On the hyperbolicity of Grads moment system in gas kinetic theory - - PowerPoint PPT Presentation
On the hyperbolicity of Grads moment system in gas kinetic theory Yuwei Fan Department of Mathematics, Stanford University Bay Area Scientific Computing Day December 3, 2016 based on the work jointed with Zhenning Cai, Duke University, NC
Yuwei Fan Department of Mathematics, Stanford University Bay Area Scientific Computing Day December 3, 2016 based on the work jointed with Zhenning Cai, Duke University, NC Ruo Li, Peking University, China
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 1 / 25
1
Introduction Gas Kinetic Theory Boltzmann Equation Moment Method
2
Grad’s Moment System Grad’s moment method Grad’s 13 moment system
3
Hyperbolicity of Grad’s moment system 1D case 3D case
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 2 / 25
Introduction
1
Introduction Gas Kinetic Theory Boltzmann Equation Moment Method
2
Grad’s Moment System Grad’s moment method Grad’s 13 moment system
3
Hyperbolicity of Grad’s moment system 1D case 3D case
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 3 / 25
Introduction Gas Kinetic Theory
Kn = Mean free path λ typical length scale L free path collision convection
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 4 / 25
Introduction Gas Kinetic Theory
Kn = Mean free path λ typical length scale L free path collision convection
Euler equation N-S equation no-slip BC. N-S equation slip BC. DSMC
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 4 / 25
Introduction Gas Kinetic Theory
Kn = Mean free path λ typical length scale L
Euler equation N-S equation no-slip BC. N-S equation slip BC. DSMC
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25
Introduction Gas Kinetic Theory
Euler equation N-S equation no-slip BC. N-S equation slip BC. DSMC
densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25
Introduction Gas Kinetic Theory
densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ↓ δ(xi(t), pi(t))
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25
Introduction Gas Kinetic Theory
densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ↓ f(t, x, p) = ∑
i
δ(xi, pi) ← δ(xi(t), pi(t)) Distribution function: f(t, x, ξ), (ξ = p/m)
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25
Introduction Gas Kinetic Theory
densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ⇑ ↓ ∫
R3(1, p, |p|2)f dp
← f(t, x, p) = ∑
i
δ(xi, pi) ← δ(xi(t), pi(t)) Distribution function: f(t, x, ξ), (ξ = p/m)
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25
Introduction Boltzmann Equation
free path collision convection Boltzmann equation (Boltzmann 1872) reads: ∂f ∂t + ξ · ∇xf = Q(f, f), ↓ ↓ Convection Collision Q(f, f) is collision term, and (t, x, ξ) ∈ R+ × RD × RD.
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 6 / 25
Introduction Boltzmann Equation
Boltzmann equation (Boltzmann 1872) reads: ∂f ∂t + ξ · ∇xf = Q(f, f), Q(f, f) is collision term, and (t, x, ξ) ∈ R+ × RD × RD. Notations: ρ → density u → macroscopic velocity T → tempurature σij → stress tensor ρTij = ρTδij + σij → press tensor qi → heat flux. Local equilibrium : (Maxwell 1860) M(t, x, ξ) = ρ(t, x) √ 2πT(t, x)
D exp
( − |ξ − u(t, x)|2 2T(t, x) )
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 6 / 25
Introduction Boltzmann Equation
Boltzmann equation (Boltzmann 1872) reads: ∂f ∂t + ξ · ∇xf = Q(f, f), Q(f, f) is collision term, and (t, x, ξ) ∈ R+ × RD × RD.
1
Complex collision term Q(f, f), e.g. binary collision term: Q(f, f) = ∫
R3
∫
S+(f′f′ 1 − ff1)B(|ξ − ξ1|, σ) dξ1 dn;
2
High-order variable: 1 (t) + D(x) + D(ξ) = 2D+1;
3
ξ ∈ R3.
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 7 / 25
Introduction Moment Method
Euler equation N-S equation no-slip BC. N-S equation slip BC. DSMC Boltzmann equation
densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ⇑ ↓ ∫
R3(1, ξ, |ξ|2)f dξ
← f(t, x, ξ) = ∑
i
δ(xi, ξi) ← δ(xi(t), ξi(t))
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 8 / 25
Introduction Moment Method
Euler equation N-S equation no-slip BC. N-S equation slip BC. DSMC Boltzmann equation
densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ⇑ ↓ ∫
R3(1, ξ, |ξ|2)f dξ
← f(t, x, ξ) = ∑
i
δ(xi, ξi) ← δ(xi(t), ξi(t)) ↓ ↓ ∫
R3(1, ξ, |ξ|2)f dξ
→ ∫
R3 fξα dξ
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 8 / 25
Introduction Moment Method
densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ⇑ ↓ ∫
R3(1, ξ, |ξ|2)f dξ
← f(t, x, ξ) = ∑
i
δ(xi, ξi) ← δ(xi(t), ξi(t)) ↓ ↓ ∫
R3(1, ξ, |ξ|2)f dξ
→ ∫
R3 fξα dξ
Boltzmann equation = ⇒ Hydrodynamic equations ↑ Moment method
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 8 / 25
Introduction Moment Method
M → finite-dimensional subspace of D-variate polynomials {mi(ξ)}M
i=0
→ a basis of M, m = (m0, · · · , mM)T µi = ⟨fmi⟩ → moments, concerned with in the issue µ → (µ0, . . . , µM)T Moment equations: ∂µi ∂t +
D
∑
d=1
∂⟨ξdmi(ξ)f⟩ ∂xd = ⟨miQ(f, f)⟩ (1) Moment closure: Give the state equations of ⟨ξdmi(ξ)f⟩ and ⟨miQ(f, f)⟩, d = 1, . . . , D, i = 0, . . . , M by µ. ∂µ ∂t +
D
∑
d=1
∂F d(µ) ∂xd = Q(µ)
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 9 / 25
Introduction Moment Method
Well-posedness of the model: Hyperbolicity, Stability,· · · Preserving of physics: Conservation, H-theorem, Galilean invariance, · · · Approximation efficiency: # DOF vs Accuracy Implementation: BC, Easy to implement,. . .
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 10 / 25
Introduction Moment Method
Definition (Globally Hyperbolic) The first-order equations ∂w ∂t + A(w) ∂w ∂x = 0 is globally hyperbolic if the coefficient matrix A(w) is diagonalizable with real eigenvalues for any admissible w. What if the system is not hyperbolic?
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 11 / 25
Introduction Moment Method
Example The initial value problem ∂ ∂t (u v ) + (0 a 1 ) ∂ ∂x (u v ) = 0, (u(x, 0) v(x, 0) ) = (u0(x) v0(x) ) . The characteristic speeds of the system is √a and −√a, and the system is hyperbolic if and only if a > 0. This system can be reduced as utt − auxx = 0, u(x, 0) = u0(x), ut(x, 0) = −av0,t(x). If a is negative, for example a = −1, the system turns to be elliptic equation with two boundary conditions, resulting in the inexistence of weak solution of the system.
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 12 / 25
Grad’s Moment System
1
Introduction Gas Kinetic Theory Boltzmann Equation Moment Method
2
Grad’s Moment System Grad’s moment method Grad’s 13 moment system
3
Hyperbolicity of Grad’s moment system 1D case 3D case
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 13 / 25
Grad’s Moment System Grad’s moment method
Idea: Assume f is not far from the local equilibrium feq, f ∼ feq. Grad’s expansion: expand the distribution around feq: f = feq ∑
α∈I
gαHe[u,T ]
α
(ξ) = ∑
α∈I
fαH[u,T ]
α
(ξ). Moment closure: fα = T |α|
α!
⟨ fHe[u,T ]
α
(ξ) ⟩ Orthogonal polynomial: Hermite polynomial He[u,T ]
α
(ξ) = (−1)|α| ω[u,T ](ξ) ∂αω[u,T ](ξ) ∂ξα , ω[u,T ](ξ) = feq ρ , Basis function: H[u,T ]
α
(ξ) = ω[u,T ](ξ)He[u,T ]
α
(ξ).
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 14 / 25
Grad’s Moment System Grad’s moment method
Grad’s expansion: f(t, x, ξ) = ∑
α∈I
fαH[u,T ]
α
(ξ). Properties of Grad’s expansion: Constraints: f0 = ρ, fed = 0, d = 1, · · · , D,
D
∑
d=1
f2ed = 0. First term is Maxwellian: f0H[u,T ] (ξ) = feq. Basis functions depend on t, x. ∂H[u,T ]
α
(ξ) ∂s =
D
∑
i=1
H[u,T ]
α+ei(ξ) ∂ui
∂s + 1 2 ∂T ∂s
D
∑
i=1
H[u,T ]
α+2ei(ξ),
s = t, xd.
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 14 / 25
Grad’s Moment System Grad’s moment method
Grad’s expansion: f = ∑
α∈I
fαH[u,T ]
α
(ξ) substitute into Boltzmann Equation: ∂f ∂t + ξ∇f = Q(f, f) matching coefficients(Galerkin) Grad’s Moment Equations: ∂wM ∂t +
D
∑
d=1
Ad,M(wM) ∂wM ∂xd = QM
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 15 / 25
Grad’s Moment System Grad’s 13 moment system
Grad’s 13 moment expansion [Grad 1949]: f|G13 = feq [ 1 + Tij − δijT 2T 2 ( CiCj − δijC2) + 2 5 qk ρT 2 Ck ( C2 2T − 5 2 )] , where Ci = ξi − ui is the relative velocity. M = {1, ξ, ξ ⊗ ξ, ξ|ξ|2}. Substituting the expansion into the Boltzmann equation, and matching the coefficients of the polynomials, we can obtain the well-known Grad’s 13 moment system
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 16 / 25
Grad’s Moment System Grad’s 13 moment system
dρ dt + ρ∂uk ∂xk = 0, dui dt + Tik ρ ∂ρ ∂xk + ∂Tik ∂xk = 0, dTij dt + 2Tk(i ∂uj) ∂xk + 1 ρ ( 4 5 ∂q(i ∂xj) + 2 5 δij ∂qk ∂xk ) = Q(Tij), dqi dt − (TijTjk − 2TTik + T 2δik) ∂ρ ∂xk + 7 5 qi ∂uk ∂xk + 7 5 qk ∂ui ∂xk + 2 5 qk ∂uk ∂xi − ρTik (∂Tjk ∂xj − 7 6 ∂Tjj ∂xk ) + 2ρT ( ∂Tik ∂xk − 1 3 ∂Tjj ∂xi ) = Q(qi).
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 17 / 25
Hyperbolicity of Grad’s moment system
1
Introduction Gas Kinetic Theory Boltzmann Equation Moment Method
2
Grad’s Moment System Grad’s moment method Grad’s 13 moment system
3
Hyperbolicity of Grad’s moment system 1D case 3D case
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 18 / 25
Hyperbolicity of Grad’s moment system 1D case
1D case: Grad’s 13 moment system degenerates into dw dt + A(w) ∂w ∂x = Q(w) (2) where w = (ρ, u1, T11, T22, q1)T ,
A(w) = ρ T11/ρ 1 2T11 6 5ρ 2 5ρ −4(T11 − T22)2/9 16q1/5 ρ(11T11 + 16T22)/18 ρ(17T11 − 8T22)/9 .
The characteristic polynomial is det(λI − A) = λ [ λ4 − 2 45 (101T11 + 16T22)λ2 − 96 25 q1 ρ λ + 1 15(53T 2
11 − 16T11T22 + 8T 2 22)
] , which only depends on σ11
ρ
and q1
ρ . (T11 = T + σ11/ρ, T22 = T − σ11/2ρ)
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 19 / 25
Hyperbolicity of Grad’s moment system 1D case
1D case: Grad’s 13 moment system degenerates into dw dt + A(w) ∂w ∂x = Q(w) (2)
−2 −1.5 −1 −0.5 0.5 1 1.5 2 −1 −0.5 0.5 1 1.5 2
Maxwellian Hyperbolicity region Non-hyperbolicity region
Figure 1: x-axis:
q1 ρT 3/2 , y-axis: σ11 ρT Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 19 / 25
Hyperbolicity of Grad’s moment system 1D case
1D case: Grad’s 13 moment system degenerates into dw dt + A(w) ∂w ∂x = Q(w) (2) Result: (2) is not globally hyperbolic (I. Muller 1998) Maxwellian is an inner point of hyperbolicity region
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 19 / 25
Hyperbolicity of Grad’s moment system 3D case
3D case: Write Grad’s 13 Moment System in quasi-linear form: dw dt +
3
∑
d=1
Ad ∂w ∂xd = Q(w), where w = (ρ, u1, u2, u3, T11, T22, T33, T12, T13, T23, q1, q2, q3)T . It is enough to examine A1 due to the rotational invariance.
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 20 / 25
Hyperbolicity of Grad’s moment system 3D case
For the Gaussian distribution f = ρ √ det(2πΘ) exp ( − 1 2 CT Θ−1C ) , Θ = T ϵT ϵT T T , where |ϵ| < 1, the characteristic polynomial of A1 is det(λI − A1) = (λ − u1)3 125 [5(λ − u1)2 − 7T] · g ( (λ − u1)2 T ) , g(x) = 25x4 − 165x3 + ( 257 + 48ϵ2) x2 + ( 8ϵ2 − 105 ) x−28ϵ2. g(x) has at least one negative zero ⇓ A1 has at least two complex eigenvalues ⇓ Grad 13 is NOT hyperbolic near Maxwellian
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 21 / 25
Hyperbolicity of Grad’s moment system 3D case
Hyperbolicity region of Grad 13 moment system on the T12 − q1 plane is:
−1.5 −1 −0.5 0.5 1 1.5 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25
Hyperbolicity region
Non-hyperbolicity region Maxwellian
Figure 1: x-axis:
q1 ρT 3/2 , y-axis: T12 T Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 22 / 25
Hyperbolicity of Grad’s moment system 3D case
1D case: The system is not globally hyperbolic (I. Muller 1998) Maxwellian is an inner point of hyperbolicity region 3D case: (Cai, Fan and Li, KRM 2014) Grad’s 13 moment system is not globally hyperbolic Maxwellian is NOT an inner point of hyperbolicity region
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 23 / 25
Hyperbolicity of Grad’s moment system 3D case
1D case: The system is not globally hyperbolic (I. Muller 1998) Maxwellian is an inner point of hyperbolicity region 3D case: (Cai, Fan and Li, KRM 2014) Grad’s 13 moment system is not globally hyperbolic Maxwellian is NOT an inner point of hyperbolicity region Question:
1
Why we care about the hyperbolicity?
2
Why Grad’s moment systems is not globally hyperbolic?
3
How to fix the loss of hyperbolicity of Grad’s moment system?
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 23 / 25
Hyperbolicity of Grad’s moment system 3D case
Discrete velocity method for BGK: discrete points ξk = k∆v, ∆v = L/N, k = −N, ..., N, fk = f(ξk), ∂fk ∂t + ξk ∂fk ∂x = 1 τ (feq
k
− fk) Constraints on the velocity grid(S. Brull et.al 2014) Large enough bounds L ≥ max
t,x (u(t, x) + c
√ T(t, x)), −L ≤ min
t,x (u(t, x) − c
√ T(t, x)) small enough grid step ∆v ≤ min
t,x
√ T(t, x)
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 24 / 25
Hyperbolicity of Grad’s moment system 3D case
Discrete velocity method for BGK: discrete points ξk = k∆v, ∆v = L/N, k = −N, ..., N, fk = f(ξk), ∂fk ∂t + ξk ∂fk ∂x = 1 τ (feq
k
− fk) Adaptive discrete velocity(S. Brull et.al 2014) discrete points: ξk = u(t, x) + k∆v √ T(t, x) Transformation: ˜ f(t, x, v) = √ T(t, x) ρ(t, x) f(t, x, u(t, x) + √ T(t, x)v) Governing equation: (equivalent to solve the following equation by DVM) ( dln(ρ) dt − 3 2 dln(T) dt ) ˜ f + d ˜ f dt −
3
∑
j=1
∂ ˜ f ∂vj ( 1 √ T duj dt + vj 2 dln(T) dt ) +
3
∑
d=1
√ Tvd ( ∂ln(ρ) ∂xd − 3 2 ∂ln(T) ∂xd ) ˜ f + ∂ ˜ f ∂xd −
3
∑
j=1
∂ ˜ f ∂vj ( 1 √ T ∂uj ∂xd + vj 2 ∂ln(T) ∂xd ) = 1 τ ( ˜ feq − ˜ f)
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 24 / 25
Hyperbolicity of Grad’s moment system 3D case
Regularized moment method: parabolic equations, not first-order equations R13(H. Struchtrup, M. Torrilhon 2003) R26(X. Gu et. al 2009) NRxx(Z. Cai, R. Li 2012) Hyperbolic regularized moment method: globally hyperbolic Globally hyperbolic moment system (Z. Cai, Y. Fan, R. Li, 2013, 2014) Generalized framework for kinetic equation(Z. Cai, Y. Fan, R. Li, 2015; Y. Fan et. al 2016) Approximate moment method: properties V.S implement Approximate 14 moment system (J. McDonald and M. Torrilhon, 2013) Bi-Gaussian based moment system (in preparation)
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 25 / 25
Hyperbolicity of Grad’s moment system 3D case
Regularized moment method: parabolic equations, not first-order equations R13(H. Struchtrup, M. Torrilhon 2003) R26(X. Gu et. al 2009) NRxx(Z. Cai, R. Li 2012) Hyperbolic regularized moment method: globally hyperbolic Globally hyperbolic moment system (Z. Cai, Y. Fan, R. Li, 2013, 2014) Generalized framework for kinetic equation(Z. Cai, Y. Fan, R. Li, 2015; Y. Fan et. al 2016) Approximate moment method: properties V.S implement Approximate 14 moment system (J. McDonald and M. Torrilhon, 2013) Bi-Gaussian based moment system (in preparation)
Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 25 / 25