On a New Diagram Notation for the Derivation of Hyperbolic Moment Models

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1 On a New Diagram Notation for the Derivation of Hyperbolic Moment Models Julian Koellermeier, Manuel Torrilhon, Yuwei Fan March 17th, 2017 Stanford University J. Koellermeier 1 / 57 of Hyperbolic Moment Models

2 Germany s largest technical university 45k students, 10k staff J. Koellermeier 2 / 57 of Hyperbolic Moment Models

3 Outline Conclusion J. Koellermeier 3 / 57 of Hyperbolic Moment Models

4 Introduction Motivation Rarefied Gas Dynamics Boltzmann Equation Aim Derive hyperbolic PDE systems for rarefied gas flows Extension of standard fluid dynamic equations Reentry flows Micro channel flows J. Koellermeier 4 / 57 of Hyperbolic Moment Models

5 Introduction Motivation Rarefied Gas Dynamics Boltzmann Equation Aim Derive hyperbolic PDE systems for rarefied gas flows Extension of standard fluid dynamic equations Reentry flows Micro channel flows Importance of Hyperbolicity Physical solutions with bounded propagation speeds Well-posedness and stability of the solution J. Koellermeier 4 / 57 of Hyperbolic Moment Models

6 Rarefied Gas Dynamics Motivation Rarefied Gas Dynamics Boltzmann Equation Goal solve and simulate flow problems involving rarefied gases Knudsen number distinguish flow regimes by orders of the Knudsen number Kn = mean free path length reference length = λ L J. Koellermeier 5 / 57 of Hyperbolic Moment Models

7 Boltzmann Transport Equation Motivation Rarefied Gas Dynamics Boltzmann Equation t f (t, x, c) + c i f (t, x, c) = S(f ) x i PDE for particles probability density function f (t, x, c) Describes change of f due to transport and collisions Collision operator S Usually a 7-dimensional phase space Free motion Collision Free path J. Koellermeier 6 / 57 of Hyperbolic Moment Models

8 Boltzmann Transport Equation 1D Motivation Rarefied Gas Dynamics Boltzmann Equation f (t, x, c) + c f (t, x, c) = S(f ) t x PDE for particles probability density function f (t, x, c) Describes change of f due to transport and collisions Collision operator S 3-dimensional phase space Free motion Collision Free path J. Koellermeier 7 / 57 of Hyperbolic Moment Models

9 Macroscopic Quantities Motivation Rarefied Gas Dynamics Boltzmann Equation f (t, x, c) is related to the macroscopic quantities density ρ(t, x), velocity v(t, x), temperature θ(t, x) ρ(t, x) = ρ(t, x)v(t, x) = 1 2 ρ(t, x)θ(t, x) ρ(t, x)v(t, x)2 = R R R f (t, x, c) dc cf (t, x, c) dc 1 2 c2 f (t, x, c) dc J. Koellermeier 8 / 57 of Hyperbolic Moment Models

10 Discretization in Velocity Space Grad HME QBME Review of Hyperbolic Moment Models J. Koellermeier 9 / 57 of Hyperbolic Moment Models

11 A Transformed Velocity Variable Discretization in Velocity Space Grad HME QBME f c ξ(t, x, c) := c v(t, x) θ(t, x) = f (ξ) = ρ ) exp ( ξ2 2πθ 2 10 f (c) = ρ ) (c v)2 exp ( 2πθ 2θ c J. Koellermeier 10 / 57 of Hyperbolic Moment Models

12 A Transformed Velocity Variable Discretization in Velocity Space Grad HME QBME f c ξ(t, x, c) := c v(t, x) θ(t, x) = f (ξ) = ρ ) exp ( ξ2 2πθ 2 10 f (c) = ρ ) (c v)2 exp ( 2πθ 2θ c Transformed velocity space reduces numerical complexity J. Koellermeier 10 / 57 of Hyperbolic Moment Models

13 A Transformed Velocity Variable Discretization in Velocity Space Grad HME QBME f c ξ(t, x, c) := c v(t, x) θ(t, x) = f (c) = ρ ) (c v)2 exp ( 2πθ 2θ c f (ξ) = ρ ) exp ( ξ2 2πθ 2 Transformed velocity space reduces numerical complexity J. Koellermeier 10 / 57 of Hyperbolic Moment Models

14 Discretization in Velocity Space Grad HME QBME A Transformed Velocity Variable f HcL c v (t, x) ξ(t, x, c) := p θ(t, x) = f HxL ρ (c v )2 f (c) = exp 2θ 2πθ c ρ ξ f (ξ) = exp 2 2πθ Transformed velocity space reduces numerical complexity J. Koellermeier 11 / 57 of Hyperbolic Moment Models x

15 Model Order Reduction Discretization in Velocity Space Grad HME QBME Ansatz: Expansion f (t, x, c) = M α=0 f α (t, x)φ v,θ α (ξ) Basis function: is weighted Hermite polynomial φ v,θ α (ξ) = 1 ) θ α+1 2 exp ( ξ2 H α (ξ) 2π 2 J. Koellermeier 12 / 57 of Hyperbolic Moment Models

16 Model Order Reduction Discretization in Velocity Space Grad HME QBME Ansatz: Expansion f (t, x, c) = M α=0 f α (t, x)φ v,θ α (ξ) Basis function: is weighted Hermite polynomial φ v,θ α (ξ) = 1 ) θ α+1 2 exp ( ξ2 H α (ξ) 2π 2 Reduction of Complexity One PDE for f (t, x, c) that is high-dimensional System of PDEs for ρ(t, x), v(t, x), θ(t, x), f α (t, x) that is low-dimensional J. Koellermeier 12 / 57 of Hyperbolic Moment Models

17 Grad s Method [Grad, 1949] Discretization in Velocity Space Grad HME QBME Galerkin Approach Standard method Multiplication with test function and integration Grad result t u M + A Grad x u M = 0, A Grad = u 4 = (ρ, v, θ, f 3, f 4 ) T v ρ θ ρ v θ v ρ 0 ρθ 0 4f 3 2 v 4 f 3θ 3f ρ 5f θ v Loss of hyperbolicity J. Koellermeier 13 / 57 of Hyperbolic Moment Models

18 Discretization in Velocity Space Grad HME QBME Hyperbolic Moment Equations (HME) [Cai et al., 2012] Modification of equations Based on Grad s method Modification of last equation to achieve hyperbolicity HME result t u M + A HME x u M = 0, A HME = u 4 = (ρ, v, θ, f 3, f 4 ) T v ρ θ ρ v θ v ρ 0 ρθ 0 4f 3 2 v 4 ρ 0 f 3 θ v f 3θ Globally hyperbolic for every state vector u M J. Koellermeier 14 / 57 of Hyperbolic Moment Models

19 Discretization in Velocity Space Grad HME QBME Quadrature-Based Moment Equations (QBME) [JK, 2013] Quadrature-Based Projection Approach Based on Grad s method Substitution of integrals by Gaussian quadrature QBME result t u M + A QBME x u M = 0, A QBME = u 4 = (ρ, v, θ, f 3, f 4 ) T v ρ θ ρ v θ v ρ 0 ρθ 0 4f f 4 θ v 4 f 3θ ρ 5f 4 f 3 θ+ 15f 4 ρθ v Globally hyperbolic for every state vector u M J. Koellermeier 15 / 57 of Hyperbolic Moment Models

20 Additions and Developments Discretization in Velocity Space Grad HME QBME Hyperbolic Moment Models Extension to n-d Framework for hyperbolic models... Applications Relativistic Boltzmann equation Quantum Gas... J. Koellermeier 16 / 57 of Hyperbolic Moment Models

21 Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations J. Koellermeier 17 / 57 of Hyperbolic Moment Models

22 Derivation of Moment System Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations f (t, x, c) = ( fα t v f α 1 f α 2 2 M α=0 f α (t, x)φ v,θ α (ξ) f (t, x, c) + c f (t, x, c) = 0 t x + v f α x + θ f α 1 + (α + 1) f α+1 x x + t + (vf α 1 + θf α 2 + (α + 1)f α ) v x + θ t +1 2 (vf α 2 + θf α 3 + (α + 1)f α 1 ) θ x ) φ v,θ α (ξ) = 0 J. Koellermeier 18 / 57 of Hyperbolic Moment Models

23 Derivation of Moment System (2) Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations J. Koellermeier 19 / 57 of Hyperbolic Moment Models

24 Towards a Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations Previous derivations Concise and mathematically elegant way to derive models Very theoretical and no direct insight into equations Diagram-Based Notation Better understanding of hyperbolic regularization Derivation of new models J. Koellermeier 20 / 57 of Hyperbolic Moment Models

25 Derivation 1 Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations Equation and ansatz f (t, x, c) = t f (t, x, c) + c x f (t, x, c) = 0 M α=0 f α (t, x)φ v,θ c v(t, x) α (ξ), ξ = θ(t, x) s (f α φ α (ξ)) = s f α φ α (ξ) + f α s φ α (ξ) = s f α φ α (ξ) + f α ( θ φ α (ξ) s θ + ξ φ α (ξ) s ξ) ( = s f α φ α (ξ) + f α ( θ φ α (ξ) s θ + ξ φ α (ξ) 1 s v ξ ) θ 2θ sθ J. Koellermeier 21 / 57 of Hyperbolic Moment Models

26 Derivation 2 Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations Equation and ansatz f (t, x, c) = t f (t, x, c) + c x f (t, x, c) = 0 M α=0 f α (t, x)φ v,θ c v(t, x) α (ξ), ξ = θ(t, x) ( s (f α φ α (ξ)) = s f α φ α (ξ) + f α ( θ φ α (ξ) s θ + ξ φ α (ξ) ) c f α φ α (ξ) = ( v + θξ f α φ α (ξ) 1 θ s v ξ 2θ sθ ) J. Koellermeier 22 / 57 of Hyperbolic Moment Models

27 Choice of Basis Function Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations weighted Hermite polynomial φ α (ξ) = 1 ) θ α+1 2 H α (ξ) exp ( ξ2 2π 2 1. ξ derivative: ξ φ α(ξ) = θφ α+1 (ξ) 2. θ derivative: θ φ α(ξ) = α+1 2θ φ α(ξ) 3. multiplication with ξ: ξφ α (ξ) = θφ α+1 (ξ) + α θ φ α 1 (ξ) J. Koellermeier 23 / 57 of Hyperbolic Moment Models

28 ξ derivative diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations ξ φ α(ξ) = θφ α+1 (ξ) J. Koellermeier 24 / 57 of Hyperbolic Moment Models

29 θ derivative diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations θ φ α(ξ) = α + 1 2θ φ α(ξ) J. Koellermeier 25 / 57 of Hyperbolic Moment Models

30 ξ multiplication diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations ξφ α (ξ) = θφ α+1 (ξ) + α θ φ α 1 (ξ) J. Koellermeier 26 / 57 of Hyperbolic Moment Models

31 Diagram Notation Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations Aim 1 Extend diagram notation from basis function to whole equation 2 Split terms of equation into steps 3 Use single diagram for each step t f (t, x, c) + c x f (t, x, c) = 0 s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 2a + 2b ( c f α φ α (ξ) = v + ) θξ f α φ α (ξ) 3 ( 1 s v ξ ) θ 2θ sθ J. Koellermeier 27 / 57 of Hyperbolic Moment Models

32 1 diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations ( f α ξ φ α (ξ) = f α ) θ φ α+1 (ξ) J. Koellermeier 28 / 57 of Hyperbolic Moment Models

33 2 diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations ( f α φ α 1 s v ξ ) ( ) θ 2θ sθ = f α α sθ 2 θ 3 φ α 1 sv φ α sθ θ 2 θ φ α+1 J. Koellermeier 29 / 57 of Hyperbolic Moment Models

34 2a diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations ( s f α ) φ α (ξ) = s f α φ α (ξ) J. Koellermeier 30 / 57 of Hyperbolic Moment Models

35 2b diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations ( f α θ φ α (ξ) s θ = f α α + 1 ) 2θ sθ s φ α (ξ) J. Koellermeier 31 / 57 of Hyperbolic Moment Models

36 3 diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations ( v + ) θξ f α φ α (ξ) = f α (θφ α 1 (ξ) + vφ α (ξ) + αφ α+1 (ξ)) J. Koellermeier 32 / 57 of Hyperbolic Moment Models

37 Grad diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 2a + 2b ( 1 s v ξ ) θ 2θ sθ J. Koellermeier 33 / 57 of Hyperbolic Moment Models

38 Grad diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 2a + 2b ( 1 s v ξ ) θ 2θ sθ J. Koellermeier 33 / 57 of Hyperbolic Moment Models

39 Grad diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 2a + 2b ( 1 s v ξ ) θ 2θ sθ J. Koellermeier 33 / 57 of Hyperbolic Moment Models

40 Grad diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 2a + 2b ( 1 s v ξ ) θ 2θ sθ J. Koellermeier 33 / 57 of Hyperbolic Moment Models

41 Grad diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 2a + 2b ( 1 s v ξ ) θ 2θ sθ J. Koellermeier 33 / 57 of Hyperbolic Moment Models

42 Grad diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 2a + 2b ( 1 s v ξ ) θ 2θ sθ J. Koellermeier 33 / 57 of Hyperbolic Moment Models

43 Single Basis Function Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations ( (1) : f α φ α ( (2) : f α φ α θ α + 1 ) 2θ xθ u f α φ α, ) ( ) α θ 3 xθ u f α φ α J. Koellermeier 34 / 57 of Hyperbolic Moment Models

44 Grad diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations J. Koellermeier 35 / 57 of Hyperbolic Moment Models

45 HME diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations J. Koellermeier 36 / 57 of Hyperbolic Moment Models

46 QBME diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations J. Koellermeier 37 / 57 of Hyperbolic Moment Models

47 Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations Simplified Hyperbolic Moment Equations (SHME) QBME and HME cut off terms during derivation Corresponds to new formulas, different adaptivity How much adaptivity is necessary? ( s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 1 s v ξ ) θ 2θ sθ s f α φ α (ξ) ( c f α φ α (ξ) = v + ) θξ f α φ α (ξ) Neglect adaptivity of s ( ) Keep adaptivity of c ( ) J. Koellermeier 38 / 57 of Hyperbolic Moment Models

48 SHME diagram Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations s (f α φ α (ξ)) = s f α φ α (ξ) + f α θ φ α (ξ) s θ + f α ξ φ α (ξ) 2a + 2b ( 1 s v ξ ) θ 2θ sθ J. Koellermeier 39 / 57 of Hyperbolic Moment Models

49 Preliminaries Diagram Notation Simplified Hyperbolic Moment Equations Simplified Hyperbolic Moment Equations (SHME) Simplified derivation Neglecting parts of adaptivity SHME result v ρ t u M + A SHME x u M = 0, θ ρ v A SHME = 0 2θ v u 4 = (ρ, v, θ, f 3, f 4 ) T ρ 0 ρθ v θ v Globally hyperbolic (equilibrium) J. Koellermeier 40 / 57 of Hyperbolic Moment Models

50 Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock J. Koellermeier 41 / 57 of Hyperbolic Moment Models

51 Conservative PDE systems Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock Standard conservative PDE system t u + x F(u) = 0 J. Koellermeier 42 / 57 of Hyperbolic Moment Models

52 Conservative PDE systems Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock Standard conservative PDE system Basic Finite Volume scheme u n+1 i t u + x F(u) = 0 = u n i t x Numerical flux F needed i+ 1 2 Conservation property by design ( ) F F i+ 1 i J. Koellermeier 42 / 57 of Hyperbolic Moment Models

53 Non-conservative PDE systems Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock Non-conservative PDE system t u + A(u) x u = 0 J. Koellermeier 43 / 57 of Hyperbolic Moment Models

54 Non-conservative PDE systems Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock Non-conservative PDE system t u + A(u) x u = 0 Standard conservative PDE system t u + x F(u) = 0 Can be written in conservative form iff A(u) = F(u) u In general no flux function available What about partially conservative systems? Special numerical methods are needed J. Koellermeier 43 / 57 of Hyperbolic Moment Models

55 Numerical Methods Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock Wave Propagation scheme [LeVeque, 1997] 2 nd order upwind type scheme Castro scheme [Castro, Pares, 2004] 1 st order upwind type scheme PRICE-C scheme [Canestrelli, 2009] 1 st order centered scheme J. Koellermeier 44 / 57 of Hyperbolic Moment Models

56 1D Shock Tube Test Case Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock Riemann problem with BGK collision operator t w + A(w) x w = 1 Pw, x [ 2, 2] τ ρ L = 7, ρ R = 1 Variable vector w = (ρ, u, θ, f 3, f 4 ) Relaxation time τ = Kn ρ non-linear J. Koellermeier 45 / 57 of Hyperbolic Moment Models

57 Model Equations Grad model HME model A Grad = A HME = Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock v ρ θ ρ v θ v ρ 0 ρθ 0 4f 3 2 v 4 f 3θ 3f ρ 5f θ v v ρ θ ρ v θ v ρ 0 ρθ 0 4f 3 2 v 4 ρ 0 f 3 θ v f 3θ J. Koellermeier 46 / 57 of Hyperbolic Moment Models

58 Model Equations 2 Grad model QBME model A Grad = A QBME = Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock v ρ θ ρ v θ v ρ 0 ρθ 0 4f 3 2 v 4 f 3θ 3f ρ 5f θ v v ρ θ ρ v θ v ρ 0 ρθ 0 4f f 4 θ v 4 f 3θ ρ 5f 4 f 3 θ+ 15f 4 ρθ v J. Koellermeier 47 / 57 of Hyperbolic Moment Models

59 Model Equations 3 Grad model SHME model A Grad = Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock v ρ θ ρ v θ v ρ 0 ρθ 0 4f 3 2 v 4 f 3θ 3f ρ 5f θ v v ρ θ ρ v A SHME = 0 2θ v ρ 0 ρθ v θ v J. Koellermeier 48 / 57 of Hyperbolic Moment Models

60 Method Comparison, Kn = 0.05 Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock 7 PRICE UPRICE Castro WP J. Koellermeier 49 / 57 of Hyperbolic Moment Models

61 Method Comparison, Kn = 0.5 Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock 7 PRICE UPRICE Castro WP J. Koellermeier 50 / 57 of Hyperbolic Moment Models

62 Model Comparison, Kn = 0.05 Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock 7 HME SHME DVM QBME J. Koellermeier 51 / 57 of Hyperbolic Moment Models

63 Model Comparison, Kn = 0.5 Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock 7 HME SHME DVM QBME J. Koellermeier 52 / 57 of Hyperbolic Moment Models

64 Averaged Solution, Kn = 0.5 Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock 7 QBME8 QBME9 QBME8+9 DVM J. Koellermeier 53 / 57 of Hyperbolic Moment Models

65 Model Convergence, Kn = 0.5 Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock 10 1 rho u p theta Figure: HME and QBME Figure: SHME J. Koellermeier 54 / 57 of Hyperbolic Moment Models

66 2D bow shock,ma = 3, Kn = Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock ρ Figure: 1st order Figure: 2nd order J. Koellermeier 55 / 57 of Hyperbolic Moment Models

67 2D bow shock,ma = 3, Kn = 0.5 Non-conservative Numerical Methods 1D Shock Tube Test Case 2D Bow Shock ρ Figure: 1st order Figure: 2nd order J. Koellermeier 56 / 57 of Hyperbolic Moment Models

68 Summary and Further Work Summary Hyperbolic Moment Equations Hierarchical models for rarefied gases Different models are available Diagram notation Reveal details of derivation Allow derivation of new models Further work M-adaptivity J. Koellermeier 57 / 57 of Hyperbolic Moment Models

69 Summary and Further Work Summary Hyperbolic Moment Equations Hierarchical models for rarefied gases Different models are available Diagram notation Reveal details of derivation Allow derivation of new models Further work M-adaptivity Thank you for your attention J. Koellermeier 57 / 57 of Hyperbolic Moment Models

70 References Summary 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), J. Koellermeier, M. Torrilhon Hyperbolic Moment Equations Using Quadrature-Based Projection Methods, Proceedings of the 29th Symposium on Rarefied Gas Dynamics, AIP Conf. Proc (2014), , Y. Fan, J. Koellermeier, J. Li, R. Li, M. Torrilhon A framework on the globally hyperbolic moment method for kinetic equations using operator projection method J. Stat. Phys., 162(2) (2016), Z. Cai, Y. Fan and R. Li Globally hyperbolic regularization of Grad s moment system, Comm. Pure Appl. Math., 67(3) (2014), H. Grad On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2(4) (1949), J. Koellermeier 58 / 57 of Hyperbolic Moment Models

Numerical Simulation of Rarefied Gases using Hyperbolic Moment Equations in Partially-Conservative Form

Numerical Simulation of Rarefied Gases using Hyperbolic Moment Equations in Partially-Conservative Form Julian Koellermeier, Manuel Torrilhon May 18th, 2017 FU Berlin J. Koellermeier 1 / 52 Partially-Conservative