[mihirmakwana6]

I am solving a 1D Advection equation :

du/dt + c*du/dx = 0 ;

c is constant and greater than 0 ( i.e a right moving wave )



At time , t = 0 , i.e Initial condition is a square wave



I took ' c ' such that the wave completes one full revolution (loop) of the domain.


I used FOU for advcetion term and forward euler for time integration :

u_new(i,1) = u_old(i,1) - (c*del_t/del_x)*(u_old(i,1)- u_old(i-1,1))

where i : grid point

where del_t is obtained by CFL.




for periodic B.C , I do the following



for i=imax:-1:1 % imax is max. grid points

if i == 1

u_new(i,1) = u_new(imax,1); % periodic b.c

elseif i== 2

u_new(i,1) = u_old(i,1) - (c*del_t/del_x)*(u_old(i,1) -
u_old(imax,1)); % since u(1,1) = u(imax,1);

else
u_new(i,1) = u_old(i,1) - (c*del_t/del_x)*(u_old(i,1) - u_old(i-1,1));

end

end




so whenever i = 1 appears , I replace it with imax .


Is the implementation of PERIODIC B.C correct?

Thanks in advance

- Mihir

[mihirmakwana6]

Can someone please confirm if the periodic B.C is properly implemented?


Thanks in advance

- Mihir

[FMDenaro]

assuming the periodicity is posed at nodes i=1 and i=N, for c>0 you do the cycle for i =2 to N computing unew(i) from the FTUS.
At the end, you simply set unew(1)=unew(N) ando go again in the time integration.
Nothing else to do...

[mihirmakwana6]

Originally Posted by FMDenaro

assuming the periodicity is posed at nodes i=1 and i=N, for c>0 you do the cycle for i =2 to N computing unew(i) from the FTUS.
At the end, you simply set unew(1)=unew(N) ando go again in the time integration.
Nothing else to do...

thankyou for the reply.

for the FOU scheme

Sir, if you consider the equation for i = 2 , it will have a term u_old(1,1) .

so are you saying that I should NOT change it to u_old(N,1)

[FMDenaro]

Originally Posted by mihirmakwana6

thankyou for the reply.

for the FOU scheme

Sir, if you consider the equation for i = 2 , it will have a term u_old(1,1) .

so are you saying that I should NOT change it to u_old(N,1)

a t=0 uold MUST be known for any i as it corresponds to an initial condition for the Cauchy problem...

[mihirmakwana6]

Originally Posted by FMDenaro

a t=0 uold MUST be known for any i as it corresponds to an initial condition for the Cauchy problem...

sir, I have defined uold @ t = 0 for every i

I just wanted to ask that since we have the periodic condition
u(1) = u(N)

so is it required to change uold(1) to uold(N) in the equation for 2nd grid point

[FMDenaro]

Originally Posted by mihirmakwana6

sir, I have defined uold @ t = 0 for every i

I just wanted to ask that since we have the periodic condition
u(1) = u(N)

so is it required to change uold(1) to uold(N) in the equation for 2nd grid point

here an example of coding

uold(1:N) = initial condition f(x,0)
For t= ... to ...
for i=2 to N
unew(i)= .... FTUS ...
end
unew(1)=unew(N)
uold(1:N)=unew(1:N)
end

[mihirmakwana6]

Originally Posted by FMDenaro

here an example of coding

uold(1:N) = initial condition f(x,0)
For t= ... to ...
for i=2 to N
unew(i)= .... FTUS ...
end
unew(1)=unew(N)
uold(1:N)=unew(1:N)
end

1) sir, I did the same thing as yours. But I thought that since b.c are periodic hence I need to change the equation for i = 2 using an if statement

uold(1:N) = initial condition f(x,0)
For t= ... to ...
for i=2 to N
if i==2
unew(i) = uold(2)-(c*dt/dx)*(uold(2)-uold(N))
else
unew(i) = uold(i)-(c*dt/dx)*(uold(i)-uold(i-1))
end
end
unew(1)=unew(N)
uold(1:N)=unew(1:N)
end

I did this way and I find slight variation in the profile as compared to what you suggested

2) if I use FTCS ( i.e CDS scheme)

that for i = N , I get

unew(N) = uold(N)-(c*dt/dx/2)*(uold(N+1)-uold(N-1))

now , I don't have the point N+1 on the grid ... so what to do

[FMDenaro]

you don't need any IF ....

periodicity is a BC.s, not an initial condition, therefore you have f(1)=f(N) but also f(2)=f(N+1), f(3)=f(N+2) and so on

[mihirmakwana6]

Originally Posted by FMDenaro

you don't need any IF ....

periodicity is a BC.s, not an initial condition, therefore you have f(1)=f(N) but also f(2)=f(N+1), f(3)=f(N+2) and so on

ok.

I am getting the results for FOU and CDS

Sir, now I am trying to implement MUSCL scheme

I used the equation (4.20) with minmod slope limiter given by equation (4.28) & (4.29)
from the following link

http://www.mpia.de/homes/dullemon/le...vection_II.pdf


Is the formula correct

[FMDenaro]

FTCS is unconditionally unstable on the linear advection equation......

[mihirmakwana6]

Originally Posted by FMDenaro

FTCS is unconditionally unstable on the linear advection equation......

Sir, yes .

Ideally the wave should just get advected without any change in the shape , however that does not happen with CDS ( FTCS ) and FOU ( FTUS ) .

CDS scheme is unstable , and results in oscillations.
FOU scheme does give the solution but the square waves gets smooth as it gets advected

I just compared the trend of my solutions with that given on the site
https://en.wikipedia.org/wiki/MUSCL_scheme

So, now I am trying to use MUSCL scheme as it can handle the jump as in the square wave.

I reffered to the site

http://www.mpia.de/homes/dullemon/le...vection_II.pdf

I used the equation (4.20) with minmod slope limiter given by equation (4.28) & (4.29)

So are these equations correct?

[FMDenaro]

Check the FTUS, it must provide the exact solution at courant number=1.

MUSCL is quite good but you can find some more accurate method in the book of LeVeque on FV for hyperbolic equations

[mihirmakwana6]

Originally Posted by FMDenaro

Check the FTUS, it must provide the exact solution at courant number=1.

MUSCL is quite good but you can find some more accurate method in the book of LeVeque on FV for hyperbolic equations

I substituted courant number=1 for FTUS and yes it gives exact solution.

Sir I actually want to implement MUSCL and the book by leveque does not give much insight into it.

So, I referred the site mentioned above. Just wanted to jnow whether the eqautions are right or not.

[FMDenaro]

have also a look here

http://weberknecht.uni-muenster.de/n...vol_script.pdf

[mihirmakwana6]

Originally Posted by FMDenaro

have also a look here

http://weberknecht.uni-muenster.de/n...vol_script.pdf

ok. thankyou for the help.

[mihirmakwana6]

Originally Posted by FMDenaro

here an example of coding

uold(1:N) = initial condition f(x,0)
For t= ... to ...
for i=2 to N
unew(i)= .... FTUS ...
end
unew(1)=unew(N)
uold(1:N)=unew(1:N)
end

sir, say you have 5 cells in 1D with 5 points at the cell centres.

They are named as 1,2,3,4,5

i.e i am using FINITE VOLUME method

then, does the above coding example hold true for periodic B.C. i.e solving from i = 2 to i = 5 and then at the end saying that

unew(1)=unew(5)

[FMDenaro]

Originally Posted by mihirmakwana6

sir, say you have 5 cells in 1D with 5 points at the cell centres.

They are named as 1,2,3,4,5

i.e i am using FINITE VOLUME method

then, does the above coding example hold true for periodic B.C. i.e solving from i = 2 to i = 5 and then at the end saying that

unew(1)=unew(5)

yes, you can use such setting in general and if you have higher order schemes that use more nodes you can set unew(0)=unew(N-1) and so on

[mihirmakwana6]

Originally Posted by FMDenaro

yes, you can use such setting in general and if you have higher order schemes that use more nodes you can set unew(0)=unew(N-1) and so on

Sir, so say now i have 20 grid points i = 1 to 20 ( & thus 20 C.V ) ..

Now say i use fifth order upwind scheme for node 20 ( last node ).
if the flow is from right to left then i will need 3 nodes upstream (this depends on the upwind scheme) of the node 20.
then can i say that

u(21) = u(2)
u(22) = u(3)
u(23) = u(4)

similarly for node 2 with flow from left to right the 3 upstream nodes will be

u(1) --- i know its value
u(0) = u(19)
u(-1) = u(18)


is this correct ???

[FMDenaro]

Originally Posted by mihirmakwana6

Sir, so say now i have 20 grid points i = 1 to 20 ( & thus 20 C.V ) ..

Now say i use fifth order upwind scheme for node 20 ( last node ).
if the flow is from right to left then i will need 3 nodes upstream (this depends on the upwind scheme) of the node 20.
then can i say that

u(21) = u(2)
u(22) = u(3)
u(23) = u(4)

similarly for node 2 with flow from left to right the 3 upstream nodes will be

u(1) --- i know its value
u(0) = u(19)
u(-1) = u(18)


is this correct ???

yes, correct