Limiters for LBM

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					        Limiters for LBM
         Alexander N. Gorban
        University of Leicester
  Centre for Mathematical Modelling
            Joint work with
Robert A. Brownlee and Jeremy Levesley
•   Dispersive Oscillations
•   Idea of Limiters
•   Point-wise limiters for LBM
•   Nonlinear Median versus linear Mean filters
•   Viscosity adjustment
•   Entropy calculation
•   Conclusion
                  Dispersive oscillations:
              a Price for high-order accuracy

    S.K. Godunov
                                                                    Tadmor&Zhong, 2006
To prevent nonphysical oscillations, most upwind schemes
employ limiters that reduce the spatial accuracy to first order
through shock waves. In fact, Godunov showed that the capturing
of a discontinuity without oscillation required that the spatial
accuracy of the scheme reduce to first order.
                                                   C.J. Roy, 2003
     The difference operators produce
     systems with non-decaying
   In 1944, von Neumann conjectured
that the mesh-scale oscillations in
velocity can be interpreted as heat
energy produced by the irreversible
action of the shock wave, and that as
step h→0, the approximations converge
in the weak sense to the appropriate
discontinuous solution of the equations.

  P.D. Lax and demonstrated that this is
wrong because of unavoidable
(integrable) oscillation of finite amplitude
in the differential approximation to the
difference scheme.
   For weak, but not strong limits
     Levermore & Liu modulation equation

The Hopf initial-value problem

The semidiscrete difference scheme

The central averages

The modulation equations`

They describe the weak limit of the
modulated period-two oscillations so long as
a solution of these equations remains
sufficiently regular and inside the strictly   Numerical solutions of the given
hyperbolic region.                             scheme with initial data uin(x) = -x1/3
Entropy Conservation does not help. I

                 We construct a new family of entropy stable
                 difference schemes which retain the precise
                 entropy decay of the Navier–Stokes equations,
Entropy conservation does not help. II

                              Entropic LBGK collision
                              (Gorban&Karlin LNP660, 2005)
                     If somebody would like to claim that he
                     suppressed the dispersive oscillations,
                     then the best advice is: please look for
                     the hidden additional dissipation.
Idea of limiters
                               LBM Limiters.
    Distribution f
                          1. Idea and notations
                         Limiter in action
                                                   Non-equilibrium part of distribution f

                     Equilibrium f*M             f*m(f)

                                                               Macroscopic variables M

Limiter in action:

Limiters do not change the macroscopic variables (moments)
Let us recall…
Positivity Rule
     There is a simple recipe for positivity preservation: to substitute
     nonpositive result of collisions F(f) by the closest nonnegative state that
     belongs to the straight line
Positivity rule was tested for lid-driven
       cavity and shock tube by:

 It works (stabilises the system), but why?
      Histograms of Nonequlibrium Entropy

Histograms of nonequilibrium entropy ΔS for the 1:2 athermal shock tube simulation with
(a) ν=0.066, (b) ν=0.0066 and (c) ν=0.00066 after 400 time steps using LBGK without any
limiter. Entropic equilibria with perfect entropy are used. The x-axis interval is from zero to
(a) 450E(ΔS), (b) 97E(ΔS) and (c) 32E(ΔS), respectively. It is divided into 20 bins. The tails are
much heavier than exponential ones: (a) p < 10−170, (b) p < 10−21 and (c) p < 10−8
              Ehrenfests’ coarse-graining
                                          P. Ehrenfest, T. Ehrenfest-Afanasyeva. The
                                          Conceptual Foundations of the
                                          Statistical Approach in Mechanics, In: Enziklopädie
                                          der Mathematischen Wissenschaften, vol. 4.
                                          (Leipzig 1911). Reprinted: Dover, Phoneix 2002.

The Ehrenfests’ coarse-graining: two                    The Ehrenfests’ chain.
“motion – coarse-graining” cycles in 1D
(a, values of probability density are
presented by the height of the columns)
and one such cycle in 2D (b).
                Ehrenfests’ coarse-graining
                     limiters for LBM
We proposed a LBM in which the difference
between microscopic current and
macroscopic equilibrium entropy is monitored
in the simulation

the populations are returned to their
equilibrium states (Ehrenfests’ step) if a
threshold value is exceeded.
                                               Density profile of the isothermal 1:2
                                               shock tube simulation after 300 time
                                               steps using (a) LBGK 3; (c) LBGK-ES 7
                                               with threshold δ=10−3. Sites where
                                               Ehrenfests’ steps are employed are
                                               indicated by crosses.
    LBM Limiters.
2. Monotonic Limiters
                                   Entropic filters
        We trim nonequilibrium entropy: ΔS→ΔSt

      here, ΔSt is the filtered field of the nonequilibrium entropy ΔS.
      How should we select the proper filter?
      There are two “first choices”: the mean filter and the median filter
Mean filter                                                       Median filter
The mean filter is a simple sliding-window spatial filter         The median filter is also a sliding-window spatial filter,
that replaces the center value in the window with the             but it replaces the center value in the window with the
average (mean) of all the pixel values in the window.             median of all the pixel values in the window. As for
The window, or kernel, is usually square but can be any           the mean filter, the kernel is usually square but can
shape. An example of mean filtering of a single 3x3               be any shape. An example of median filtering of a
window of values is shown below                                   single 3x3 window of values is shown below.

 unfiltered values     mean filtered                               unfiltered values        median filtered
  5     3      6       *     *     *                                6      2      0         *       *     *
  2     1      9       *     5     *                                3     97      4         *       4     *
  8     4      7       *     *     *                               19      3     10         *       *     *

 5 + 3 + 6 + 2 + 1 + 9 + 8 + 4 + 7 = 45                              in order:
 45 / 9 = 5                                                          0, 2, 3, 3, 4, 6, 10, 15, 97
                                           From online textbook
                   Median versus Mean
•Here we use window sizes
of 3 and 5 samples.
•The first two columns
show a step function,
degraded by some random
•The two last columns show
a noisy straight line, and in
addition one and two
samples, which are
considerably different from
the neighbor samples.

                           From online textbook
           Entropic median filter
The nonequilibrium entropy ΔS field: the highly nonequilibrium impulse
noise should be erased. The first choice gives the median filter
    One point median filtering
If we change the maximal value (ΔS-ΔSmed) (at one point!) it still works:

                       no filtering                            no filtering

           One point median filtering (a) ν=1/3·10−1; (b) ν=10−9.
  Lid-driven cavity on 100×100 grid
                            First Hopf bifurcation
In literature, for the first bifurcation
Re = 7400–8500

                                           Energy of the signal at one observation point
  Spectrum for Re = 7375
Adjustable viscosity

Here “we” stands
for Paul J. Dellar
            Filters with adjustable viscosity

The second idea is to filter the non-equilibrium parts of the distribution functions.

 What is the difference? Here, not only the norm of the nonequilibrium part
 of the distribution function changes but the direction also.
 The possible choice of coefficients is rich even for linear filters.

 Which filter is better? It depends on the purposes and criteria.
      Dissipation control

Additional dissipation:

at one point
ΔS( f ) –ΔS( f*+φ( ΔS( f ))(f – f*))≈
ΔS( f ) (1 –φ2( ΔS( f )))

in total:
ΣxΔS( f ( x )) (1 –φ2( ΔS( f ( x ))))
         How can we calculate ΔS ?

Do we need other entropies?
The collection is here:
• Dispersive oscillations are unavoidable for high-order
  schemes in areas with steep gradients.
• Hence, we should choose between spurious oscillation in
  high order non-monotone methods and additional
  dissipation in first-order methods.
• LBMs are very convenient for constructing of limiters and
  filters we can change the nonequilibrium part of the
  distribution and do not touch the macroscopic fields.
• The family of nonequilibrium limiters and filters is rich
  and it may be possible to construct a proper limiter and
  filter to resolve many known difficulties.
• In areas with steep gradients the entropic median limiters
  erase spurious oscillations without blurring of shocks, and
  do not affect smooth solutions.
• The first choice for ΔS gives the Kullback divergence.
It is useful and interesting to
construct limiters and filters
           for LBM!

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