Comparison of Hyperbolic Moment Models for the Boltzmann ...€¦ · Three Gorges University,...

IntroductionTransformed Boltzmann Equation

Hyperbolic Quadrature-Based Projection MethodConclusion

Comparison of Hyperbolic Moment Modelsfor the Boltzmann Equation

Julian Kollermeier, Prof. Dr. Manuel Torrilhon

August 8th, 2014

Three Gorges University, Yichang, China

Julian Kollermeier, Prof. Dr. Manuel Torrilhon 1 / 26 Hyperbolic Moment Models for the Boltzmann Equation



RWTH Aachen

largest technical university in Germany (40,000 students)

strong focus on engineering and computational engineering science




RWTH Aachen

largest technical university in Germany (40,000 students)

strong focus on engineering and computational engineering science

MathCCES

Math division of Center for Computational Engineering Science

∼ 25 PhD and Post-Doc positions, 3 ProfessorsHead: Manuel Torrilhon




Outline

1 Introduction

2 Transformed Boltzmann Equation

3 Hyperbolic Quadrature-Based Projection Method

4 Conclusion




Rarefied Gas DynamicsBoltzmann EquationSolution Methods

Introduction





Rarefied Gas Dynamics I

Goal

solve and simulate flow problems involving rarefied gases

Knudsen number

distinguish flow regimes by orders of the Knudsen number Kn = λL

λ is the mean free path length

L is a reference length

Flow regimes

Kn ≤ 0.1: continuum model; Navier-Stokes Equation and extensions

Kn ≥ 0.1: rarefied gas; Boltzmann Equation or Monte-Carlosimulations





Rarefied Gas Dynamics II

Applications for large Kn = λL

large λ: rarefied gases, atmospheric reentry flights

small L: micro-scale applications, Knudsen pump, MEMS

Tasks

computation of mass flow rates

calculation of shock layer thickness

accurate prediction of heat flux





Boltzmann Transport Equation

∂

∂tf (t, x, c) + ci

∂

∂xif (t, x, c) = S(f )

PDE for particles’ probability density function f (t, x, c)

Describes change of f due to transport and collisions

Collision operator S

x ∈ Rd , c ∈ Rd

No external force





Solution Methods

Stochastic and Deterministic Methods

Direct Simulation Monte Carlo (DSMC) [Bird, 1963]

stochastic methodsimulates flight and collisions of many particles

Discrete Velocity Method (DVM) e.g. [Mieussens, 1998]

deterministic methodpoint evaluations of BTE

Unified Gas-Kinetic Scheme [Xu, 2010]

deterministic methodused for the entire range of Kn

Moment Methods (e.g. R13 [Torrilhon, 2003])

deterministic methodderive equations for moments of f




Transformation of Velocity VariableCompatibility ConditionsAnsatz

Transformed Boltzmann Equation





Transformation of Velocity Variable (1D)

ξ(t, x , c) :=c − v(t, x)√

θ(t, x)

=⇒

-10 -5 0 5 10c

0.2

0.4

0.6

0.8

1.0fHcL

f (c) =ρ√2πθ

exp

(−(c − v)2

2θ

)f (ξ) =

ρ√2πθ

exp

(−ξ

2

2

)

Lagrangian velocity space reduces numerical complexity





Transformation of Velocity Variable (1D)

ξ(t, x , c) :=c − v(t, x)√

θ(t, x)

=⇒

-10 -5 0 5 10c

0.2

0.4

0.6

0.8

1.0fHcL

f (c) =ρ√2πθ

exp

(−(c − v)2

2θ

) -4 -2 0 2 4Ξ

0.2

0.4

0.6

0.8

1.0

fHΞL

f (ξ) =ρ√2πθ

exp

(−ξ

2

2

)

Lagrangian velocity space reduces numerical complexity





Transformation of Boltzmann Equation

∂

∂tf (t, x, c) + ci

∂

∂xif (t, x, c)

⇓

Dt f +√θξj∂xj f +∂ξj f

(− 1√

θ

(Dtvj +

√θξi∂xi vj

)− 1

2θξj

(Dtθ +

√θξi∂xi θ

))= 0

Additional terms from chain rule for f , with ξi (t, x, c) := ci−vi (t,x)√θ(t,x)

Convective time derivative Dt := ∂t + vi∂xiAdditional variables vi and θ ⇒ need additional equations





Compatibility Conditions

Macroscopic variables ρ, v, θ are moments of the PDF f

ρ(t, x) =

∫Rd

f (t, x, c) dc,

ρ(t, x)v(t, x) =

∫Rd

cf (t, x, c) dc,

d · ρ(t, x)θ(t, x) =

∫Rd

|c− v|2 f (t, x, c) dc.

d + 2 compatibility conditions close the system





Ansatz

Expansion

f (t, x, ξ) =ρ

√2πθ

dexp

(−ξ2i

2

) n∑i=0

αi (t, x)Φi (ξ)

Maxwellian times expansion in basis functions

Flexible basis functions, e.g. polynomial or spherical harmonics




Hyperbolicity and Existing ModelsQuadrature-based Projection MethodComparison to Grad and Cai in 1D

Hyperbolic Quadrature-BasedProjection Method





Hyperbolicity

Definition: Hyperbolicity

The PDE system of the form

∂

∂tu + A(u)

∂

∂xu = 0

is globally hyperbolic if matrix A is diagonalizable with real eigenvalues forevery u.

Hyperbolicity necessary for

Well-posedness of the initial value problem and stable solutions

Physical solutions with real-valued propagation speed





Discrete Velocity Method

Ansatz: Discrete point values of distribution functionf (t, x, ξ)→ f (t, x, ξ0), . . . , f (t, x, ξn)

Projection: Point evaluations of equation∫Rd

· δ(ξ − ξj)dξ

Application to transformed Boltzmann Equation

Discretization of derivative ∂ξj f (t, x, ξ)

Reduce unknowns using compatibility conditions

Only locally hyperbolic, loss of hyperbolicity possible





Grad’s Method [Grad, 1949]

Ansatz: Hermite basis functions

f (t, x, ξ) =1√2π

exp

(−ξ2

2

) n∑i=0

αi (t, x)Hei (ξ)

Projection: Multiplication with Hermite test function and integration∫Rd

· Hej(ξ)dξ


Analytical derivation using orthogonality and recursions

Reduce unknowns using compatibility conditions

χA(λ) =√

(n + 1)!Hen+1(λ) + γ1

(1− λ2αn−1

)+ γ2αn

Only locally hyperbolic, loss of hyperbolicity possible





Quadrature-based Projection [Koellermeier, 2014]

Ansatz: Arbitrary basis functions

f (t, x, ξ) =1√2π

exp

(−ξ2

2

) n∑i=0

αi (t, x)Φi (ξ)

Projection: Substitute integration by quadrature∫Rd

· Ψj(ξ)dξ ≈n∑

k=0

wk · |ξk Ψj(ξk)


Analytical derivation using abstract framework

Incorporate compatibility conditions analytically

χA(λ) =√

(n + 1)!Hen+1(λ)

Globally hyperbolic under simple conditions on Φi ,Ψi , ξk and wk





Comment on Quadrature-based Projection Method

Development of abstract framework

Special treatment of compatibility conditions

Useful for arbitrary basis and test functions

Applicable to general d-dimensional case

Hyperbolicity

Conditions for global hyperbolicity are easy to verify

Proof relies on properties of quadrature rule and test/basis functions

Eigenvalues are directly obtained





Quadrature-based Projection Method in 1D

Ansatz: Hermite basis functionsΦi (ξ) = Hei (ξ), Ψi (ξ) = Hei (ξ)

Projection: Gauss-Hermite quadraturePoints ξk are roots of HeN+1(ξ),wk are corresponding weights

Global Hyperbolicity

Global hyperbolicity has been analytically proven in 1D and generald-dimensional case

Eigenvalues contain roots of Hermite polynomials: λk =√θ ξk + v





Visualization of Velocity Transformation (1D, N=4)

ξ =c − v√

θ

=⇒

c =√θ ξ + v

-4 -2 2 4x

0.2

0.4

0.6

0.8

1.0f HxL

transformed velocity space ξ

physical velocity space c

Adaptive discretization of velocity space






ξ =c − v√

θ

=⇒

c =√θ ξ + v

-4 -2 2 4x

0.2

0.4

0.6

0.8

1.0f HxL



Adaptive discretization of velocity spaceJulian Kollermeier, Prof. Dr. Manuel Torrilhon 21 / 26 Hyperbolic Moment Models for the Boltzmann Equation





ξ =c − v√

θ

=⇒

c =√θ ξ + v

-4 -2 2 4x

0.2

0.4

0.6

0.8

1.0f HxL


-10 -5 5 10c

0.2

0.4

0.6

0.8

1.0f HcL


Adaptive discretization of velocity spaceJulian Kollermeier, Prof. Dr. Manuel Torrilhon 21 / 26 Hyperbolic Moment Models for the Boltzmann Equation




Comparison to Grad and Cai in 1D

Grad’s Method [Grad, 1949]

Ansatz: Expansion in Hermite ansatz functionsProjection: Exact integration

Cai’s Method [Cai et al. 2014]

Ansatz: Arbitrary basis functionRegularization: Cut-off certain terms during computationsAdditional terms only in last equation

Quadrature-based Projection Method [Koellermeier et al., 2014]

Ansatz: Arbitrary basis functionRegularization: Corresponding Gaussian-quadrature ruleAdditional terms in last two equations, includes Cai’s regularization terms





Comparison of regularizations

Result of comparison

Cut-off method and Quadrature-based projection method are verysimilar

Regularization includes different terms

Different methods lead to regularization of Grad’s system

Only the last (cut-off) or the last two (quadrature) equations change

Combinations of regularization terms do not lead to hyperbolic system




Summary

Conclusion




Summary

Summary and Further Work

Results

Abstract framework and hyperbolicity proof

Hyperbolicity conditions and eigenvalues

Understanding of different regularizations

Summary

Quadrature-based projection methods derive globally hyperbolic momentequations for an adaptive solution of the transformed Boltzmann Equation

Further Work

Generalization of the different regularizations

Numerics for the (non-conservative) hyperbolic PDE system




Summary

References

Thank you for your attention

J. Koellermeier, R.P. Schaerer and M. Torrilhon

A Framework for Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection Methods,Kinet. Relat. Mod. 7(3) (2014), 531-549

J. Koellermeier, M. Torrilhon

Hyperbolic Moment Equations Using Quadrature-Based Projection Methods,29th Rarefied Gas Dynamics, Xi’an (2014)

Z. Cai, Y. Fan and R. Li

Globally hyperbolic regularization of Grad’s moment system,Comm. Pure Appl. Math., 67(3) (2014), 464–518.

H. Grad

On the kinetic theory of rarefied gases,Comm. Pure Appl. Math., 2(4) (1949), 331–407.


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