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
a Price for high-order accuracy
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
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
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…
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
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).
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.
2. Monotonic Limiters
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
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.
at one point
ΔS( f ) –ΔS( f*+φ( ΔS( f ))(f – f*))≈
ΔS( f ) (1 –φ2( ΔS( f )))
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