modeling of fluctuations in algorithm refinement methods · modeling of fluctuations in algorithm...

Modeling of Fluctuations in AlgorithmRefinement Methods

John Bell

Lawrence Berkeley National Laboratory

DSMC09 ConferenceSanta Fe, New Mexico

Sept. 13-16, 2009

Collaborators: Aleksandar Donev, Eric Vanden-Eijnden,Alejandro Garcia, Jonathan Goodman, SarahWilliams

Bell, et. al., LBNL AMAR

Outline

Hybrid algorithmsMotivationApproach

Role of fluctuationsBurgers’ equation example

Landau-Lifshitz Navier Stokes equationsNumerical methods for stochastic PDE’sHybrid algorithms for fluidsExamplesConclusions


Multi-scale models of fluid flow

Most computations of fluid flows use a continuum representation(density, pressure, etc.) for the fluid.

Dynamics described by set of PDEs.

Well-established numerical methods (finite difference, finiteelements, etc.) for solving these PDEs.

Hydrodynamic PDEs are accurate over a broad range of lengthand time scales.

But at some scales the continuum representation breaks down andmore physics is needed

When is the continuum description of a gas not accurate?

Discreteness of collisions and fluctuations are important

Micro-scale flows, surface interactions, complex fluidsParticles / macromolecules in a flowBiological / chemical processes


Hybrid methods

Look at fluid mechanics problems that span kinetic andhydrodynamics scales

Approaches

Molecular description –correct but expensive

Continuum CFD – cheap butdoesn’t model correct physics

Hybrid – Use different modelsfor the physics in differentparts of the domain

Molecular model onlywhere neededCheaper continuummodel in the bulk of thedomain


Hybrid approach

Develop a hybrid algorithm for fluid mechanics that couples aparticle description to a continuum description

AMR provides a framework for such a couplingAMR for fluids except change to a particle description at thefinest level of the heirarchy

Use basic AMR design paradigm for development of ahybrid method

How to integrate a levelHow to synchronize levels


DSMC

Discrete Simulation Monte Carlo (DSMC) is a leadingnumerical method for molecular simulations of dilute gases

Initialize system with particlesLoop over time steps

Create particles at openboundariesMove all the particlesProcess particle/boundaryinteractionsSort particles into cellsSelect and execute randomcollisionsSample statistical values

Example of flow past asphere


Adaptive mesh and algorithm refinement

AMR approach to constructing hybrids –Garcia et al., JCP 1999

Hybrid algorithm – 2 levelAdvance continuum CNS solver

Accumulate flux FC at DSMCboundary

Advance DMSC regionInterpolation – Sampling fromChapman-Enskog distributionFluxes are given by particlescrossing boundary of DSMC region

SynchronizeAverage down – momentsReflux δF = −∆tAFC +

∑p Fp

DSMC

Buffer cells

Continuum

DSMC boundaryconditions

A B C

D E F

1 23

DSMC flux


Hydrodynamic Fluctuations

Given particle positions and velocities, measure macroscopicvariables

These quantities naturally fluctuateParticle scheme:

Capture variance of fluctuationsPredict time-correlations at hydrodynamic scalePredict non-equilibrium fluctuations hydrodynamic scale


Fluctuations at continuum level

How do molecular-scale fluctuations interact with thecontinuum?

Can / should we capture fluctuations at the continuum level?

Do they matter?

This has been investigated for a number of modelsDiffusion”Train” modelBurgers’ equation

Example: Burgers’ equation – Bell et al. JCP, 2007.


AERW / Burgers’

Asymmetric Excluded Random Walk:

uj

A

B

C

ujE

D

Continuum Grid

Particle Patch

Bou

ndar

y S

ites

Bou

ndar

y S

ites

ADVANCE

SYNCHRONIZE

E

B

Probability of jump to the right sets the “Reynolds” number

Mean field given by viscous Burgers’ equationStochastic flux models fluctuation behavior

ut + c((u(1− u))x = εuxx + Sx

Stochastic flux is zero-mean Gaussian uncorrelated in spaceand time with magnitude from fluctuation dissipation theorem


AERW / Burgers’

Simple numerical ideas work wellSecond-order Godunov for advective fluxStandard finite difference approximation of viscosityExplicitly add stochastic flux


AERW / Burgers’

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Mean

Hybrid AERW/SPDEHybrid AERW/DPDEAERWSPDE

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

1

2x 10

−3 Variance

−0.4 −0.2 0 0.2 0.4−5

−4

−3

−2

−1

x 10−5 Correlation

Rarefaction

0 1 2 3 4 5 6 7 8 9 10

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−3

Var

ianc

e of

sho

ck lo

catio

n

time

Hybrid w/ px=0.55

Hybrid w/px=0.7

Hybrid w/ px=0.8

AERW w/px=0.7

Hybrid w/no noise in SPDE region and px=0.7

Figure 9: Variance of shock location 〈δs(t)2〉 versus time t for a determin-istically steady shock (uL = 0.1, uR = 0.9). Methods used are: AERWwith p→ = 0.7 (green); SPDE/AERW hybrid with p→ = 0.55 (blue);SPDE/AERW hybrid with p→ = 0.7 (red); SPDE/AERW hybrid withp→ = 0.8 (black); DPDE/AERW hybrid with p→ = 0.7 with 8 cells ofadditional buffering (magenta).

method we have demonstrated that it is necessary to include the effect offluctuations, represented as a stochastics flux, in the mean field equations toensure that the hybrid preserved key properties of the system. As expected,not representing fluctuations in the continuum regime leads to a decay inthe variance of the solution that penetrates into the particle region. Some-what more surprising is that the failure to include fluctuations was shownto introduce spurious correlations of fluctuations in equilibrium simulationsand for rarefactions. Even more troubling is the observation that using adeterministic PDE solver coupled to the random walk model suppresses thedrift of shock location seen with the pure random walk model and with theAR hybrid using a stochastic PDE solver.

We plan to extend this basic hybrid framework to the solution of the com-pressible Navier Stokes equations in multiple dimension. For that extension

24

Shock Drift

Failure to include the effectof fluctuations at the contin-num level disrupts correla-tions in the particle region


Landau-Lifshitz fluctuating Navier Stokes

What about fluid dynamics?Can we model fluctuations at the continuum level?Do they matter in this context?

∂U/∂t +∇ · F = ∇ · D +∇ · S where U =

ρJE

F =

ρvρvv + PI(E + P)v

D =

0τ

κ∇T + τ · v

S =

0S

Q+ v · S

,

〈Sij (r, t)Sk`(r′, t ′)〉 = 2kBηT(δK

ik δKj` + δK

i`δKjk − 2

3δKij δ

Kk`

)δ(r− r′)δ(t − t ′),

〈Qi (r, t)Qj (r′, t ′)〉 = 2kBκT 2δKij δ(r− r′)δ(t − t ′),


LLNS Discretization

Experience with Burger’s equations suggests adding astochastic flux to a standard second-order scheme capturesfluctuations reasonably well.

This is not the case with LLNS

〈δρ2〉〈δJ2〉〈δE2〉Exact value 2.35× 10−8 13.34 2.84× 1010

MacCormack 2.01× 10−8 13.31 2.61× 1010

PPM 1.97× 10−8 13.27 2.58× 1010

DSMC 2.35× 10−8 13.21 2.78× 1010

Error MacCormack −14.3% −0.3% −8.4%Error PPM −16.0% −0.5% −9.4%Error DSMC 0.0% −1.0% −2.1%

Mass conservation is microscopically exact. There is no diffusion orfluctuation term in the mass conservation equation


Numerical methods for stochastic PDE’s

Accurately capturing fluctuations in a hybrid algorithm requiresaccurate methods for PDE’s with a stochastic flux.

∂tU = LU + KW

where W is spatio-temporal white noise

We can characterize the solution of these types of equations interms of the invariant distribution, given by the covariance

S(k , t) =< U(k , t ′)U∗(k , t ′ + t) >=

∫ ∞−∞

eiωtS(k , ω)dω

whereS(k , ω) =< U(k , ω)U∗(k , ω) >

is the dynamic structure factorWe can also define the static structure factor

S(k) =

∫ ∞−∞

S(k , ω)dω


Fluctuation dissipation relation

For∂tU = LU + KW

ifL + L∗ = −KK ∗

then the equation satisfies a fluctuation dissipation relation and

S(k) = I

The linearized LLNS equations are of the form

∂tU = −∇ · (AU − C∇U − BW )

When BB∗ = 2C, then the fluctuation dissipation relation issatisfied and the equilibrium distribution is spatially white withS(k) = 1


Discretization design issues

Consider discretizations of

∂tU = −∇ · (AU − C∇U − BW )

of the form∂tU = −D(AU − CGU − BW )

Scheme design criteria1 Discretization of advective component DA is skew adjoint;

i.e., (DA)∗ = −DA2 Discrete divergence and gradient are skew adjoint:

D = −G∗

3 Discretization without noise should be relatively standard4 Should have “well-behaved” discrete static structure factor

S(k) ≈ 1 for small k ; i.e. S(k) = 1 + αkp + h.o.tS(k) not too large for all k . (Should S(k) ≤ 1 for all k?)


Example: Stochastic heat equation

ut = µuxx +√

2µWx

Explicit Euler discretizaton

un+1j = un

j +µ∆t∆x2

(un

j−1 − 2unj + un

j+1

)+√

2µ∆t1/2

∆x3/2

(W n

j+ 12−W n

j− 12

)Predictor / corrector scheme

unj = un

j +µ∆t∆x2

(un

j−1 − 2unj + un

j+1

)+√

2µ∆t1/2

∆x3/2

(W n

j+ 12−W n

j− 12

)

un+1j =

12

[un

j + unj +

µ∆t∆x2

(un

j−1 − 2unj + un

j+1

)+

√2µ

∆t1/2

∆x3/2

(W n

j+ 12−W n

j− 12

)]


Structure factor for stochastic heat equation

0 0.2 0.4 0.6 0.8 1

k / kmax

0

0.5

1

1.5

2

Sk

b=0.125 E ulerb=0.25 E ulerb=0.125 PCb=0.25 PCb=0.5 PCb=0 (I deal)

Euler

S(k) = 1 + βk2/2

Predictor/Corrector

S(k) = 1− β2k4/4

PC2RNG:

S(k) = 1 + β3k6/8

How stochastic fluxes are treated can effect accuracy


Elements of discretization of LLNS – 1D

Spatial discretizationStochastic fluxes generated at facesStandard finite difference approximations for diffusion

Fluctuation dissipation

Higher-order reconstruction based on PPM

UJ+1/2=

712

(Uj + Uj+1) −1

12(Uj−1 + Uj+2)

Evaluate hyperbolic flux using Uj+1/2Adequate representation of fluctuations in density flux

Temporal discretizationLow storage TVD 3rd order Runge KuttaCare with evaluation of stochastic fluxes can improveaccuracy


Structure factor for LLNS in 1D

0 0.2 0.4 0.6 0.8 1

k / kmax

0.9

0.925

0.95

0.975

1

Sk

Sr

(1R NG)

1 + (Su-1) / 3

ST

Sr

(2R NG)

Small k theory

0 0.2 0.4 0.6 0.8 1

k / kmax

0

0.05

0.1

0.15

Sk

| Sr u

| (1R NG)

| Sr T

|

| SuT

|

| Sr u

| (2R NG)


Discretization for LLNS

〈δρ2〉〈δJ2〉〈δE2〉Exact value 2.35× 10−8 13.34 2.84× 1010

PPM 1.97× 10−8 13.27 2.58× 1010

RK3 2.29× 10−8 13.54 2.82× 1010

Error PPM −16.0% −0.5% −9.4%Error for RK3 −2.5% 1.5% −0.7%

Basic scheme has been generalized to 3D and two componentmixtures

Additional complication is correlation between elements ofstochastic stress tensor

Several standard discretization approaches do not correctlyrespect these correlations

Do not satisfy discrete fluctuation dissipation relationLeads to spurious correlations

Alternative approach based on randomly selecting faces onwhich to impose correlation

Preserves correlation structure in 2D and 3DBell, et. al., LBNL AMAR

SPDE versus DSMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

1. 5

1

0. 5

0

0.5

1

1.5x 10

5

x/L

<∂

ρ(x

) ∂

J(x

*)>

DSMCRK3

Correlation of density andmomentum in a thermal

gradient

0 0.5 1

x 107

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

10

Time<

∂ σ

P(t

)2 >

Mach 1

.2

Mach 1.4

Mach 2.0

Shock drift


Hybrid Issues

� ��

��

� ��

�� 3b

3a

2a

2b1

Maxwellian versus Chapman EnskogMeasuring gradients for C-E is problematicMaxwellian typically gives better results in hybrid

Refinement criterionUse smoothed gradient – width based on noise magnitudePotentially use breakdown parameter to guide refinement


EquilibriumDoes deterministic PDE treatment affect fluctuations for fluidsimulations?

Energy Statistics

5 10 15 20 25 30 35 401.514

1.516

1.518x 10

6⟨ E

⟩

5 10 15 20 25 30 35 400

1

2

3x 10

10

⟨ δ E

2 ⟩ Stoch. PDE onlyDeter. hybridStoch. hybridTheory

5 10 15 20 25 30 35 40−3

−2

−1

0x 10

9

⟨ δ E

i δ E

20 ⟩


Non-equilibrium

Look at correlations in the presence of a thermal gradient

5 10 15 20 25 30 35 402.5

2

1.5

1

0.5

0

0.5

1

1.5

x 10� 5

⟨ δ ρ

j δ J

20 ⟩

DSMCStoch. hybridRefinement region

Stochastic hybrid5 10 15 20 25 30 35 40

2.5

2

1.5

1

0.5

0

0.5

1

1.5

x 10�5

DSMCDeter. hybridRefinement region

Deterministic hybrid


Shock propagation

t0

Ensemble of stochastic PDE runsEnsemble of stochastic hybrid runsSingle stochastic hybrid run

t1

t2

25 50 75 100 125

t3

Propagation of refined shock


Piston problem

T1, ρ1 T2, ρ2

Piston

ρ1T1 = ρ2T2

Wall and piston are adiabatic boundariesDynamics driven by fluctuations


Piston dynamics

Hybrid simulation of PistonSmall DSMC region near the pistonI-DSMC – see Donev talk


Piston position vs. time

Piston versus time

0 2500 5000 7500 10000t

6

6.25

6.5

6.75

7

7.25

7.5

7.75x(

t)ParticleStoch. hybridDet. hybrid

0 250 500 750 10006

6.5

7

7.5

8

Note: Error associated with deterministic hybrid enhanced forheavier pistons


Averge piston position vs. time

Ensemble piston motion versus time

0.0 5.0×104

1.0×105

1.5×105

2.0×105

Time

8

9

10

11

12P

isto

n po

sitio

n

Stoch. hybrid (4x40)Det. hybrid (4x40)Stoch. hybrid (8x10)Particle

0 5×104

1×105

0.5

1

2

4P

isto

n of

fset

Note: Some sensitivity to mesh used for continuum simulationBell, et. al., LBNL AMAR

Piston velocity autocorrelation

Start with piston at equilibrium location

0 1 2 3

0

5×10-4

1×10-3

VACF

ParticleStoch. wP=2

Det. wP=2

Det. wP=4

Det. wP=8

0 50 100 150 200 250

Time-5.0×10

-4

-2.5×10-4

0.0

2.5×10-4

5.0×10-4

7.5×10-4

1.0×10-3

VACF

ParticleStoch. hybridDet. hybridDet. hybrid x10

Velocity Autocorrelation FunctionBell, et. al., LBNL AMAR

Summary

Failing to include fluctuations at the continuum level in a hybrid model can pollute themicroscopic model

Hybrid methodology for capturing fluctuations

Microscopic model – DSMC

Continuum model – discretization of LLNS equationsRK3 centered schemeCaptures equilibrium fluctuations

Hybridization based on adaptive mesh refinement constructs

Future issues

Numerics / MathematicsMathematical structure of systemsCriterion for evaluating schemes / hybridsStochastic analogs of limiting – robustnessFluctuations in low Mach number flows

PhysicsReacting systemsComplex fluids


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