On the hyperbolicity of Grads moment system in gas kinetic theory - PowerPoint PPT Presentation

On the hyperbolicity of Grad’s 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 Ruo Li, Peking University, China Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 1 / 25

Outline Introduction 1 Gas Kinetic Theory Boltzmann Equation Moment Method Grad’s Moment System 2 Grad’s moment method Grad’s 13 moment system Hyperbolicity of Grad’s moment system 3 1D case 3D case Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 2 / 25

Introduction Outline Introduction 1 Gas Kinetic Theory Boltzmann Equation Moment Method Grad’s Moment System 2 Grad’s moment method Grad’s 13 moment system Hyperbolicity of Grad’s moment system 3 1D case 3D case Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 3 / 25

Introduction Gas Kinetic Theory Gas kinetic theory Mean free path λ Kn = typical length scale L collision convection free path Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 4 / 25

Introduction Gas Kinetic Theory Gas kinetic theory Mean free path λ Kn = typical length scale L collision convection free path N-S equation ? Euler N-S equation slip BC. DSMC no-slip BC. equation 0 ← Kn 10 − 3 10 − 2 10 − 1 1 10 hydrodynamics gas kinetic theory Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 4 / 25

Introduction Gas Kinetic Theory Gas kinetic theory Mean free path λ Kn = typical length scale L ? N-S equation Euler N-S equation slip BC. DSMC no-slip BC. equation 10 − 3 10 − 2 10 − 1 0 ← Kn 1 10 hydrodynamics gas kinetic theory Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25

Introduction Gas Kinetic Theory Gas kinetic theory ? N-S equation Euler N-S equation slip BC. DSMC no-slip BC. equation 10 − 3 10 − 2 10 − 1 0 ← Kn 1 10 hydrodynamics gas kinetic theory 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 Gas kinetic theory densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ↓ δ ( x i ( t ) , p i ( t )) Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25

Introduction Gas Kinetic Theory Gas kinetic theory densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ↓ f ( t, x , p ) = ∑ δ ( x i , p i ) ← δ ( x i ( t ) , p i ( t )) i Distribution function: f ( t, x , ξ ) , ( ξ = p /m ) Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25

Introduction Gas Kinetic Theory Gas kinetic theory densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ⇑ ↓ R 3 (1 , p , | p | 2 ) f d p ∫ ← f ( t, x , p ) = ∑ δ ( x i , p i ) ← δ ( x i ( t ) , p i ( t )) i Distribution function: f ( t, x , ξ ) , ( ξ = p /m ) Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 5 / 25

Introduction Boltzmann Equation Boltzmann Equation collision convection free path Boltzmann equation (Boltzmann 1872) reads: ∂f ∂t + ξ · ∇ x f = Q ( f, f ) , ↓ ↓ Convection Collision Q ( f, f ) is collision term, and ( t, x , ξ ) ∈ R + × R D × R D . Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 6 / 25

Introduction Boltzmann Equation Boltzmann Equation Boltzmann equation (Boltzmann 1872) reads: ∂f ∂t + ξ · ∇ x f = Q ( f, f ) , Q ( f, f ) is collision term, and ( t, x , ξ ) ∈ R + × R D × R D . Notations: ρ → density u → macroscopic velocity T → tempurature σ ij → stress tensor ρT ij = ρTδ ij + σ ij → press tensor q i → heat flux . Local equilibrium : (Maxwell 1860) − | ξ − u ( t, x ) | 2 ( ) ρ ( t, x ) M ( t, x , ξ ) = D exp 2 T ( t, x ) √ 2 πT ( t, x ) Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 6 / 25

Introduction Boltzmann Equation Difficulties in Solving Boltzmann Equation Boltzmann equation (Boltzmann 1872) reads: ∂f ∂t + ξ · ∇ x f = Q ( f, f ) , Q ( f, f ) is collision term, and ( t, x , ξ ) ∈ R + × R D × R D . Complex collision term Q ( f, f ) , e.g. binary collision term: 1 ∫ ∫ S + ( f ′ f ′ Q ( f, f ) = 1 − ff 1 ) B ( | ξ − ξ 1 | , σ ) d ξ 1 d n ; R 3 High-order variable: 1 ( t ) + D( x ) + D( ξ ) = 2D+1; 2 ξ ∈ R 3 . 3 Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 7 / 25

Introduction Moment Method Start Point of Moment Method Boltzmann N-S equation Euler N-S equation slip BC. equation DSMC no-slip BC. equation 0 ← Kn 10 − 3 10 − 2 10 − 1 1 10 hydrodynamics gas kinetic theory densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ⇑ ↓ R 3 (1 , ξ , | ξ | 2 ) f d ξ ∫ f ( t, x , ξ ) = ∑ ← δ ( x i , ξ i ) ← δ ( x i ( t ) , ξ i ( t )) i Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 8 / 25

Introduction Moment Method Start Point of Moment Method Boltzmann N-S equation Euler N-S equation slip BC. equation DSMC no-slip BC. equation 0 ← Kn 10 − 3 10 − 2 10 − 1 1 10 hydrodynamics gas kinetic theory densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ⇑ ↓ R 3 (1 , ξ , | ξ | 2 ) f d ξ ∫ f ( t, x , ξ ) = ∑ ← δ ( x i , ξ i ) ← δ ( x i ( t ) , ξ i ( t )) i ↓ ↓ R 3 f ξ α d ξ ∫ R 3 (1 , ξ , | ξ | 2 ) f d ξ ∫ → Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 8 / 25

Introduction Moment Method Start Point of Moment Method densy gas → rarefied gas → very rarefied gas ↓ ↓ ↓ hydrodynamics ? molecular dynamics ↓ ↓ ↓ Continuum mechanics → ? ← molecular motion ↓ ⇑ ↓ R 3 (1 , ξ , | ξ | 2 ) f d ξ ∫ f ( t, x , ξ ) = ∑ ← δ ( x i , ξ i ) ← δ ( x i ( t ) , ξ i ( t )) i ↓ ↓ R 3 f ξ α d ξ ∫ R 3 (1 , ξ , | ξ | 2 ) f d ξ ∫ → Boltzmann equation = ⇒ Hydrodynamic equations ↑ Moment method Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 8 / 25

Introduction Moment Method Moment method → finite-dimensional subspace of D -variate polynomials M { m i ( ξ ) } M m = ( m 0 , · · · , m M ) T → a basis of M , i =0 µ i = ⟨ fm i ⟩ → moments, concerned with in the issue ( µ 0 , . . . , µ M ) T µ → Moment equations: D ∂µ i ∂ ⟨ ξ d m i ( ξ ) f ⟩ ∑ ∂t + = ⟨ m i Q ( f, f ) ⟩ (1) ∂x d d =1 Moment closure: Give the state equations of ⟨ ξ d m i ( ξ ) f ⟩ and ⟨ m i Q ( f, f ) ⟩ , d = 1 , . . . , D, i = 0 , . . . , M by µ . D ∂ µ ∂ F d ( µ ) ∑ ∂t + = Q ( µ ) ∂x d d =1 Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 9 / 25

Introduction Moment Method Good Model or Bad Model 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 Hyperbolicity 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 Hyperbolicity Example The initial value problem ) ∂ ( u ) ( 0 ( u ) ( u ( x, 0) ) ( u 0 ( x ) ) ∂ a + = 0 , = . v 1 0 v v ( x, 0) v 0 ( x ) ∂t ∂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  u tt − au xx = 0 ,   u ( x, 0) = u 0 ( x ) ,  u t ( x, 0) = − av 0 ,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 Outline Introduction 1 Gas Kinetic Theory Boltzmann Equation Moment Method Grad’s Moment System 2 Grad’s moment method Grad’s 13 moment system Hyperbolicity of Grad’s moment system 3 1D case 3D case Yuwei Fan (Mathematics, Stanford) Moment Method December 3, 2016 13 / 25