: xin(nx)
real(r_kind),intent(in ) :: xin9(nx)
real(r_kind),intent(in ) :: para(nx)
real(r_kind),intent(in ) :: tend(nx)
real(r_kind),intent(inout) :: tend9(nx)
!tend(1) = -para(1)*( xin(1) - xin(2) )
!tend(2) = para(2)*xin(1) - xin(2) - xin(1)*xin9(3) - xin9(1)*xin(3)
!tend(3) = xin9(1)*xin(2) + xin(1)*xin9(2) - para(3)*xin(3)
!tend(3) = xin9(1)*xin(2) + xin(1)*xin9(2) - para(3)*xin(3)
xin(1) = tend(3)*xin9(2) + xin(1)
xin(2) = xin9(1)*tend(3) + xin(2)
xin(3) = -para(3)*tend(3) + xin(3)
!tend(2) = para(2)*xin(1) - xin(2) - xin(1)*xin9(3) - xin9(1)*xin(3)
xin(1) = para(2)*tend(2) + xin(1)
xin(2) = -tend(2) + xin(2)
xin(1) = -tend(2)*xin9(3) + xin(1)
xin(3) = -xin9(1)*tend(2) + xin(3)
!tend(1) = -para(1)*( xin(1) - xin(2) ) = -para(1)*xin(1)+para(1)*xin(2)
xin(1) = -para(1)*tend(1) + xin(1)
xin(2) = para(1)*tend(1) + xin(2)
endsubroutine
endmodule