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 Numerics

Outline 1 Rarefied Gases and Hyperbolic Moment Equation 2 3 4 J. Koellermeier 2 / 52 Partially-Conservative Numerics

Rarefied Gas Dynamics Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations Aim Modeling and numerical simulation of rarefied gases Knudsen number Distinguish flow regimes by orders of the Knudsen number Kn = mean free path length reference length = λ L J. Koellermeier 3 / 52 Partially-Conservative Numerics

Rarefied Gas Dynamics (2) Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations Applications for large Kn = λ L large λ: high altitude flights or atmospheric reentry small L: micro-scale applications, Knudsen pump, MEMS J. Koellermeier 4 / 52 Partially-Conservative Numerics

Rarefied Gas Dynamics (2) Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations Applications for large Kn = λ L large λ: high altitude flights or atmospheric reentry small L: micro-scale applications, Knudsen pump, MEMS Tasks computation of mass flow rates calculation of shock layer thickness accurate prediction of heat flux J. Koellermeier 4 / 52 Partially-Conservative Numerics

Boltzmann Transport Equation Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations 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 5 / 52 Partially-Conservative Numerics

Boltzmann Transport Equation 1D Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations 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 6 / 52 Partially-Conservative Numerics

Macroscopic Quantities Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations 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) + 1 2 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 7 / 52 Partially-Conservative Numerics

Model Order Reduction Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations Problem: Direct discretization of f (t, x, c) is extremely costly Ansatz: Expansion Reduction of Complexity f (t, x, c) = M α=0 f α (t, x)φ v,θ α (c) 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 8 / 52 Partially-Conservative Numerics

Grad s Method (Grad, 1949 [5]) Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations 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 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 J. Koellermeier 9 / 52 Partially-Conservative Numerics

Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations Hyperbolic Moment Equations (Cai et al., 2012 [1]) 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 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 J. Koellermeier 10 / 52 Partially-Conservative Numerics

Introduction to Rarefied Gases Boltzmann Equation Hyperbolic Moment Equations Quadrature-Based Moment Equations (JK, 2014 [6]) 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 J. Koellermeier 11 / 52 Partially-Conservative Numerics

Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Non-conservative Numerical Methods J. Koellermeier 12 / 52 Partially-Conservative Numerics

Conservative PDE systems Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Standard conservative PDE system t u + x F(u) = 0 J. Koellermeier 13 / 52 Partially-Conservative Numerics

Conservative PDE systems Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation 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 1 2 2 J. Koellermeier 13 / 52 Partially-Conservative Numerics

Non-conservative PDE systems Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Non-conservative PDE system t u + A(u) x u = 0 Can be written in conservative form iff A(u) = F(u) u In general no flux function available J. Koellermeier 14 / 52 Partially-Conservative Numerics

Non-conservative PDE systems Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Non-conservative PDE system t u + A(u) x u = 0 Can be written in conservative form iff A(u) = F(u) u In general no flux function available Hyperbolic moment equations are all given in partially-conservative form Special numerical methods are needed J. Koellermeier 14 / 52 Partially-Conservative Numerics

Non-conservative Numerics Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation How do we evaluate A at a cell boundary? J. Koellermeier 15 / 52 Partially-Conservative Numerics

Non-conservative Numerics Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation How do we evaluate A at a cell boundary? Roe matrix for conservative systems A Roe (u L, u R ) (u R u L ) = F (u R ) F (u L ) J. Koellermeier 15 / 52 Partially-Conservative Numerics

Non-conservative Numerics Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation How do we evaluate A at a cell boundary? Roe matrix for conservative systems Non-conservative case A Roe (u L, u R ) (u R u L ) = F (u R ) F (u L ) A Roe depends on a path ψ between u L and u R (DLM, 1995 [2]) Example: ψ (s, u L, u R ) = u L + s (u R u L ), s [0, 1] Extension: Generalized Roe matrix A ψ (u L, u R ) (u R u L ) = 1 0 A (ψ (s, u L, u R )) ψ s ds J. Koellermeier 15 / 52 Partially-Conservative Numerics

Generalization of Roe matrix Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Reduces to standard Roe matrix for conservative case A ψ (u L, u R ) (u R u L ) = = 1 0 1 0 1 A (ψ (s, u L, u R )) ψ s ds F (ψ (s, u L, u R )) ψ ψ s ds F (ψ (s, u L, u R )) = ds 0 s = F (ψ (1, u L, u R )) F (ψ (0, u L, u R )) = F (u R ) F (u L ) J. Koellermeier 16 / 52 Partially-Conservative Numerics

Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Computation of generalized Roe matrix A ψ (u L, u R ) (u R u L ) = 1 0 A (ψ (s, u L, u R )) ψ s ds Choose linear path ψ (s, u L, u R ) = u L + s (u R u L ), s [0, 1] Use Gaussian quadrature to compute integral J. Koellermeier 17 / 52 Partially-Conservative Numerics

Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Computation of generalized Roe matrix A ψ (u L, u R ) (u R u L ) = 1 0 A (ψ (s, u L, u R )) ψ s ds Choose linear path ψ (s, u L, u R ) = u L + s (u R u L ), s [0, 1] Use Gaussian quadrature to compute integral Computation A ψ (u L, u R ) (u R u L ) = 1 0 A (ψ (s, u L, u R )) (u R u L ) ds = A ψ (u L, u R ) = 1 0 A (ψ (s, u L, u R )) ds M ω j A (ψ (s j, u L, u R )) j=1 J. Koellermeier 17 / 52 Partially-Conservative Numerics

Different paths Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation linear path ψ (s, u L, u R ) = u L + s (u R u L ) polynomial path ψ N (s, u L, u R ) = u L + s N (u R u L ) ψ N + (s, u L, u R ) = u R + (1 s) N (u L u R ) J. Koellermeier 18 / 52 Partially-Conservative Numerics

Different paths Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Tuomi path ψ T (s, u L, u R ) = ( {( u 1, u 2) T u 1 = L + 2s ( ur 1 ) ) u1 L, u 2 T L, s 1 ( 2, u 1 R, ul 2 + (2s 1) ( ur 2 )) T u2 L, s > 1 2 J. Koellermeier 19 / 52 Partially-Conservative Numerics

Non-Conservative Variables Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation 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 J. Koellermeier 20 / 52 Partially-Conservative Numerics

Non-Conservative Variables Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation 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 ) J. Koellermeier 20 / 52 Partially-Conservative Numerics

Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Upwind scheme [Castro, Pares, 2004] First order scheme u n+1 i = u n i t x ( A ( i+ u n 1 i+1 u n ) ( i + A + i u n 1 i u n ) ) i 1 2 2 Upwind type scheme, uses eigenvalue information A ± i+ 1 2 = A ψ (u i, u i+1 ) ± = R ψ Λ ± ψ R 1 ψ J. Koellermeier 21 / 52 Partially-Conservative Numerics

PRICE-C scheme [Canestrelli, 2009] Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation First order scheme u n+1 i = u n i t x ( A ( i+ u n 1 i+1 u n ) ( i + A + i u n 1 i u n ) ) i 1 2 2 Same notation as Castro scheme PRImitive CEntered scheme, uses no eigenvalue information Reduces to FORCE scheme in the conservative case FORCE scheme u n+1 i = u n i t ( F FORCE x i+ 1 2 ) F FORCE i 1 2 J. Koellermeier 22 / 52 Partially-Conservative Numerics

Complete PRICE-C scheme Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation First order scheme u n+1 i = u n i t x ( A ( i+ u n 1 i+1 u n ) ( i + A + i u n 1 i u n ) ) i 1 2 2 A i+ 1 2 A + i 1 2 = 1 ( ( 2A ψ u n 4 i, u n ) x i+1 t I t ) ( ( Aψ u n x i, u n )) 2 i+1 = 1 ( ( 2A ψ u n 4 i 1, u n ) x i t I t ) ( ( Aψ u n x i 1, u n )) 2 i J. Koellermeier 23 / 52 Partially-Conservative Numerics

Higher order extension Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation WENO reconstruction in space u i u i (x) ADER approach in time u i (x, t) = u (x i, t n ) + (x x i ) u x + (t tn ) u t u t = A(u) xu Integration of PDE over time-space volume and computation of integrals using Gaussian quadrature and reconstruction J. Koellermeier 24 / 52 Partially-Conservative Numerics

Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Wave Propagation scheme [LeVeque, 1997] First order scheme u n+1 i = u n i t ( A + u i + A ) u i+1 x A u i is called fluctuation Fluctuations are split A u i = A u i + A + u i Similar to flux difference splitting, but without a flux function J. Koellermeier 25 / 52 Partially-Conservative Numerics

Solution of local Riemann problem Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation A(u ) = R Λ R 1 i 1 2 Wave speeds λ j = Λ jj Waves W j = α j R j Left and right going fluctuations A u i = p (λ p ) W p A + u i = p (λ p ) + W p J. Koellermeier 26 / 52 Partially-Conservative Numerics

Second order extension Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Add correction term u n+1 i = u n i t x Second order corrections F i = 1 2 ( A + u i + A ) t u i+1 ( Fi+1 x F ) i p ( λ p i 1 t ) x λp i W p i Limiter for stability W p i = φ ( θ p i ) W p i, θp i = Wp i 1 Wp i W p i W p i J. Koellermeier 27 / 52 Partially-Conservative Numerics

Numerical Schemes Non-conservative Numerics Upwind scheme PRICE-C Wave Propagation Castro scheme [Castro, Pares, 2004] Arbitrary order upwind type scheme Implemented on 2D unstructured grids PRICE-C scheme [Canestrelli, 2009] Arbitrary order centered scheme Implemented on 2D unstructured grids Wave Propagation scheme [LeVeque, 1997] 2 nd order upwind type scheme Implemented on 2D uniform cartesian grids J. Koellermeier 28 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock J. Koellermeier 29 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 1D Shock Tube Test Case 2D Forward Facing Step 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 30 / 52 Partially-Conservative Numerics

Model Equations Grad model HME model A Grad = A HME = 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 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 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 J. Koellermeier 31 / 52 Partially-Conservative Numerics

Model Equations 2 Grad model QBME model A Grad = A QBME = 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 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 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 J. Koellermeier 32 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 1. Euler vs Moment Equations, Kn = 0.05 7 exact HME12 Euler 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 33 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 2. Roe matrix, linear path Kn = 0.5 7 linear 1 linear 3 linear 80 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 34 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 2. Roe matrix, polynomial path Kn = 0.5 7 polynomial 20 polynomial 40 polynomial 80 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 35 / 52 Partially-Conservative Numerics

2. Roe matrix, paths Kn = 0.5 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 7 linear 3 polynomial 80 Tuomi 4 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 36 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 3. Non-Conservative Variables, Kn = 0.05 7 primitive part.convective convective 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 37 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 3. Non-Conservative Variables, Kn = 0.5 7 primitive part.convective convective 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 38 / 52 Partially-Conservative Numerics

4. Scheme Comparison, Kn = 0.05 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 7 PRICE UPRICE Castro WP2 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 39 / 52 Partially-Conservative Numerics

4. Scheme Comparison, Kn = 0.5 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 7 PRICE UPRICE Castro WP2 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 40 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 5. High resolution scheme convergence Kn = 0.5 30 2nd 10000 2nd 4000 2nd 1000 2nd 400 1.16 20 0.76 10 0.36 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 41 / 52 Partially-Conservative Numerics

6. Model Comparison Kn = 0.05 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 7 HME QBME Grad DVM 0.6 5 0.4 3 0.2 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 42 / 52 Partially-Conservative Numerics

6. Model Comparison Kn = 0.5 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 7 HME QBME DVM 0.75 5 0.5 3 0.25 1 0 1 0.5 0 0.5 1 1.5 J. Koellermeier 43 / 52 Partially-Conservative Numerics

7. Model Convergence, Kn = 0.5 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 7 QBME8 QBME9 QBME8+9 DVM 0.75 10 1 rho u p theta 5 0.5 10 2 3 0.25 1 0 1 0.5 0 0.5 1 1.5 4+5 5+6 6+7 7+8 8+99+10 J. Koellermeier 44 / 52 Partially-Conservative Numerics

2D Forward Facing Step 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock Test setup rarefied supersonic flow over a forward facing step slip boundary conditions unstructured quad grid, 31k cells J. Koellermeier 45 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 2D Forward Facing Step, Ma = 3, Euler J. Koellermeier 46 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 2D Forward Facing Step, Ma = 3, QBME, Kn = 0.001 J. Koellermeier 47 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 2D Forward Facing Step, Ma = 3, QBME, Kn = 0.01 J. Koellermeier 48 / 52 Partially-Conservative Numerics

1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock 2D Forward Facing Step, Ma = 3, QBME, Kn = 0.1 J. Koellermeier 49 / 52 Partially-Conservative Numerics

2D bow shock, Ma = 3, ρ plotted 1D Shock Tube Test Case 2D Forward Facing Step 2D Bow Shock Figure: Kn = 0.0005 Figure: Kn = 0.5 J. Koellermeier 50 / 52 Partially-Conservative Numerics

Summary Summary Further work Hyperbolic Moment Equations Extension of Euler equations for rarefied gases Hierarchical model Partially-conservative systems Non-conservative numerics Accurate computation of non-conservative products Tests showed convergence of schemes and models J. Koellermeier 51 / 52 Partially-Conservative Numerics

Further work Summary Further work Aim Adaptive numerical method to couple standard fluid dynamics and rarefied gases J. Koellermeier 52 / 52 Partially-Conservative Numerics

Further work Summary Further work Aim Adaptive numerical method to couple standard fluid dynamics and rarefied gases Model adaptivity Hierarchical moment equations Coupling strategies Refinement and coarsening Error estimation J. Koellermeier 52 / 52 Partially-Conservative Numerics

Further work Summary Further work Aim Adaptive numerical method to couple standard fluid dynamics and rarefied gases Model adaptivity Hierarchical moment equations Coupling strategies Refinement and coarsening Error estimation Thank you for your attention J. Koellermeier 52 / 52 Partially-Conservative Numerics

References Summary Further work Z. Cai, Y. Fan, and R. Li. Globally hyperbolic regularization of Grad s moment system. Communications on Pure and Applied Mathematics, 67(3):464 518, 2014. G. Dal Maso, P. G. LeFloch, and F. Murat. Definition and weak stability of nonconservative products. J. Math. Pures Appl., 74(6):483 548, 1995. Y. Di, Y. Fan, and R. Li. 13-moment system with global hyperbolicity for quantum gas. Journal of Statistical Physics, 167(5):1280 1302, 2017. Y. Fan, J. Koellermeier, J. Li, R. Li, and M. Torrilhon. Model reduction of kinetic equations by operator projection. Journal of Statistical Physics, 162(2):457 486, 2016. H. Grad. On the kinetic theory of rarefied gases. Communications on Pure and Applied Mathematics, 2(4):331 407, 1949. J. Koellermeier, R. P. Schaerer, and M. Torrilhon. A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods. Kinetic and Related Models, 7(3):531 549, 2014. J. Koellermeier and M. Torrilhon. Hyperbolic moment equations using quadrature-based projection methods. AIP Conference Proceedings, 1628(1):626 633, 2014. J. Koellermeier 53 / 52 Partially-Conservative Numerics

Additions and Developments Hyperbolic Moment Models Extension to n-d (JK, 2014 [7]) Framework for hyperbolic models (Fan, JK et al., 2016 [4]) Numerical simulation (JK, 2017 [8])... Applications Relativistic Boltzmann equation (Tang, 2017 [9]) Quantum Gas (Di et al., 2017 [3])... J. Koellermeier 54 / 52 Partially-Conservative Numerics