[petsc-users] Oscillations in finite difference solution

[Mohammad Mirzadeh]

Along the same lines, also have a look at Essentially Non-Oscillatory (ENO)
and Weighted ENO schemes which basically constructed oscillation-free,
high-order, 'upwind' approximations to grad(f). Any standard text on FD
(such as the one by LeVeque or the one by Strikwerda) have chapters on why
central finite differences fail and why you would need an upwind-based
method.

Mohammad

On Wed, Feb 22, 2012 at 4:24 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:

> On Wed, Feb 22, 2012 at 18:15, Patrick Alken <patrick.alken at colorado.edu>wrote:
>
>> Yes I mean the direct solver residual is around 10e-15. The PETSc
>> residual is 4e00
>>
>
> Did you try -pc_type lu?
>
>>  What sort of PDE are you solving?
>>
>>
>> The PDE is:
>>
>> grad(f) . B = g
>>
>> where B is a known vector field, g is a known scalar function, and f is
>> the unknown scalar function to be determined (I am discretizing this
>> equation for f in spherical coords)
>>
>
> Also known as steady-state advection. This is the most famous test problem
> in which centered differences fails miserably. If you use an upwind method,
> it will be stable, but will also be first order. To get higher than first
> order accuracy, you need a nonlinear spatial discretization (see Godunov's
> Theorem, "total variation diminishing", or "total variation bounded").
>
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