Fortran 三维球面极坐标阵列的数值积分

我想在r，θ和φ（球面极坐标）的空间上集成一个3d阵列。对于1d，我使用辛普森的1/3规则，但对于3d，我感到困惑。另外，您是否愿意为集成或子例程推荐其他方法？我正在使用Fortran 95。

我已经编写了用于3d集成的Fortran代码，我想我应该与您分享。 计算三维函数积分的代码为：

!This program uses Simpson's 1/3 method to calulate volume 
integral in r,theta & phi.
program SimpsonInteg3d
implicit none
integer::i,j,k
integer, parameter :: N=10,M=360,L=180
integer, parameter:: rmin=0,rmax=N,phimin=0,phimax=M,&
thetamin=0,thetamax=L
double precision,&
dimension(rmin:rmax,thetamin:thetamax,phimin:phimax)::U
real*8, parameter :: pi = 4*atan(1.0),dr=1./N,&
dtheta=pi/(L),dphi=2*pi/M
real*8 :: r(rmin:rmax)=(/(i*dr,i=rmin,rmax)/),&
theta(thetamin:thetamax)=(/(j*dtheta,j=thetamin,thetamax)/),&
p(phimin:phimax)=(/(k*dphi,k=phimin,phimax)/)
real*8::intg
do i=rmin,rmax
do j=thetamin, thetamax
do k=phimin,phimax
    !The function which has to be integrated.
    U(i,j,k)=r(i)* (sin((p(k)))**2) *sin(theta(j))
enddo 
enddo 
enddo

call Integration(Intg,U,r,theta,p)
print*,"Integration of function U using simpson's 1/3=", Intg

end program
!===============================================================!
!Subroutine for calculating integral of a function in 3d.
subroutine Integration(Intg,U,r,theta,p)
implicit none
integer::i,j,k  
integer, parameter :: N=10,M=360,L=180
integer, parameter ::rmin=0,rmax=N,&
phimin=0,phimax=M,thetamin=0,thetamax=L
double precision,& 
dimension(rmin:rmax,thetamin:thetamax,phimin:phimax):: U
real*8:: 
r(rmin:rmax),theta(thetamin:thetamax),p(phimin:phimax),Intg,Ia
double precision,dimension(rmin:rmax)::Itheta
real*8, parameter :: pi = 4*atan(1.0),dr=1./N,&
dtheta=pi/(L),dphi=2*pi/M

Intg=0
Ia=0
do i=rmin+1,rmax-1
call Integtheta(Itheta,i,U,r,theta,p)
if(mod(i,2).eq.0) then
Ia = Ia + 2*Itheta(i)*r(i)**2
else
Ia = Ia + 4*Itheta(i)*r(i)**2
endif
end do
call Integtheta(Itheta,rmin,U,r,theta,p)
call Integtheta(Itheta,rmax,U,r,theta,p)
Intg=(dr/3)*(Itheta(rmin)+Itheta(rmax)+ Ia)

end subroutine Integration

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!Subroutine for calculating integral of U along theta and phi 
subroutine Integtheta(Itheta,i,U,r,theta,p)
implicit none
integer::i,j,k
integer, parameter :: N=10,M=360,L=180
integer, parameter ::rmin=0,rmax=N,&
phimin=0,phimax=M,thetamin=0,thetamax=L
double precision,& 
dimension(rmin:rmax,thetamin:thetamax,phimin:phimax)::U
real*8:: r(rmin:rmax),theta(thetamin:thetamax),p(phimin:phimax)
double precision,dimension(rmin:rmax)::Itheta,Itha
double precision,dimension(rmin:rmax,thetamin:thetamax)::Ip
real*8, parameter :: pi = 4*atan(1.0),dr=1./N,&
dtheta=pi/(L),dphi=2*pi/M


Itheta(i)=0
Itha(i)=0
do j=thetamin+1,thetamax-1
call Integphi(Ip,i,j,U,r,theta,p)
if(mod(j,2).eq.0) then
Itha(i) = Itha(i) + 2*Ip(i,j)*sin(theta(j))
else
Itha(i) = Itha(i) + 4*Ip(i,j)*sin(theta(j))
endif
end do
call Integphi(Ip,i,thetamin,U,r,theta,p)
call Integphi(Ip,i,thetamax,U,r,theta,p)
Itheta(i)=(dtheta/3)*(Ip(i,thetamin)+Ip(i,thetamax)+ Itha(i))

end subroutine Integtheta
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!Subroutine for calculating integral of U along phi
subroutine Integphi(Ip,i,j,U,r,theta,p)
implicit none
integer::i,j,k
integer, parameter :: N=10,M=360,L=180
integer, parameter ::rmin=0,rmax=N,&
phimin=0,phimax=M,thetamin=0,thetamax=L
double precision,& 
dimension(rmin:rmax,thetamin:thetamax,phimin:phimax)::U
real*8:: r(rmin:rmax),theta(thetamin:thetamax),p(phimin:phimax)
double precision,dimension(rmin:rmax,thetamin:thetamax)::Ip,Ipa
real*8, parameter :: pi = 4*atan(1.0),dr=1./N,&
dtheta=pi/(L),dphi=2*pi/M

Ipa(i,j)=0
do k=phimin+1,phimax-1
if(mod(k,2).eq.0) then
Ipa(i,j) = Ipa(i,j) + 2*U(i,j,k)
else
Ipa(i,j)= Ipa(i,j) + 4*U(i,j,k)
endif
end do
Ip(i,j)=(dphi/3)*(U(i,j,phimin)+U(i,j,phimax)+ Ipa(i,j))
end subroutine Integphi

---

它首先计算函数

U

沿φ的积分，然后使用函数

Ip

沿θ计算积分。最后，函数Itheta用于计算沿

r

的积分。请显示您的代码并使用标记。阅读并考虑在其他地方询问你是否想要数学，而不是FORTRAN编码。也许是scicomp.stackexchange.com。如果你在那里问，删除这个问题。对不起，我同意，我应该在数学或scicomp中问这个问题。顺便说一下，我已经算出了，我将发布用于计算三维积分的Fortran代码。