[saurabh007]

I discretized an unsteady state convection diffusion equation and solved it with both CDS and upwind scheme and Crank Nicholson time step. But the solution of CDS and upwind are same till certain time interval (exactly speaking till 60 iterations) but after that upwind tends to give wrong solution. Is there something wrong with my code or is it because of less accurate upwind scheme.

Please help.
Thanks in advance.

[FMDenaro]

Originally Posted by saurabh007

I discretized an unsteady state convection diffusion equation and solved it with both CDS and upwind scheme and Crank Nicholson time step. But the solution of CDS and upwind are same till certain time interval (exactly speaking till 60 iterations) but after that upwind tends to give wrong solution. Is there something wrong with my code or is it because of less accurate upwind scheme.

Please help.
Thanks in advance.

your question is too general... you used first order upwind? what about your problem (initial and boundary condition). What about your CFL and diffusivity parameters? What about your cell Peclet?
Provide also a graph of the solution

[Martin Hegedus]

Unless artificial dissipation has been added, the central difference method will be unstable.

[FMDenaro]

Originally Posted by Martin Hegedus

Unless artificial dissipation has been added, the central difference method will be unstable.

Actually he wrote that the problem is convection- diffusion, thus even for central difference exists a finite stability region in the (Peh, cfl) plane.

[saurabh007]

The problem is as follows:

• Given a squre block of side length 1m
• Make 101*101 partition

[(∂∅)/∂t+ u (∂∅)/∂x+ v (∂∅)/∂y=D((∂^2∅)/(∂x^2 )+ (∂^2∅)/(∂y^2 ))
u = cos((pi*(x - 0.5))*sin(pi*(y - 0.5))
v = -sin(pi*(x - 0.5))*cos(pi*(y - 0.5))
D=1

Boundary condition: phi = 0 at boundaries.

Initial Condition: phi = cos(pi*(x - 0.5))*cos(pi*(y - 0.5))

I am not able to attach the results because of size constraint but what happens is that after 67 time iterations the result in case of upwind begins to diverge from the exact solution. But in CDS, it is exactly the same.

[Martin Hegedus]

Originally Posted by FMDenaro

Actually he wrote that the problem is convection- diffusion, thus even for central difference exists a finite stability region in the (Peh, cfl) plane.

I could be wrong on all of it. All I can say for sure is that if I don't use enough of it in my 2nd order central difference N.S. code, it blows up and it is independent of the CFL number.

[FMDenaro]

Originally Posted by Martin Hegedus

I could be wrong on all of it. All I can say for sure is that if I don't use enough of it in my 2nd order central difference N.S. code, it blows up and it is independent of the CFL number.

the stability region depends on both CFL and Reh, if you use a small cfl and a small Reh ( about 2-3). You can see an example in the Hirsch book (Green book) for the convection-diffusion problem.

I worked using CN+AB time integration and second order central discretization and obtained a stability region (see attachment).

When you add artificial dissipation you are just working at small Reh, the same effect should be obtained by increasing the mesh resolution

[saurabh007]

Once the steady state is reached, does the upwind scheme has the problem ov overestimation or something like that if we continue for more period of time.

[saurabh007]

I have attached an image where clearly some perturbation is seen when the upwind is run for more time.

[FMDenaro]

Originally Posted by saurabh007

I have attached an image where clearly some perturbation is seen when the upwind is run for more time.

your solution is of order 10^-9, practically you have a vanished solution, the oscillation you see are simply due to the fact that you are approaching the limit of the threshold for the convergence...
You have no numerical instability

[Martin Hegedus]

Originally Posted by FMDenaro

the stability region depends on both CFL and Reh, if you use a small cfl and a small Reh ( about 2-3). You can see an example in the Hirsch book (Green book) for the convection-diffusion problem.

I worked using CN+AB time integration and second order central discretization and obtained a stability region (see attachment).

When you add artificial dissipation you are just working at small Reh, the same effect should be obtained by increasing the mesh resolution

Can't say I'm well versed in the ins and outs of this, but I'm surmising that this depends on what is being diffused by the equation set. For example, I believe the low Mach number (incompressible) N.S. equations diffuse vorticity but not pressure. So some sort of dissipation must be numerically added.

Regardless, the original posters equation set seems to be manufactured so I guess it's set up to work correctly.