Comparison of Numerical Solutions for the Boltzmann Equation and Different Moment Models

Comparison of Numerical Solutions for the Boltzmann Equation and Different Moment Models Julian Koellermeier, Manuel Torrilhon October 12th, 2015 Chinese Academy of Sciences, Beijing Julian Koellermeier, Manuel Torrilhon 1 / 27 Numerical Solutions for the Boltzmann Equation

Outline Introduction 1 Introduction 2 3 4 5 Summary Julian Koellermeier, Manuel Torrilhon 2 / 27 Numerical Solutions for the Boltzmann Equation

Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction Introduction Julian Koellermeier, Manuel Torrilhon 3 / 27 Numerical Solutions for the Boltzmann Equation

Introduction Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction Aim Derive hyperbolic PDE systems for rarefied gas flows Extension of standard fluid dynamic equations Reentry flows Micro channel flows Julian Koellermeier, Manuel Torrilhon 4 / 27 Numerical Solutions for the Boltzmann Equation

Introduction Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction Aim Derive hyperbolic PDE systems for rarefied gas flows Extension of standard fluid dynamic equations Reentry flows Micro channel flows Importance of Hyperbolicity Well-posedness and stability of the solution Julian Koellermeier, Manuel Torrilhon 4 / 27 Numerical Solutions for the Boltzmann Equation

Rarefied Gas Dynamics I Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction Goal solve and simulate flow problems involving rarefied gases Knudsen number distinguish flow regimes by orders of the Knudsen number Kn = λ L 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-Carlo simulations Julian Koellermeier, Manuel Torrilhon 5 / 27 Numerical Solutions for the Boltzmann Equation

Rarefied Gas Dynamics II Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction Applications for large Kn = λ L 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 Julian Koellermeier, Manuel Torrilhon 6 / 27 Numerical Solutions for the Boltzmann Equation

Boltzmann Transport Equation Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction 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 Julian Koellermeier, Manuel Torrilhon 7 / 27 Numerical Solutions for the Boltzmann Equation

Model Order Reduction Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction Ansatz f (t, x, c) = M i=0 f i (t, x)h ρ,v,θ i (c) Julian Koellermeier, Manuel Torrilhon 8 / 27 Numerical Solutions for the Boltzmann Equation

Model Order Reduction Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction Ansatz f (t, x, c) = M i=0 f i (t, x)h ρ,v,θ i (c) Reduction of Complexity One PDE for f (t, x, c) that is 7-dimensional Julian Koellermeier, Manuel Torrilhon 8 / 27 Numerical Solutions for the Boltzmann Equation

Model Order Reduction Rarefied Gas Dynamics Boltzmann Equation Model Order Reduction Ansatz f (t, x, c) = M i=0 f i (t, x)h ρ,v,θ i (c) Reduction of Complexity One PDE for f (t, x, c) that is 7-dimensional System of PDEs for ρ(t, x), v(t, x), θ(t, x), f i (t, x) that is 4-dimensional Julian Koellermeier, Manuel Torrilhon 8 / 27 Numerical Solutions for the Boltzmann Equation

Grad HME QBME OP Julian Koellermeier, Manuel Torrilhon 9 / 27 Numerical Solutions for the Boltzmann Equation

Grad s Method [Grad, 1949] Grad HME QBME OP Galerkin Approach Standard method Multiplication with test function and integration Julian Koellermeier, Manuel Torrilhon 10 / 27 Numerical Solutions for the Boltzmann Equation

Grad s Method [Grad, 1949] Grad HME QBME OP Galerkin Approach Standard method Multiplication with test function and integration Grad result t u M + A Grad x u M = 0, Julian Koellermeier, Manuel Torrilhon 10 / 27 Numerical Solutions for the Boltzmann Equation

Grad s Method [Grad, 1949] Grad HME QBME OP Galerkin Approach Standard method Multiplication with test function and integration Grad result t u M + A Grad x u M = 0, u 4 = (ρ, v, θ, f 3, f 4 ) T Julian Koellermeier, Manuel Torrilhon 10 / 27 Numerical Solutions for the Boltzmann Equation

Grad s Method [Grad, 1949] Galerkin Approach Standard method Grad HME QBME OP 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 0 0 0 θ ρ v 1 0 0 6 0 2θ v ρ 0 ρθ 0 4f 3 2 v 4 f 3θ 3f ρ 5f 3 4 2 θ v Julian Koellermeier, Manuel Torrilhon 10 / 27 Numerical Solutions for the Boltzmann Equation

Grad s Method [Grad, 1949] Galerkin Approach Standard method Grad HME QBME OP 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 0 0 0 θ ρ v 1 0 0 6 0 2θ v ρ 0 ρθ 0 4f 3 2 v 4 f 3θ 3f ρ 5f 3 4 2 θ v Loss of hyperbolicity Julian Koellermeier, Manuel Torrilhon 10 / 27 Numerical Solutions for the Boltzmann Equation

Grad HME QBME OP (HME) [Cai et al., 2012] Modification of equations Based on Grad s method Modification of terms in last equation to achieve hyperbolicity Julian Koellermeier, Manuel Torrilhon 11 / 27 Numerical Solutions for the Boltzmann Equation

Grad HME QBME OP (HME) [Cai et al., 2012] Modification of equations Based on Grad s method Modification of terms in 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 0 0 0 θ ρ v 1 0 0 6 0 2θ v ρ 0 ρθ 0 4f 3 2 v 4 ρ 0 f 3 θ v f 3θ Julian Koellermeier, Manuel Torrilhon 11 / 27 Numerical Solutions for the Boltzmann Equation

Grad HME QBME OP (HME) [Cai et al., 2012] Modification of equations Based on Grad s method Modification of terms in 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 0 0 0 θ ρ v 1 0 0 6 0 2θ 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 Julian Koellermeier, Manuel Torrilhon 11 / 27 Numerical Solutions for the Boltzmann Equation

Grad HME QBME OP 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 0 0 0 θ ρ v 1 0 0 6 0 2θ v ρ 0 ρθ 0 4f 3 2 10f 4 θ v 4 f 3θ ρ 5f 4 f 3 θ+ 15f 4 ρθ v Julian Koellermeier, Manuel Torrilhon 12 / 27 Numerical Solutions for the Boltzmann Equation

Grad HME QBME OP 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 0 0 0 θ ρ v 1 0 0 6 0 2θ v ρ 0 ρθ 0 4f 3 2 10f 4 θ v 4 f 3θ ρ 5f 4 f 3 θ+ 15f 4 ρθ v Globally hyperbolic for every state vector u M Julian Koellermeier, Manuel Torrilhon 12 / 27 Numerical Solutions for the Boltzmann Equation

Operator Projection Framework Grad HME QBME OP Moment equations D t w + MD x w = 0 P projection matrix w := P w projected flow variables D := P DP T projected derivative matrix M := P MP T projected multiplication matrix Julian Koellermeier, Manuel Torrilhon 13 / 27 Numerical Solutions for the Boltzmann Equation

Operator Projection Framework Grad HME QBME OP Moment equations t w + D 1 MD x w = 0 P projection matrix w := P w projected flow variables D := P DP T projected derivative matrix M := P MP T projected multiplication matrix Globally hyperbolic for every state vector w Julian Koellermeier, Manuel Torrilhon 14 / 27 Numerical Solutions for the Boltzmann Equation

Model Summary Introduction Grad HME QBME OP Properties globally hyperbolic system multiple spatial dimensions rotational invariance single framework includes all theories Problems analysis of system including collision operator numerical simulations Julian Koellermeier, Manuel Torrilhon 15 / 27 Numerical Solutions for the Boltzmann Equation

Non-Conservative PDE systems Non-Conservative Non-Conservative Variables Julian Koellermeier, Manuel Torrilhon 16 / 27 Numerical Solutions for the Boltzmann Equation

Non-Conservative PDE systems Non-Conservative PDE systems Non-Conservative Non-Conservative Variables Standard conservative PDE system t u + x F(u) = 0 Julian Koellermeier, Manuel Torrilhon 17 / 27 Numerical Solutions for the Boltzmann Equation

Non-Conservative PDE systems Non-Conservative PDE systems Non-Conservative Non-Conservative Variables Standard conservative PDE system t u + x F(u) = 0 Non-conservative PDE system t u + A(u) x u = 0 Can be written in conservative form iff A(u) = F(u) u HME and QBME are partially-conservative systems Julian Koellermeier, Manuel Torrilhon 17 / 27 Numerical Solutions for the Boltzmann Equation

Non-Conservative PDE systems Non-Conservative PDE systems Non-Conservative Non-Conservative Variables Standard conservative PDE system t u + x F(u) = 0 Non-conservative PDE system t u + A(u) x u = 0 Can be written in conservative form iff A(u) = F(u) u HME and QBME are partially-conservative systems Special numerical methods are needed Julian Koellermeier, Manuel Torrilhon 17 / 27 Numerical Solutions for the Boltzmann Equation

Non-Conservative Non-Conservative PDE systems Non-Conservative Non-Conservative Variables Wave Propagation scheme [LeVeque, 1997] Upwind type scheme First order and almost second order Implemented on 2D uniform cartesian grids PRICE-C scheme [Canestrelli, 2009] Centered scheme First order and arbitrary order Implemented on 2D unstructured grids Julian Koellermeier, Manuel Torrilhon 18 / 27 Numerical Solutions for the Boltzmann Equation

Non-Conservative Variables Non-Conservative PDE systems Non-Conservative Non-Conservative Variables Non-conservative PDE system t u + A(u) x u = 0 Julian Koellermeier, Manuel Torrilhon 19 / 27 Numerical Solutions for the Boltzmann Equation

Non-Conservative Variables Non-Conservative PDE systems Non-Conservative Non-Conservative Variables Non-conservative PDE system Variable transformation t u + A(u) x u = 0 w = B 1 (u) u = B (w) Transformed non-conservative PDE system t w + 1 B (w) B (w) A (B (w)) w w xw = 0 Primitive vs partially conserved variables Primitive variables: u = (ρ, v, θ, f 3, f 4 ) Partially conserved variables: w = ( ρ, ρv, ρ ( v 2 + θ ), f 3, f 4 ) Julian Koellermeier, Manuel Torrilhon 19 / 27 Numerical Solutions for the Boltzmann Equation

Shock Tube Test Case Model Comparison Method Comparison Variable Set Comparison Julian Koellermeier, Manuel Torrilhon 20 / 27 Numerical Solutions for the Boltzmann Equation

Shock Tube Test Case Shock Tube Test Case Model Comparison Method Comparison Variable Set Comparison -2 0 2 Julian Koellermeier, Manuel Torrilhon 21 / 27 Numerical Solutions for the Boltzmann Equation

Shock Tube Test Case Shock Tube Test Case Model Comparison Method Comparison Variable Set Comparison -2 0 2 Riemann problem with BGK collision operator t u + A x u = 1 Pu, x [ 2, 2] τ ρ L = 7, ρ R = 1 Variable vector u = (ρ, v, θ, f 3, f 4 ) Relaxation time τ = Kn ρ non-linear Julian Koellermeier, Manuel Torrilhon 21 / 27 Numerical Solutions for the Boltzmann Equation

QBME vs Grad, Kn = 0.05 Shock Tube Test Case Model Comparison Method Comparison Variable Set Comparison ρ Grad 7 ρ QBME p Grad p QBME 0.6 u Grad u QBME 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 Julian Koellermeier, Manuel Torrilhon 22 / 27 Numerical Solutions for the Boltzmann Equation

QBME vs HME, Kn = 0.5 Shock Tube Test Case Model Comparison Method Comparison Variable Set Comparison ρ HME 7 ρ QBME p HME p QBME 0.6 u HME u QBME 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 Julian Koellermeier, Manuel Torrilhon 23 / 27 Numerical Solutions for the Boltzmann Equation

PRICE vs WP, Kn = 0.5 Shock Tube Test Case Model Comparison Method Comparison Variable Set Comparison ρ PRICE 7 ρ WP p PRICE p WP 0.6 u PRICE u WP 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 Julian Koellermeier, Manuel Torrilhon 24 / 27 Numerical Solutions for the Boltzmann Equation

PRICE vs WP2, Kn = 0.5 Shock Tube Test Case Model Comparison Method Comparison Variable Set Comparison ρ PRICE 7 ρ WP2 p PRICE p WP2 0.6 u PRICE u WP2 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 Julian Koellermeier, Manuel Torrilhon 25 / 27 Numerical Solutions for the Boltzmann Equation

Shock Tube Test Case Model Comparison Method Comparison Variable Set Comparison Primitive vs Partially Conserved, Kn = 0.5 ρ prim 7 ρ partconv p prim p partconv 0.6 u prim u partconv 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 Julian Koellermeier, Manuel Torrilhon 26 / 27 Numerical Solutions for the Boltzmann Equation

Summary and Further Work Summary Equations: QBME, HME Numerics: Wave Propagation, PRICE Results: Comparison of models, numerical schemes, variable sets Differences between models are larger than between numerical schemes Further Work More simulations and test cases Higher order PRICE scheme Julian Koellermeier, Manuel Torrilhon 27 / 27 Numerical Solutions for the Boltzmann Equation

Summary and Further Work Summary Equations: QBME, HME Numerics: Wave Propagation, PRICE Results: Comparison of models, numerical schemes, variable sets Differences between models are larger than between numerical schemes Further Work More simulations and test cases Higher order PRICE scheme Thank you for your attention! Julian Koellermeier, Manuel Torrilhon 27 / 27 Numerical Solutions for the Boltzmann Equation

References Introduction 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. Using Quadrature-Based Projection Methods, 29th Rarefied Gas Dynamics, Xi an (2014) Y. Fan, J. Koellermeier, J. Li, R. Li. A framework on the globally hyperbolic moment method for kinetic equations using operator projection method accepted 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. A. Canestrelli. Numerical Modelling of Alluvial Rivers by Shock Capturing Methods, Universita Degli Studi di Padova, (2009). R. LeVeque. Wave Propagation Algorithms for Multidimensional Hyperbolic Systems, J. Comput. Phys., 131 (1997), 327 353. Julian Koellermeier, Manuel Torrilhon 28 / 27 Numerical Solutions for the Boltzmann Equation