QUESTION: Heat diffusion equation is u_t= (D(u)u_x)_x.
Modify this program to investigate the following developments:
Allow for the diffusivity D(u) to change discontinuously, with initial data as u(x,0)= (1+x)(1-x)^2. boundary values u(+-1,t)=0.
Treat in detail the case D(u)=1 when x<1/2 and D(u)=1/2 otherwise.
=> This is my normal code:
program crank_nicolson
implicit none
real, allocatable :: x(:),u(:),a(:),b(:),c(:),d(:)
real:: m,dx,dt,tmax
integer:: j,ni,ji
print*, 'enter the total number of time steps'
read*, ni
print*, 'enter the final time'
read*, tmax
dt=tmax/ni !the size of timestep
print*, 'this gives stepsize of time dt=',dt
dx= 0.1 !delta x =0.1
ji= 20
m = dt/(2*dx**2)
allocate (x(0:ji),u(0:ji+1),a(ji),b(ji),c(ji),d(ji))
open(10,file='crank_nicolsan.m')
!initial condition
do j=1,ji
x(j)= -1+j*dx
u(j)=(1+x(j))*(1-x(j))**2 !x=-1+j*dx
end do
x(0)= -1
x(JI)= 1
!boundary condition
u(0)=0
u(ji+1)=0
do j = 1, ji ! a,b,c are the coefficients of c-n scheme and d is right part
a(j) = -m
b(j) = 1+2.0*m
c(j) = -m
d(j) = m*u(j+1)+(1-2.0*m)*u(j)+m*u(j-1)
end do
call thomas(a,b,c,d)
do j=1,ji-1
u(j)=d(j)
end do
!Print out the Approximate solution in matlab file
write(10,*) 'ApproximateSolution =[',x(0),u(0)
do j =1, JI-1
write(10,*) x(j),u(j)
end do
write(10,*)x(JI),u(JI),']'
write(10,*) "plot(ApproximateSolution(:,1),ApproximateSolution(:,2),'r')"
write(10,*) "xlabel('x'),ylabel('temperature'),legend('Approximate C-N Scheme')"
close(10)
contains
subroutine thomas (a,b,c,d)
real, intent(inout) :: a(:),b(:),c(:),d(:)
integer j
do j = 2,ji !combined decomposition and forward substitution
! a - sub-diagonal (means it is the diagonal below the main diagonal)
! b - the main diagonal
! c - sup-diagonal (means it is the diagonal above the main diagonal)
! d - right part
a(j) = a(j)/b(j-1)
b(j) = b(j)-a(j)*c(j-1)
d(j) = d(j)-a(j)*d(j-1)
end do
!back substitution
d(ji) = d(ji)/b(ji)
do j = ji-1,1,-1
d(j) = (d(j)-c(j)*d(j+1))/b(j)
end do
end subroutine thomas
end program crank_nicolson
(THERE ARE NO ERRORS IN THE CODE. JUST NEED TO MODIFY THE ABOVE CODE Allowing for the diffusivity D(u) to change discontinuously WITH the case D(u)=1 when x<1/2 and D(u)=1/2 otherwise.)