Towards an ultra efficient kinetic scheme : High-Performance Computing G. Dimarco 1 J. Narski 2 ere 2 R. Loub` 1 Department of Mathematics and Computer Science, University of Ferrara, Italy. 2 Institut de Math´ ematiques de Toulouse, Universit´ e de Toulouse 31062 Toulouse, France. SHARK-FV Ofir 2014 G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Motivation Motivations Modeling of non equilibrium gas flows (plasma, hypersonic flow) Kinetic equations extremely difficult to solve numerically (7 dimensions) FKS G. Dimarco, R. Loub` ere , Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation , J. Comput. Phys., Vol. 255, 2013, pp 680-698. G. Dimarco, R. Loub` ere , Towards an ultra efficient kinetic scheme. Part II: the high order case , J. Comput. Phys., Vol. 255, 2013, pp 699-719. Goal Parallelize FKS G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Problem formulation Boltzmann-BGK equation ∂ t f + V · ∇ X f = 1 τ ( M f − f ) , f = f ( X , V , t ) distribution od particles, τ relaxation time. Collisions modeled by relaxation towards the local thermodynamical equilibrium defined by the Maxwellian distribution � −� U − V � 2 � ρ M f = M f [ ρ, U , T ] ( V ) = (2 πθ ) 3 / 2 exp , 2 θ θ = TR ρ, U , T , R — density, mean velocity, temperature and gas constant Macroscopic moments � � � U = 1 θ = 1 R 3 � V − U � 2 f d V , ρ = R 3 f d V , R 3 V f d V , ρ 3 ρ Total energy � E = 1 R 3 � V � 2 f d V = 1 2 ρ � U � 2 + 3 2 ρθ 2 G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
The limit of τ → 0 If number of collisions goes to infinity, then τ → 0 and f → M f . One retrieves compressible Euler equations ∂ρ ∂ t + ∇ X · ( ρ U ) = 0 , ∂ ( ρ U ) + ∇ X · ( ρ U ⊗ U + pI ) = 0 , ∂ t ∂ E ∂ t + ∇ X · (( E + p ) U ) = 0 , E = 3 2 ρθ + 1 2 ρ � U � 2 , p = ρθ, where I is the identity and p the pressure given by a perfect equation of state with gas constant γ = 5 / 3 in 3D. This is the fluid/macroscopic model. G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Fast Kinetic Scheme Semi-Lagrangian scheme for Discrete Velocity Model (DVM) approximation of Boltzmann-BGK equation. DVM Let K be a bounded set of N 3 v multi-indices of N 3 . Let V be a Cartesian grid given by V = { V k = k ∆ v + W , k ∈ K} , where ∆ v is the grid step in the velocity space. The generic cell in the velocity space is ω k +1 / 2 = [ V k ; V k +1 ]. We denote the discrete collision invariants on V by � � t 1 , V k , 1 2 � V k � 2 m k = G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Fast Kinetic Scheme Semi-Lagrangian scheme for Discrete Velocity Model (DVM) approximation of Boltzmann-BGK equation. DVM The continuous distribution function f is replaced by a vector f K ( X , t ) = ( f k ( X , t )) k , f k ( X , t ) ≈ f ( X , V k , t ) . Fluid quantities: � F ( X , t ) = m k f k ( X , t ) ∆ v . k ∈K G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Fast Kinetic Scheme Set of N 3 v evolution equations ∂ t f k + V k · ∇ X f k = 1 τ ( E k [ F ] − f k ) Space and time discretization Cartesian uniform grid X = { X j = j ∆ x + Y , j ∈ J } , ∆ x is the grid step, Y is a vector in R 3 and J is a subset of N 3 . t n +1 = t n + ∆ t , time step ∆ t defined by a CFL condition. Splitting − → ∂ t f k + V k · ∇ X f k = 0 , Transport stage ∂ t f k = 1 Relaxation stage − → τ ( E k [ F ] − f k ) . G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Fast Kinetic Scheme V k+2 V k+2 k k n+1 f k+2,j V k+1 V k+1 n f k+2,j j j n+1 f k+1,j V k V k n f k+1,j j j n n+1 f k,j f k,j j j j , k be the pointwise approximation at discrete time t n of the distribution f : Let f n f n j , k = f ( X j , V k , t n ) and E n j , k [ F ] be the equilibrium distribution approximation of M n j , k = M f ( X j , V k , t n ) defined at any point X j of space at discrete time t = t n . n k be a piecewise constant function associated with velocity V k at time t n Let f defined at each space cell by 1 � n f ( X , V k , t n ) d X f k , j = | Ω j | Ω j Exact transport during ∆ t : ∗ , n +1 n = f k ( X − V k ∆ t ) , ∀ X ∈ Ω f k G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Fast Kinetic Scheme Relaxation step ∂ t f j , k = 1 τ ( E j , k [ F ] − f j , k ) Initial data is given by the result of the transport step ∗ , n +1 f ∗ , n +1 = f ( X j ). j , k k Maxwellian computed using macroscopic quantities � = F ∗ , n +1 m k f ∗ , n +1 F n +1 = ∆ v j j j , k k ∈K Preservation of macroscopic quantities: moments before ( F ∗ , n +1 ) and j after ( F n +1 ) unchanged. Then j = exp( − ∆ t /τ ) f ∗ , n +1 f n +1 + (1 − exp( − ∆ t /τ )) E k [ F n +1 ] , j , k j , k j New value of f n +1 only in the cell centers, we need f n +1 in whole domain for the transport step. G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Fast Kinetic Scheme Relaxation step ∂ t f j , k = 1 τ ( E j , k [ F ] − f j , k ) Define E k as the equilibrium function with the discontinuities located in the same positions as f k n +1 ∗ , n +1 ∗ , n +1 ( X )[ F ] = E n +1 E j , k [ F ] , ∀ X such that f ( X ) = f ( X j ) k k k Finally n +1 ∗ n +1 ( X ) = f k ( X , t n +∆ t ) = exp( − ∆ t /τ ) f f k ( X )+(1 − exp( − ∆ t /τ )) E ( X )[ F ] k k G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
HOFKS - high order extension Second order in time Time splitting with Strang splitting strategy. CFL: || V k || ∆ t max ≤ 1 L c K performs well in collisionless regimes scheme stable for ∆ t > CFL projection over the equilibrium of first order → loss of accuracy close to fluid regime G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
HOFKS - high order extension n +1 ∗ n +1 ( X ) = exp( − ∆ t /τ ) f k ( X ) + (1 − exp( − ∆ t /τ )) E ( X )[ F ] f k k Solve the equilibrium part with of the distribution function with a macroscopic scheme instead of kinetic. Moments from the transport stage are replaced by a solution of Euler equations. We use MUSCL scheme. Stability condition: ∆ t < 1 ∆ x 2 α max In the limit of τ → 0 scheme corresponds to a high order shock capturing scheme. Second order close to the fluid limit. G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Efficient implementation j+ δ particle leaving cell j for cell y particle entering cell j from cell j− δ p p p leaving for j+ δ p j p’ entering from j− δ p p x Particle implementation: Initially N 3 v particles are positioned at the cell center X 0 p = (∆ x / 2 , ∆ y / 2 , ∆ z / 2) t each particle has a unique constant velocity V p from the velocity space, p = 1 , · · · , N 3 v . The transport of these particles during ∆ t follows n +1 � = X n X p + ∆ t V p . p Same set of particles in every space cell, only positions and velocities of particles in generic cell kept in memory. G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
Particle mass Each particle p in cell j carries its “mass” which is updated defined at t n thanks to the previous mass m n − 1 j , U n and updated moments ρ n j , θ n j as j , p j , p = exp( − ∆ t /τ ) m n − 1 j , U n m n j , p + (1 − exp( − ∆ t /τ )) M f [ ρ n j , θ n j ]( V p ) � � (∆ v ) 3 ρ n −� V p − U n j � 2 j = exp( − ∆ t /τ ) m n − 1 j , p + (1 − exp( − ∆ t /τ )) j ) 3 / 2 exp (2 πθ n 2 θ n j Because the fluid quantities are obtained through discrete summations on particles in cell j : N 3 � v F n m n j = j , p ∆ v p =1 the updated fluid quantities are therefore obtained after the transport step following � � F n +1 = F n m n m n j − j , p ∆ v + j − δ, p ′ ∆ v . j n +1 n +1 p , � p ′ , � X ∈ Ω j / X p ′ ∈ Ω j p � �� � � �� � Leaving particles Entering particles G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing Recall that these conservative cell centered fluid quantities are
Generic algorithm Relaxation step. Compute masses of N 3 v particles, store them in an 1 array of the size N 3 v × N 3 Transport of particles. Displace N 3 v particles, produce a list of N out 2 particles escaping the generic cell and store the δ determining the destination and provenance of associated sister particles. Update conservative variables F n +1 3 j Update primitive variables 4 G. Dimarco, J. Narski, R. Loub` ere Towards an ultra efficient kinetic scheme : High-Performance Computing
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