为了求解受诺依曼边界条件约束的隐式方案的热方程，我们可以写成：

$$T_i^{j+1}-T_i^{j}=\alpha (T_{i+1}^{j+1}-2T_{i}^{j+1}+T_{i-1}^{j+1})$$ $$\textbf{A} T^{n+1} = T^n$$ $$-\alpha T_{i-1}^{n+1}+(1+2 \alpha)T_i^{n+1}-\alpha T_{i+1}^{n+1} =T_i^n$$

初始条件和边界条件为：

$$T(L/4,t)=100 \quad , \frac{\partial T(0,t)}{\partial x} =\frac{\partial T(L,t)}{\partial x} =0$$

其中 alpha 是，是导热系数，L 是棒的长度。从我们的边界条件：$kdt/dx^2$$k$

T(2,j)= T(0,j)
T(n+1,j)=T(n-1,j)

所以我们有一个三对角矩阵，其主对角线为 ( )，下对角线和超对角线为。Neumann.BC 将矩阵的第一行和最后一行更改为$1+2\alpha$$-\alpha$

A(1,1)    A(1,2)  :    1+2\apha    -2\alpha
A(n,n-1)  A(n,n)  :   -2\alpha     1+2\alph

这是我在 FORTRAN 中对问题的实现：

program heat_rod
!
! To solve the heat eauation for a rod with 
! Tt(x,t) =sigma* Txx(x,t)
! T(x,t) = 0 and T(L/4,t)= 100, 
! N.B.C : Tx(0,t) = Tx(L,t) = 0
! x  over [0,1] and t over [0,0.1]
! Variables :
! dl, dd, du subdiagonal, maindiagonal and superdiagonal elements respectively
! To plot the result:
! echo 'splot "data" nonuniform matrix with lines' | gnuplot --persist

implicit none

integer,parameter :: n = 11, m = 31      ! n=21, m=81
integer :: i,j,index_x
real(8) :: alph,l,time
real(8) :: dl(n),dd(n),du(n),d_x,d_t,sigma
real(8) :: T_new(n),T_old(n)


open(unit=7, file="T.txt")

l = 1.d0
time = 1.d-1
d_x = l/real(n-1)
d_t  = time/real(m-1)
sigma = 1.d0 ! coefficient of termal conductivity
alph = (sigma * d_t)/(d_x*d_x)
!print *, alph
if (alph > 0.5) then
print*, "Solution may be unstable"
!stop
endif

du = -alph
dl = -alph

!du(1) = 0.d0         !Dirichlet.B.C
du(1) = -2.d0 * alph  !Neumann B.C
!dl(n) = 0.d0         !Dirichlet B.C
dl(n) = -2.d0 * alph  !Neumann B.C

dd = 1.d0 + 2.d0 * alph
!dd(1) = 1.d0
!dd(n) = 1.d0
!dd(1) = 1.d0 + 2.d0 * alph
!dd(n) = 1.d0 + 2.d0 * alph

T_old = 0.d0
index_x = l/4 *n
!print *, index_x
T_old(index_x) = 100.
!print *,T_old

! print the x positions at the first row
write(7,120) n,((i-1)*d_x,i=1,n)
120 format (I3,999f12.2)

write(7,110) 1,(T_old(i),i=1,n)

do j = 1,m
   call tridag(dl,dd,du,T_old,T_new,n)
   write(7,110) (j)*d_t ,(T_new(i),i=1,n)
   T_old = T_new
enddo

! Printing the results
!write(*,120) ((i-1)*h,i=2,nx-1)
!120 format (7x,' t',7x,'x = ',f5.2,9f11.2)
!do j = 1,m
!    write(7,110) (T_plot(i,j),i=1,n)
!enddo
110 format (f6.3,999f12.6)

!close(7)

 contains

SUBROUTINE tridag(a,b,c,r,u,n)
INTEGER n,NMAX
REAL(8) :: a(n),b(n),c(n),r(n),u(n)
PARAMETER (NMAX=500)
! Solves for a vector u(1:n) of length n the tridiagonal linear set 
! a(1:n) , b(1:n) , c(1:n) , and r(1:n) are input vectors and are not modified.
! Parameter: NMAX is the maximum expected value of n .
INTEGER j
REAL(8) :: bet,gam(NMAX)      !One vector of workspace, gam is needed.
if(b(1).eq.0.)pause 'tridag: rewrite equations'
!If this happens then you should rewrite your equations as a set of order N − 1, with u2
!trivially eliminated.
bet=b(1)
u(1)=r(1)/bet
do  j=2,n
!Decomposition and forward substitution.
gam(j)=c(j-1)/bet
bet=b(j)-a(j)*gam(j)
if(bet.eq.0.)pause 'tridag failed'
!Algorithm fails; see below.
u(j)=(r(j)-a(j)*u(j-1))/bet
enddo 
do  j=n-1,1,-1
!Backsubstitution.
    u(j)=u(j)-gam(j+1)*u(j+1)
enddo 
return
END SUBROUTINE
end program

我怀疑我定义边界条件的方式。如何验证模拟结果？提前感谢您的指导或评论。