Limiters for LBM

Alexander N. GorbanUniversity of Leicester

Centre for Mathematical ModellingJoint work with

Robert A. Brownlee and Jeremy Levesley

Plan

• 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

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

Tadmor&Zhong, 2006

The difference operators produce systems with non-decaying oscillations

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 hyperbolic region.

Numerical solutions of the given 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.1. Idea and notations

Limiter in action:

Limiters do not change the macroscopic variables (moments)

Macroscopic variables M

Distribution f

Equilibrium f*M

f

f*m(f)

Non-equilibrium part of distribution f Limiter in action

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

The Ehrenfests’ coarse-graining: two “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).

The Ehrenfests’ chain.

P. Ehrenfest, T. Ehrenfest-Afanasyeva. The Conceptual Foundations of theStatistical Approach in Mechanics, In: Enziklopädie der Mathematischen Wissenschaften, vol. 4. (Leipzig 1911). Reprinted: Dover, Phoneix 2002.

Ehrenfests’ coarse-graining limiters for LBM

We proposed a LBM in which the difference between microscopic current andmacroscopic 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 1:2shock 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 filtersWe 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 filterThe mean filter is a simple sliding-window spatial filter that replaces the center value in the window with the average (mean) of all the pixel values in the window. The window, or kernel, is usually square but can be any shape. An example of mean filtering of a single 3x3 window of values is shown below

unfiltered values

5 3 6

2 1 9

8 4 7

mean filtered

* * *

* 5 *

* * *

Median filterThe median filter is also a sliding-window spatial filter, but it replaces the center value in the window with the median of all the pixel values in the window. As for the mean filter, the kernel is usually square but can be any shape. An example of median filtering of a single 3x3 window of values is shown below.

unfiltered values

6 2 0

3 97 4

19 3 10

median filtered

* * *

* 4 *

* * *

5 + 3 + 6 + 2 + 1 + 9 + 8 + 4 + 7 = 4545 / 9 = 5

in order:0, 2, 3, 3, 4, 6, 10, 15, 97

From online textbook

18

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 noise. •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 filterThe nonequilibrium entropy ΔS field: the highly nonequilibrium impulse noise should be erased. The first choice gives the median filter

One point median filteringIf we change the maximal value (ΔS-ΔSmed) (at one point!) it still works:

One point median filtering (a) ν=1/3·10−1; (b) ν=10−9.

no filtering no filtering

Lid-driven cavity on 100×100 gridRe=1000-8000First Hopf bifurcation

In literature, for the first bifurcation Re = 7400–8500

Spectrum for Re = 7375

Energy of the signal at one observation point

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:

Conclusion• 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!