Asymptotic analysis of

lattice Boltzmann methods

Martin Rheinländer

Applications of asymptotic analysisICMMES 2006

Arbeitsgruppe Numerik (LS Junk)

Fachbereich Mathematik & Statistik

Universität Konstanz

Stability and multiscale expansions

25th July 06, Hampton Virginia

Introduction Intention & Overview

Problem:

consistency analysis → relatively simple

stability analysis → complicated, tricky

Background-question: Can formal asymptotic expansions help to formulate hypotheses about the stability behavior of a numerical scheme?

Here: Case study of a model problem.

Additional motivation: complete understanding of an exemplary lattice Boltzmann algorithm with all inherent features like:

convergence, time-scales, initial layers, boundary layers, stability, consistency,

spectrum of evolution operator, etc.

1st part: Stability analysis based on diagonalization of evolution matrix

→ attempt to be mathematically rigorous

2nd part: (involving more intuition) presents twoscale expansions as

possible & desirable tool to analyze stability

→ comparison and short discussion with stability analysis

Outline:

Introduction The model algorithm

Velocity set:

)( IEJ q −= ω

[ ]kkkk xtJxtxsxtt ),F(),(F),(F +=∆+∆+Update rule:

BGK collision operator:

11 −=s 12 =sPopulation functions:

21 F,F

Equilibrium: )E(UEq =F Mass moment: 21 FF +=U

Scalinghyperbolic parabolic

hthx =∆=∆ ,

UasU kk )()(E += 121

0=∂+∂ uau xt

2hthx =∆=∆ ,

UahsU kk )()(E += 121

02

211 =∂−−∂+∂ uuau xxt )(ω

Abbreviations: ∃∀ means ‚for All‘, means ‚it Exists‘.

Stability Matrix formulation of the algorithm

=

δδ

δδ

γγ

γγ

ββ

ββ

αα

αα

...

...

...

...

...

...

...

......

...

...

...

...

...

...

...

C

=

...

...

...

...

....

....

....

........

....

....

....

...

...

...

...

1

1

1

1

1

1

1

1

T

Iteration = evolution step

• collision (nodal operation)

• transport (left/right shift)

),F()(),(F~

) xtJIxt +=1

=

),(F

),(F

),(F~

),(F~

xt

xt

xt

xt

2

1

2

1

δγβα

),(F~

),(F) hsxtxht kkk −=+2

1sasright the tohops

1sasleft the tohops

22

11

=

−=

F~

F~

Stability Characteristic polynomial of the evolution matrix

Spectrum of shift matrices L ,R ( transport matrix): unit roots Nwwww Ni2

e:...,,, 2π

=

⇒ Characteristic polynomial decomposes into product of quadratic polynomials:.

( )( )[ ]∏−

=

−−−1

0

N

m

mm ww βγλδλαλ a

( )[ ] ( )ωλϕωϕωλϕλχ −++−+= 1sinicos22 a:);(

);(N mπλχ 2

Eigenvectors discrete Fourier transform yields diagonalized transport matrix,

which respects special structure of collision matrix.

⇒

transport

=

II

II

R

LE

δγβα

0

0:

collision

Block-structure of

evolution matrix:

Stability Spectrum of the evolution matrix & spectral portraits

Eigenvalues of evolution operator associated to arbitrary grid are contained in:

( ){ }0;with20 =∈∃∈= ϕλχπϕλω ),[:),( CaS

),()(spec aE ωS⊂ samples spectral limit set uniformly w.r.t. ϕ

60.=ωunder-relaxation 10 <<ω

61.=ωover-relaxation 21 <<ω

Stability Spectral portraits & Observations

Observation: 1020 1 ≤⇔⊂∈∀ aDa )(),(:],[ ωω S

],[)(),(:],[ 20011 1 ∈⇔⊂−∈∀ ωω Daa S

],[],[)(),( 11or2001 −∉∉⇔⊄ aDa ωωS

stable

unstable

30.=a 60.=a

Stability CFL condition

CFL condition:

=

==

∆∆

≤hh

h

hh

t

xa

12

1 hyperbolic scaling

parabolic scaling 1≤⇔ ah

Sketch of the proof: Estimate the zeros of depending on :);( ϕλχ ω,a

[ ] [ ] )(sincos)(sincos)()( ωϕωϕωϕωϕωϕλ −−+−±+−−= 1i2i22

41

21 aa

Better idea: consider first special cases like or .

Discuss then the general case using the theorem of Rouché (→ complex analysis).■

1=ω πϕϕ == ,0

N.B.: The CFL-condition does not hold if the periodic boundary conditions are replaced by bounce-back like boundary conditions.

Theorem: The lattice-Boltzmann algorithm (as defined previously) respects the

CFL-condition, i.e. for .11,20 ≤≤−≤≤ aω1)( ≤Eρ

Stability Norm stability

rDf =Differential equation

FD-discretization hhh RFD =

The numeric scheme is stable w.r.t. if:

KDKh

h <∀>∃ −1 :grids0 :

h⋅

KETtnnK n <≤∆∈∀∀>∃ :,:: max0grids0 N

Especially this means for an explicit scheme like the lattice-Boltzmann algorithm:

hh

n

h KEnK <∈∀>∃∀⇒ ::0:grids N

Observation: The condition is only necessary for

stability but not sufficient. However is diagonalizable (no nontrivial Jordan blocks):

10spec 1 ≤⇔⊂ )()()( EDE ρE

Stability L2-Stability

{ }1,...,1,0ee

ee:)( 2

i-i-

ii

−∈

= NM

Nπ

ϕϕ

ϕϕ

ϕδγβα

ϕ

)(,sup,supsup1

21

21

2

2

2

2

2

∗∗ =======

AAxxAAAxAxAxAxxx

ρFact:

)(ϕM is diagonalizable and ( ) ϕϕ ϕϕρ CMCM n

n

<>∃⇒≤∈ 2

)(sup:01)(N

Each

( )N∈⇒

nnf pointwise bounded. Principle of uniform boundedness: ( )N∈nnf locally bounded

Short course (L2): All expansions hold in the 2-norm at least .

Remark: Diffusive scaling → stability result in maximum-norm (uses positivity of evolution matrix)

Due to compactness of : ]2,0[ π ( )N∈nnf globally bounded. ■

Proof: Discrete Fourier trafo (+permutation of indices) of evolution matrix → block-diagonal

matrix with 2x2 blocks:

NR ∈=→ nMff n

nn ,)(:)(,]2,0[:2

ϕϕπDefine family of continuous functions:

2]2,0[2)(max ϕ

πϕ

nn ME∈

≤⇒

invariance w.r.t. unitary transformations (e.g. discreteFourier trafo)⇒

Theorem: The evolution matrix of the LB algorithm (previously defined) satisfies the

the stability condition w.r.t. the L2-norm, if .11,20 ≤≤−≤≤ aω

Multiscale expansion Motivation: linear ↔ quadratic time scale

Observation: advection → linear time scale

deformation (flattening) → quadratic time scale

Coarse grid: 30 nodes

Fine grid: 60 nodes5.0,7.1 == aω

301

1 =h

601

2 =h

linear time: jjj hnGt =)(1 quadratic time:2

2 jjj hnGt =)(

Multiscale expansions Motivation: linear↔cubic time scale

Observation: advection → linear time scale

distortion (undulations) → cubic time scale

Coarse grid: 60 nodes

Fine grid: 120 nodes5002 .,. == aω

601

1 =h

1201

2 =h

linear time: jjj hnGt =)(1 cubic time:3

3 jjj hnGt =)(

Multiscale expansion Regular versus twoscale expansion – the idea

),(f...),(f),(f),(f),F( )()()(][ xthxthxtxtxt ααα +++== 10

Approximate grid function of LB-algorithm by regular expansion:

grid function prediction

function0‘th asymptotic

order function

1‘st asymptotic

order functionnhttihxx

n

i ====

[ ] )(),(f),(f),(f ][][][ 1+++=++ αααα hOxtJxthsxht kkkk

!!!!

residualapply Taylor expansion

Requirement

determines order functions uniquely → consistency analysis, e.g.)()()( ff 0

2

0

1

0 +=u

000 =∂+∂ )()( uau xtmust satisfy: → details: short course L2 (M. Junk)

Shortcomings: appearence of secular terms → regular expansion only valid for time

intervals of length → not capturing long time behavior over intervals.)(1O )(h

O 1

),,(f...),,(f),,(f),,(f),F( )()()(][ xhtthxhtthxhttxhttxt ααα +++== 10

Twoscale ansatz: 2 time variables to take into account observed effects.

),,(f...),,(f),,(f),,(f),F( )()()(][ xththxththxthtxthtxt 221202 ααα +++==

Multiscale expansion Connection to stability

Why do we expect a multiscale expansion to tell something about stability?

Instabilities may become noticeable near the boundary of the stability domain as background phenomena occuring in slower time scales.

�2nd time variable – formally independent but coupled if compared with grid function.

�Order functions are not uniquely determined → further assumptions & restrictions.

�Easy to compute if regular expansion is available!

1 is always eigenvalue of evolution operator independently of

associated eigenvector = constant vector (= projection of smooth function onto grid)

.,aω

Eigenvalues around 1 can be expanded w.r.t. h

0,...1 1

1 ≠+++= ++

ll

l

l

l cchchλ

Indicator for instability: ( )N

nnhch 1~,11

l

l+≈<< λ

lNn ≈⇒

Impact of instability on eigenvalues around 1 becomes only

visible in a slow time scale if 1>l

)(exp11ll

l

l

cN

cN

N

≈

+⇒>>recall

Multiscale expansion The outcome – evolution equations

Procedure to derive determining equations for order functions similar to regular case. Strategy: minimize the residual ↔ maximize order of residual → details: short course L2 (M. Junk)

2≠ω

0),,(),,( 21

)0(2

21

)0(

2=∂−∂ xttuxttu xt µ

ii) Evolution in the slow time variable:

General case : diffusion equation

0),,(~

),,( 31

)0(3

31

)0(

2=∂−∂ xttuxttu xt λ

Special case : dispersive equation2=ω

Typical effect of PDE evolution operator on initial condition: damping, flattening --- undulating, oscillating

( )261 1

~aa −−=λ

3,21

)0(

1

)0( 0),,(),,(1

==∂+∂ kkxkt xttuaxttu

i) Evolution in the fast time variable described by advection equation:

Result for the leading (0‘th) order:

( )( )2211 1 a−−= ωµCoefficients:

What do we learn? 1) possibility to get a precise quantitative prediction of grid function.

2) quantitative understanding of the observed effects └→ see later

Implications concerning stability: ⇒< 0µ backward diffusion equationill-posed IVP → instabilities expected ↔ stable behavior for 0≥µdispersive equation indifferent w.r.t. sign of no hypothesis!⇒λ

~

Deceptive cases:

1) violation of CFL condition with unstable explicit

relaxation: 01.1 ,05.2 == aω

Multiscale expansion Comparison and limits of stability prediction

( )( )2211 1 a−−= ωµ

a

ω

1

2

-1

area of

pretended

stability

domain of

stability

area of

pretended

stability

0<µ

0≥µ

0<µ

0<µ

0<µ

0≥µ

0≥µ

0≥µ

0≥µ

sign of diffusivity

coefficient

ω05.1 ,00.2 == aω

2) violation of CFL condition with limit value for

1) Near 1, only eigenvalues pertaining to

‚non-smooth‘ eigenvectors move out of

closed unit disk as leave domain

of stability.

2) Eigenvalues do not cross boundary of

unit disk inside vicinity of 1.

Hypothesis about stability based on

multiscale expansion should fail if:

a,ω

Multiscale expansion & Conclusion Test: approximation by twoscale ansatz

Snapshots 8.02 == aω

Prediction of the (numeric) mass moment:

),,(),(),(),( 2)0(

21 jhnhnhujhnhUjhnhjhnh ↔=+ FF

Nasty example: arbitrarily smooth but not analytic (n‘th derivative cannot be bounded by )nC

Observation: good approximation over

time intervals of increasing length:

( ) ( )NOOh=1 ( ) ( )21

2 NOOh=and even

Another aspect of two scale expansion:

time interval error

)1(O )( 4hO

)( 1h

O

)( 2

1

hO

)( 3hO

)( 2hO

Conclusion .

Final judgement needs more case studies.

Above all: →clarify ideas & notions being still vague

e.g. classification of instabilities → topic of future work

With exception of the corners the boundary of the actual stability

domain is accurately described. Corners seem like „magic doors“

to falsely pretended stability domains.

Some instabilities cannot be captured due to implicit

smoothness assumption of prediction function

(e.g. Taylor expansion).

High order prediction of mass moment.

Introduction Relevance of analysis -- the cycle of development .

improvement

experiments

numeric

algorithmreason ? justification?works fineunsatisfying

stability test

improvement

test with

benchmarks

systematic approachsystematic approachsystematic approachsystematic approach

analysis�consistency

�stability

analysis�consistency

�stability

� improve consistency by systematic analysis & test stability experimentally

� guidance to modify bad scheme: try to shift break down of regular expansion into higher order

� good scheme: behaves regularly well describable by regular expansion ⇒