[Nek5000-users] P101 and P103

[nek5000-users at lists.mcs.anl.gov]

These parameters control the filter that is applied to the velocity and thermal fields at the
end of each time step.

p103 controls the amplitude of the filter, 101 controls the number of modes that are filtered.

The modal decomposition is as follows:

In each direction, on each element, the solution is represented as a polynomial of degree N
in the reference element (i.e., on the interval [-1,1]).

For any Nth-order polynomial there is a modal expansion of the form:

   u(x) = \sum_k=0^N   uh_k phi_k(x)

where   phi_0 = 1, phi_1 = x, and phi_k(x)=L_k(x) - L_{k-1} (x) for k > 1, where L_k(x)=kth Legendre polynomial.

With this modal decomposition and p101=0, the effect of the filter is to construct

   ub(x) = sum_k=0^N sigma_k uh_k phi_k(x)   ~= u(x)

where sigma_k = 1 for k < N and sigma_N = 1-alpha, where alpha:=p103 is the filter strength.

alpha=1 corresponds to a sharp cut-off filter and produces a less smooth result than, say, alpha=0.05--0.25,
which are typical filter strengths.

If p101 > 0, then more modes are filtered.   In this case, one has

       sigma_k = 1  for  k < N-p101.

       sigma_k = 1-alpha*p(k)  for k = N-p101, ..., N

where p(k) is the parabola given by:

       p(k) = [ (k-kc)/(N-kc) ]^2, and kc := N-(p101+1)

The net result is a filter that is flat in modal space for k < N-p101 and which tapers off to the 
value alpha as k-->N.

The overall effect of the filter is to suppress high wavenumber instabilities.  It is easy to
show that the filter does not inhibit exponential convergence, i.e., the effect of the filter
is nil when the solution is well resolved.

References for this filter are Fischer & Mullen, CRAS 2001, and J.P. Boyd, JCP, 1998.

Paul

________________________________________
From: nek5000-users-bounces at lists.mcs.anl.gov [nek5000-users-bounces at lists.mcs.anl.gov] on behalf of nek5000-users at lists.mcs.anl.gov [nek5000-users at lists.mcs.anl.gov]
Sent: Friday, February 05, 2016 7:52 AM
To: nek5000-users at lists.mcs.anl.gov
Subject: [Nek5000-users] P101 and P103

Hi Neks,
can someone explain clearly how this parameters (101 and 103) in .rea
file, works exactly?
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