! D03PPF Example Program Text
! Mark 27.2 Release. NAG Copyright 2021.
Module d03ppfe_mod
! D03PPF Example Program Module:
! Parameters and User-defined Routines
! .. Use Statements ..
Use nag_library, Only: nag_wp
! .. Implicit None Statement ..
Implicit None
! .. Accessibility Statements ..
Private
Public :: bndary, exact, monitf, pdedef, &
uvinit
! .. Parameters ..
Real (Kind=nag_wp), Parameter :: four = 4.0_nag_wp
Real (Kind=nag_wp), Parameter :: one = 1.0_nag_wp
Real (Kind=nag_wp), Parameter :: ptone = 0.1_nag_wp
Real (Kind=nag_wp), Parameter, Public :: half = 0.5_nag_wp
Real (Kind=nag_wp), Parameter, Public :: two = 2.0_nag_wp
Real (Kind=nag_wp), Parameter, Public :: zero = 0.0_nag_wp
Integer, Parameter, Public :: itrace = 0, m = 0, nin = 5, &
nout = 6, npde = 1, nv = 0, &
nxfix = 0, nxi = 0
! .. Local Scalars ..
Real (Kind=nag_wp), Public, Save :: e
Contains
Subroutine uvinit(npde,npts,nxi,x,xi,u,nv,v)
! .. Scalar Arguments ..
Integer, Intent (In) :: npde, npts, nv, nxi
! .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: u(npde,npts), v(nv)
Real (Kind=nag_wp), Intent (In) :: x(npts), xi(nxi)
! .. Local Scalars ..
Real (Kind=nag_wp) :: a, b, c, t
Integer :: i
! .. Intrinsic Procedures ..
Intrinsic :: exp
! .. Executable Statements ..
t = zero
Do i = 1, npts
a = (x(i)-0.25_nag_wp-0.75_nag_wp*t)/(four*e)
b = (0.9_nag_wp*x(i)-0.325_nag_wp-0.495_nag_wp*t)/(two*e)
If (a>zero .And. a>b) Then
a = exp(-a)
c = (0.8_nag_wp*x(i)-0.4_nag_wp-0.24_nag_wp*t)/(four*e)
c = exp(c)
u(1,i) = (half+ptone*c+a)/(one+c+a)
Else If (b>zero .And. b>=a) Then
b = exp(-b)
c = (-0.8_nag_wp*x(i)+0.4_nag_wp+0.24_nag_wp*t)/(four*e)
c = exp(c)
u(1,i) = (ptone+half*c+b)/(one+c+b)
Else
a = exp(a)
b = exp(b)
u(1,i) = (one+half*a+ptone*b)/(one+a+b)
End If
End Do
! There are no coupled ODEs in this problem (nv = 0):
v(:) = 0._nag_wp
Return
End Subroutine uvinit
Subroutine pdedef(npde,t,x,u,ux,nv,v,vdot,p,q,r,ires)
! .. Scalar Arguments ..
Real (Kind=nag_wp), Intent (In) :: t, x
Integer, Intent (Inout) :: ires
Integer, Intent (In) :: npde, nv
! .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: p(npde,npde), q(npde), r(npde)
Real (Kind=nag_wp), Intent (In) :: u(npde), ux(npde), v(nv), vdot(nv)
! .. Executable Statements ..
p(1,1) = one
r(1) = e*ux(1)
q(1) = u(1)*ux(1)
Return
End Subroutine pdedef
Subroutine bndary(npde,t,u,ux,nv,v,vdot,ibnd,beta,gamma,ires)
! .. Scalar Arguments ..
Real (Kind=nag_wp), Intent (In) :: t
Integer, Intent (In) :: ibnd, npde, nv
Integer, Intent (Inout) :: ires
! .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)
Real (Kind=nag_wp), Intent (In) :: u(npde), ux(npde), v(nv), vdot(nv)
! .. Local Scalars ..
Real (Kind=nag_wp) :: a, b, c, ue, x
! .. Intrinsic Procedures ..
Intrinsic :: exp
! .. Executable Statements ..
beta(1) = zero
If (ibnd==0) Then
x = zero
a = (x-0.25_nag_wp-0.75_nag_wp*t)/(four*e)
b = (0.9_nag_wp*x-0.325_nag_wp-0.495_nag_wp*t)/(two*e)
If (a>zero .And. a>b) Then
a = exp(-a)
c = (0.8_nag_wp*x-0.4_nag_wp-0.24_nag_wp*t)/(four*e)
c = exp(c)
ue = (half+ptone*c+a)/(one+c+a)
Else If (b>zero .And. b>=a) Then
b = exp(-b)
c = (-0.8_nag_wp*x+0.4_nag_wp+0.24_nag_wp*t)/(four*e)
c = exp(c)
ue = (ptone+half*c+b)/(one+c+b)
Else
a = exp(a)
b = exp(b)
ue = (one+half*a+ptone*b)/(one+a+b)
End If
Else
x = one
a = (x-0.25_nag_wp-0.75_nag_wp*t)/(four*e)
b = (0.9_nag_wp*x-0.325_nag_wp-0.495_nag_wp*t)/(two*e)
If (a>zero .And. a>b) Then
a = exp(-a)
c = (0.8_nag_wp*x-0.4_nag_wp-0.24_nag_wp*t)/(four*e)
c = exp(c)
ue = (half+ptone*c+a)/(one+c+a)
Else If (b>zero .And. b>=a) Then
b = exp(-b)
c = (-0.8_nag_wp*x+0.4_nag_wp+0.24_nag_wp*t)/(four*e)
c = exp(c)
ue = (ptone+half*c+b)/(one+c+b)
Else
a = exp(a)
b = exp(b)
ue = (one+half*a+ptone*b)/(one+a+b)
End If
End If
gamma(1) = u(1) - ue
Return
End Subroutine bndary
Subroutine monitf(t,npts,npde,x,u,r,fmon)
! .. Scalar Arguments ..
Real (Kind=nag_wp), Intent (In) :: t
Integer, Intent (In) :: npde, npts
! .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: fmon(npts)
Real (Kind=nag_wp), Intent (In) :: r(npde,npts), u(npde,npts), x(npts)
! .. Local Scalars ..
Real (Kind=nag_wp) :: drdx, h
Integer :: i
! .. Intrinsic Procedures ..
Intrinsic :: abs
! .. Executable Statements ..
fmon(1) = abs((r(1,2)-r(1,1))/((x(2)-x(1))*half))
Do i = 2, npts - 1
h = (x(i+1)-x(i-1))*half
! Second derivative ..
drdx = (r(1,i+1)-r(1,i))/h
fmon(i) = abs(drdx)
End Do
fmon(npts) = fmon(npts-1)
Return
End Subroutine monitf
Subroutine exact(t,x,npts,u)
! Exact solution (for comparison purposes)
! .. Scalar Arguments ..
Real (Kind=nag_wp), Intent (In) :: t
Integer, Intent (In) :: npts
! .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: u(npts)
Real (Kind=nag_wp), Intent (In) :: x(npts)
! .. Local Scalars ..
Real (Kind=nag_wp) :: a, b, c
Integer :: i
! .. Intrinsic Procedures ..
Intrinsic :: exp
! .. Executable Statements ..
Do i = 1, npts
a = (x(i)-0.25_nag_wp-0.75_nag_wp*t)/(four*e)
b = (0.9_nag_wp*x(i)-0.325_nag_wp-0.495_nag_wp*t)/(two*e)
If (a>zero .And. a>b) Then
a = exp(-a)
c = (0.8_nag_wp*x(i)-0.4_nag_wp-0.24_nag_wp*t)/(four*e)
c = exp(c)
u(i) = (half+ptone*c+a)/(one+c+a)
Else If (b>zero .And. b>=a) Then
b = exp(-b)
c = (-0.8_nag_wp*x(i)+0.4_nag_wp+0.24_nag_wp*t)/(four*e)
c = exp(c)
u(i) = (ptone+half*c+b)/(one+c+b)
Else
a = exp(a)
b = exp(b)
u(i) = (one+half*a+ptone*b)/(one+a+b)
End If
End Do
Return
End Subroutine exact
End Module d03ppfe_mod
Program d03ppfe
! D03PPF Example Main Program
! .. Use Statements ..
Use d03ppfe_mod, Only: bndary, e, exact, half, itrace, m, monitf, nin, &
nout, npde, nv, nxfix, nxi, pdedef, two, uvinit, &
zero
Use nag_library, Only: d03pck, d03ppf, d03pzf, nag_wp
! .. Implicit None Statement ..
Implicit None
! .. Local Scalars ..
Real (Kind=nag_wp) :: con, dx, dxmesh, tout, trmesh, ts, &
x0, xmid, xratio
Integer :: i, ifail, ind, intpts, ipminf, it, &
itask, itol, itype, lenode, lisave, &
lrsave, neqn, npts, nrmesh, nwkres
Logical :: remesh, theta
Character (1) :: laopt, norm
! .. Local Arrays ..
Real (Kind=nag_wp) :: algopt(30), atol(1), rtol(1), &
xfix(nxfix), xi(nxi)
Real (Kind=nag_wp), Allocatable :: rsave(:), u(:), ue(:), uout(:,:,:), &
x(:), xout(:)
Integer, Allocatable :: isave(:)
! .. Intrinsic Procedures ..
Intrinsic :: min, real
! .. Executable Statements ..
Write (nout,*) 'D03PPF Example Program Results'
! Skip heading in data file
Read (nin,*)
Read (nin,*) npts, intpts, itype
lisave = 25 + nxfix
neqn = npde*npts + nv
nwkres = npde*(npts+3*npde+21) + 7*npts + nxfix + 3
lenode = 11*neqn + 50
lrsave = neqn*neqn + neqn + nwkres + lenode
Allocate (u(neqn),ue(intpts),uout(npde,intpts,itype),rsave(lrsave), &
x(npts),xout(intpts),isave(lisave))
Read (nin,*) itol
Read (nin,*) atol(1), rtol(1)
Read (nin,*) e
! Initialize mesh
Do i = 1, npts
x(i) = real(i-1,kind=nag_wp)/real(npts-1,kind=nag_wp)
End Do
! Set remesh parameters
remesh = .True.
nrmesh = 3
dxmesh = half
con = two/real(npts-1,kind=nag_wp)
xratio = 1.5_nag_wp
ipminf = 0
norm = 'A'
laopt = 'F'
ind = 0
itask = 1
! Set theta to .TRUE. if the Theta integrator is required
theta = .False.
algopt(1:30) = zero
If (theta) Then
algopt(1) = two
Else
algopt(1) = zero
End If
! Loop over output value of t
ts = zero
tout = zero
Do it = 1, 5
xmid = half + half*tout
tout = 0.2_nag_wp*real(it,kind=nag_wp)
! ifail: behaviour on error exit
! =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call d03ppf(npde,m,ts,tout,pdedef,bndary,uvinit,u,npts,x,nv,d03pck, &
nxi,xi,neqn,rtol,atol,itol,norm,laopt,algopt,remesh,nxfix,xfix, &
nrmesh,dxmesh,trmesh,ipminf,xratio,con,monitf,rsave,lrsave,isave, &
lisave,itask,itrace,ind,ifail)
If (it==1) Then
Write (nout,99998) atol, npts
Write (nout,99993) nrmesh
Write (nout,99992) e
Write (nout,*)
End If
! Set output points ..
dx = 0.1_nag_wp
If (tout>half) Then
dx = 0.05_nag_wp
End If
x0 = xmid - half*real(intpts-1,kind=nag_wp)*dx
Do i = 1, intpts
xout(i) = x0
x0 = x0 + dx
End Do
xout(intpts) = min(xout(intpts),x(npts))
Write (nout,99999) ts
Write (nout,99996) xout(1:intpts)
! Interpolate at output points ..
ifail = 0
Call d03pzf(npde,m,u,npts,x,xout,intpts,itype,uout,ifail)
! Check against exact solution ..
Call exact(ts,xout,intpts,ue)
Write (nout,99995) uout(1,1:intpts,1)
Write (nout,99994) ue(1:intpts)
End Do
Write (nout,99997) isave(1), isave(2), isave(3), isave(5)
99999 Format (' T = ',F6.3)
99998 Format (/,/,' Accuracy requirement =',E10.3,' Number of points = ',I3, &
/)
99997 Format (' Number of integration steps in time = ',I6,/,' Number o', &
'f function evaluations = ',I6,/,' Number of Jacobian eval', &
'uations =',I6,/,' Number of iterations = ',I6)
99996 Format (1X,'X ',5F9.4)
99995 Format (1X,'Approx sol. ',5F9.4)
99994 Format (1X,'Exact sol. ',5F9.4,/)
99993 Format (2X,'Remeshing every',I3,' time steps',/)
99992 Format (2X,'E =',F8.3)
End Program d03ppfe