Discretization of the transient term for the transport equation [NEWBIE QUESTION]

[boul...@gmail.com]

I want to use a second order implicit time integration scheme to solve
the advection-convection equation over relatively large time steps. I
found two options : Crank–Nicolson and the second order backward
differential formula (cf http://www.cfd-online.com/Wiki/Implicit_second_order_method).

For the Crank–Nicolson numerical scheme, a low CFL number is not
required for stability, but I think it is required for numerical
accuracy, so this method looks less interesting for me. Is the 2e
order BDF a better choice in this case ? I would appreciate advice on
this issue.

Thanks,

Christine

[Maciej Marek]

boul...@gmail.com pisze:

> For the Crank–Nicolson numerical scheme, a low CFL number is not
> required for stability, but I think it is required for numerical
> accuracy, so this method looks less interesting for me.

It's the same with BDF.

> Is the 2e
> order BDF a better choice in this case ? I would appreciate advice on
> this issue.

Why don't you implement both and compare the results?
If that's too much work, take a problem with analytical
solution you know and asses the error for one of the
schemes that you prefer.

It would be nice if you share with us the results
you obtain.

Regards,
Maciej Marek

[boul...@gmail.com]

On 7 sep, 09:12, Maciej Marek <archet_tou...@poczta.onet.pl> wrote:
> boulou...@gmail.com pisze:

This is probably what I will do, but I would like to know if there is
any theorical advantage to use the 2nd order BDF over the Crank–
Nicolson scheme or vice versa ?

[fiv...@gmail.com]

On Sep 7, 7:19 pm, boulou...@gmail.com wrote:
> This is probably what I will do, but I would like to know if there is
> any theorical advantage to use the 2nd order BDF over the Crank–
> Nicolson scheme or vice versa ?

Crank-Nicholson is generally cheaper for given accuracy, but if the
time step is much larger than CFL, it may allow spurious oscillations
(though these will be damped). If these are a problem, the BDF may be
preferable.

BTW, what equation are you solving? You say advection-convection, but
I assume you mean advection-diffusion? I understand advection to mean
the term

a \cdot \nabla u

where the vector field a is known and convection to mean the nonlinear

u \cdot \nabla u

where u is a vector field to be determined (or convection in the sense
of density-driven flows). Note that convection-diffusion is also
often used where I would say advection-diffusion.

Also note that cfd-online describes a stationary/Richardson iteration
for the second order implicit scheme but any solver will do (linear if
your equations are linear).

Jed