Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles
|
|
- Patience Jennings
- 5 years ago
- Views:
Transcription
1 Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3. Workshop Asymptotic-Preserving schemes, Porquerolles. May 23rd, INRIA Nancy - Grand Est & University of Strasbourg, IRMA. 2 INRIA Rennes - Bretagne Atlantique & ENS Cachan Bretagne, IRMAR. 3 University of Rennes I & ENS Cachan Bretagne, IRMAR.
2 Outline 1 2 3
3 Numerical simulation of plasmas Different scales (Knudsen number ε) = different models. Kinetic model: particles represented by a distribution function f (x, v, t), Vlasov equation (with source term S (ε)) t f +v x f + q m (E+v B) vf = S (ε) coupled to Maxwell equations or Poisson equation, (E electric field, B magnetic field, x position, v velocity, t time, q charge of particles, m masse of particles) in 3D = 7 variables: 3 in space, 3 in velocity and the time = heavy computations.
4 Fluid model: moment equations on physical quantities linked to f (density, mean velocity, temperature, etc.), smaller cost, but less precision. If two scales in the same simulation: two schemes with an interface? Difficult to deal with the interface! 1st obj. Construct an accurate but not too costly scheme. 2nd obj. Construct an asymptotic-preserving 4 (AP) scheme. 4 S. Jin, SIAM JSC 1999.
5 AP scheme Vlasov ε h Discretized Vlasov h,ε ε ε Limit h Discretized limit h Prop: Stability and consistency ε, particularly when ε. Aim: Standard schemes: h = O(ε) construct a scheme for which h is independent of ε.
6 Vlasov-BGK Vlasov with collisions = Vlasov-BGK equation: t f +v x f +E v f = 1 ε Q(f) coupled to Poisson equation: x E (x,t) = ρ(x,t) 1 = f (x,v,t)dv 1. (ε Knudsen number, ρ = ρ(x,t) charge density.) Q(f) BGK (Bhatnagar-Gross-Krook) collision operator: Q(f) = M(U) f, M(U) being the Maxwellian having the same first three moments than f, denoted by U.
7 ) M(U) = ρ 2πT exp ( v u 2 2T. (T = T (x,t) temperature, u = u(x,t) mean velocity.) U = 1 v v 2 2 = U = f dv = ρ ρu 1 2 (ρu2 +ρt). 1 v v 2 2 M(U) dv Remarks: M(U) N (L Q ) = Span { M,vM, v 2 M }, the null space of the linearized operator L Q of Q, f M(U) when ε : collisions move f closer to its Maxwellian.
8 Micro-macro model PIC method Projection of weights Finite volumes scheme AP property Decomposition 56 f = M(U)+g: t M(U)+v x M (U)+E v M(U)+ t g+v x g+e v g = 1 ε g. Transport operator T = v x +E v : t M(U)+T M(U)+ t g +Tg = 1 ε g. Hypothesis: first three moments of g must be zero = mf := m(v)f (x,v,t) dv = U(x,t), ( where m(v) = 1,v, v2 2 ) t. True at the numerical level? If not, we have to impose it. 5 M. Lemou, L. Mieussens, SIAM JSC N. Crouseilles, M. Lemou, KRM 21.
9 Micro-macro equations PIC method Projection of weights Finite volumes scheme AP property Let Π M the orthogonal projection in L 2 (M 1 dv) onto N (L Q ) 7 : [ Π M (ϕ) = 1 (v u) (v u)ϕ ρ ϕ + T ( v u ) ( ) ] v u 2 1 ϕ M. 2T 2 T Properties: (I Π M )( t M) = Π M (g) = Π M ( t g) =. Applying (I Π M ) to Vlasov-BGK = micro equation on g t g +(I Π M )T (M +g) = 1 ε g. 7 M. Bennoune, M. Lemou, L. Mieussens, JCP 28.
10 PIC method Projection of weights Finite volumes scheme AP property Applying Π M to Vlasov-BGK = macro equation on M(U) t M +Π M T (M +g) = or by taking his first three moments F (U) = S (U) = t U + x F(U)+ x vm(v)g = S(U), ρu ρu 2 +ρt ρu ( u T) ρe ρue : source term. : Euler flux
11 Algorithm PIC method Projection of weights Finite volumes scheme AP property System { t g +(I Π M )T (M +g) = 1 ε g t U + x F(U)+ x vm(v)g = S(U) equivalent to Vlasov-BGK. 1. Solving the micro part by a Particle-In-Cell (PIC) method. 2. Projection step to numerically force to zero the first three moments of g (matching procedure 8 ). 3. Solving the macro part by a finite volume scheme (mesh on x), with a source term dependent on g. 1-3 coupling: similarities with the δf method 9 but here: AP scheme. 8 P. Degond, G. Dimarco, L. Pareschi, IJNMF, S. Brunner, E. Valeo, J.A. Krommes, Phys. of Plasmas, 1999
12 PIC method PIC method Projection of weights Finite volumes scheme AP property Equation t g +(I Π M )T (M +g) = 1 ε g t g +T g = (I Π M )TM +Π M T g g ε =: S g. Model: having N p particles, with position x k, velocity v k and weight ω k, g is approximated by N p g Np (x,v,t) = ω k (t)δ(x x k (t))δ(v v k (t)). k=1
13 PIC method Projection of weights Finite volumes scheme AP property Mesh generation Fields computing on the mesh Initialization of positions and velocities of particles Computation of charge and current densities on the mesh (deposition) Emission of new particles Movement of particles Interpolation of fields on the particles
14 Shape functions PIC method Projection of weights Finite volumes scheme AP property Use of shape functions such as B-spline of order l B l (x) = (B B l 1 )(x), with In particular B (x) = { 1 x if x < x/2, else. B 1 (x) = 1 x Order : Nearest Grid Point. Order 1: Cloud In Cell. { 1 x / x, if x < x, else.
15 Deposition and interpolation PIC method Projection of weights Finite volumes scheme AP property Computation of the moment M of order p on the cell i: M p,i (t) = M p (x,t)b l (x i x) dx R = v p g (x,v,t) dv B l (x i x) dx = R N p R ω k (t)v p k (t)b l(x i x k (t)). k=1 Evaluation of the electric field on particle k: N x E (x k,t) = E (x i,t)b l (x i x k (t)). i=1
16 PIC method Projection of weights Finite volumes scheme AP property 1. Initialization: particles randomly (or quasi) distributed in phase space (x, v), weights initialized to ω k () = g (x k,v k,) LxLv N p. (L x x-length of the domain, L v v-length.) 2. Solving the electric field on the mesh: x E (x,t) = ρ(x,t) 1 = f (x,v,t)dv 1 thanks to the knowledge of ρ i (t), and then interpolation on the particles E (x k,t).
17 Solving t g +T g = PIC method Projection of weights Finite volumes scheme AP property 3. Movement of particles thanks to motion equations: dx k dt (t) = v k (t) et Verlet scheme (for example): v n+1 2 k = v n k + t 2 En (x n k ) dv k dt (t) = E (x k,t). x n+1 k = xk n + tvn+1 2 k v n+1 k = v n+1 2 k + t 2 En+1( x n+1 ) k.
18 PIC method Projection of weights Finite volumes scheme AP property Solving t g = S g 4. Evolution of weights ω k (step specific to kinetic equations with source term): dω k (t) = s k (t), dt where s k (t) = S g (x k,v k ) L xl v N p is the weight associated to the source term S g = (I Π M )T M +Π M Tg g ε. In our case: dω k (t) = α k (t) ω k (t), dt ε where α k (t) is associated to (I Π M )T M +Π M Tg.
19 PIC method Projection of weights Finite volumes scheme AP property (I Π M )T M +Π M Tg is constituted of moments of g, their derivatives and M. Moments of g are computed by deposition, derived by finite differences and evaluated on (x k,v k ) by interpolation. ρ, u and T are evaluated on (x k,v k ) by interpolation. ) M = ρ exp is now easily evaluated on (x 2πT k,v k ). ( v u 2 2T To have an AP scheme, we make the stiff term ω k(t) ε ω n+1 k = ωn k + tαn k 1+ t. ε implicit:
20 Projection step PIC method Projection of weights Finite volumes scheme AP property We now have N p g n+1 (x,v,t) k=1 We want to ensure mg n+1 =. ω n+1 k δ ( x x n+1 k We compute U g := mg n+1, N p ) ( ) δ v v n+1 k. U g (x i ) = mg n+1 xi = ω k m(v k )B l (x i x k ). k=1
21 PIC method Projection of weights Finite volumes scheme AP property We seek h N (L Q ) of the form h(x,v) = λ(x) m(v)m(x,v) s.t. U g (x i ) = mh(x i,v). We expand λ(x) on the basis of B-splines of degree l N x λ(x) = λ j B l (x x j ), λ j R 3. j=1 For example l = (l = 1 is also computed). Let p k the weights associated to M, we compute mh x i 1 x 2 m(v k ) m(v k )p k λ i. k/ x k x i x/2
22 PIC method Projection of weights Finite volumes scheme AP property We solve the 3 3 linear system U g (x i ) = A i λ i, with A i = 1 x M,i M 1,i M 2,i M 1,i M 2,i M 3,i M 2,i M 3,i M 4,i and M j,i = 1 x k/ x k x i x/2 p kv j k. We compute the weights γ k of h, N x γ k = λ j B (x k x j ) m(v k )p k = 1 x λ j k m(v k )p k, j=1 where j k is such that x k x jk < x/2.
23 PIC method Projection of weights Finite volumes scheme AP property We correct the weights of g to obtain = 1 x k/ x k x i x/2 ω new k ω k γ k, mg n+1,new xi = 1 x ω k m(v k ) 1 x Remark: order 1 = (3N x 3N x ) system. k/ x k x i x/2 k/ x k x i x/2 ω new k m(v k ) γ k m(v k ) =.
24 Macro part PIC method Projection of weights Finite volumes scheme AP property Equation t U + x F(U) = S(U,g) (= S(U) x vm(v)g ). Finite volume method U n+1 i = Ui n t ( ) Fi+1/2 n x Fn i 1/2 t S i n, with Rusanov flux Fi+1/2 n = 1 ( ( ) F U n 2 i+1 +F (U n i ) a i+1/2 (U i+1 U i ) ), where a i+1/2 = max j=i,i+1 ( abs R ( JF ( xj ))), R (JF ) being the eigenvalues of the Jacobian of F. S n i = S (U n i ) ( vmg n+1 xi+1/2 vmg n+1 xi 1/2 x where vmg n+1 xi+1/2 = 1 x k/ x k x i+1/2 x/2 ωn+1 k v k m(v k ). )
25 Analytical limit PIC method Projection of weights Finite volumes scheme AP property t g +(I Π M )T (M +g) = 1 g, ε (1) t U + x F(U)+ x vm(v)g = S(U). (2) Limit ε in (1): g = O(ε) = g = ε(i Π M )T M +O(ε 2 ), = ε(i Π M )(v x M +E v M)+O(ε 2 ), = ε(i Π M )(v x M)+O(ε 2 ), because E v M N (L Q ) and then (I Π M )(E v M) =. Injected into (2), we obtain the Navier-Stokes model t U+ x F(U) = S(U)+ε x vm(v)(i Π M )(v x M) +O(ε 2 ).
26 Numerical limit PIC method Projection of weights Finite volumes scheme AP property (1) can be discretized as ω n+1 k t ω n k ω n+1 k = = α n 1,k +αn 2,k ωn+1 kε, 1 ( ω n 1+ t/ε k tα n ) 1,k + tαn 2,k. 1 But 1+ t/ε = ε t +O(ε2 ) when ε, ωk n = O(ε) thus α n 2,k ω n+1 k = εα n 1,k +O(ε2 ). = O(ε) and We obtain the diffusion term of the Navier-Stokes model = AP property. g n+1 = ε(i Π M )T M +O(ε 2 )
27 Landau damping First test case Second test case Computational cost Initial distribution function: f (x,v,) = 1 exp( v2 )(1+αcos(kx)), x [,2π/k] 2π 2 Micro-macro initializations: 1+αcos(kx) U(x) = 1+αcos(kx) and g(x,v,t = ) =. Parameters: α =.1, k =.5. Electrical energy E(t) = E(t,x) 2 dx.
28 First test case Second test case Computational cost Convergence at the ε limit, N x = 128, N p = : log E(t) L Euler -1 MiMa, eps=.1 MiMa, eps=.1 MiMa, eps=.3 MiMa, eps= Time t
29 First test case Second test case Computational cost Heat flux of MiMa (blue) and Navier-Stokes (red) at t = 1 with N p = 1 5 for ε =.1 (left),.1 (middle) and.1 (right): eps=.1 eps=.1 eps= NS MiMa.5.4 NS MiMa.5.4 NS MiMa Heat flux Heat flux Heat flux Position x Position x Position x where heat fluxes correspond to: (3/2)ρT x T for NS and 1 ε v u 3 (M +g) for MiMa.
30 First test case Second test case Computational cost Distribution function f of PIC BGK, MiMa, g and M, at t = 2, ε = 1, N p = : f - PIC BGK f - MiMa Position x Velocity v Position x Velocity v g - MiMa M - MiMa Position x Velocity v Position x Velocity v
31 First test case Second test case Computational cost Noise reduction on ρ, ε = 1, N x = 128, N p = 5 1 5, at t =.2 (left) and t =.4 (right):.15 PIC BGK MiMa.2 PIC BGK MiMa Charge density 1-rho -.5 Charge density 1-rho Position x Position x
32 First test case Second test case Computational cost Necessary number of particles, ε = 1, N x = 128: log E(t) L PIC BGK, Npart=2 5-2 PIC BGK, Npart= 5 PIC BGK, Npart= 1 MiMa, Npart= Time t
33 First test case Second test case Computational cost Convergence in particles, ε = 1 4, N x = 128: -2 MiMa, Npart= 5 MiMa, Npart= log E(t) L Time t
34 First test case Second test case Computational cost Importance of the projection step on ρ at t = 5, ε = 1, N p = :.6 Without correction Correction of order Correction of order 1.4 Charge density 1-rho Position x
35 Test 2 First test case Second test case Computational cost Initial distribution function: f (x,v,) = v4 exp( v2 )(1+αcos(kx)), x [,2π/k] 2π 2 Micro-macro initializations: U(x) = 1+αcos(kx) 5(1+αcos(kx)) g (x,v) = 1+αcos(kx) ( v 4 2π 3 exp( v2 /2) 1 ) exp( v 2 /1). 5 Parameters: α =.5, k =.5.
36 First test case Second test case Computational cost Convergence at the ε limit, N x = 128, N p = : log E(t) L Euler MiMa, eps=.1-1 MiMa, eps=.1 MiMa, eps=.5 MiMa, eps= Time t
37 First test case Second test case Computational cost L1-norm of g, ε = 1 4 (left) and ε = 1 7 (right): 1 eps=1-4 1 eps= g(t) L1 1e-6 g(t) L1 1e-6 1e-8 1e-8 1e-1 1e-1 1e-12 1e Time t Time t
38 First test case Second test case Computational cost Computational cost Landau damping, ε =.1: MiMa N p = s. PIC-BGK N p = s log E(t) L PIC BGK, Npart=1*1 6 MiMa, Npart=1* Time t
39 Conclusions First test case Second test case Computational cost Fluid limit recovered when ε : AP scheme. Projection step to numerically force the moments of g to zero. g when ε = few particles are sufficient at the limit, whereas grid methods have a constant cost, whatever the value of ε. Noise due to PIC method reduced (because only on g) = at equivalent results, fewer particles are necessary = calculation time is reduced. Total energy preserved (about.1% ε). And perspectives: Other collision operators. 2D (4D in phase space).
40 First test case Second test case Computational cost References - M. Benoune, M. Lemou, L. Mieussens: Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys. 227, pp (28). - S. Brunner, E. Valeo, J.A. Krommes: Collisional delta-f scheme with evolving background for transport time scale simulations, Phys. of Plasmas 12, (1999). - N. Crouseilles, M. Lemou: An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, KRM 4, pp (211). - P. Degond, G. Dimarco, L. Pareschi: The moment guided Monte Carlo method, International Journal for Numerical Methods in Fluids 67, pp (211). - S. Jin: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, J. Sci. Comput. 21, pp (1999). - M. Lemou, L. Mieussens: A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, J. Sci. Comp. 31, pp (28). Thank you for your attention!
Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles
Kinetic/Fluid micro-macro numerical scheme for Vlasov-Poisson-BGK equation using particles Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3. The 8th International Conference on Computational
More informationAn asymptotic-preserving micro-macro scheme for Vlasov-BGK-like equations in the diffusion scaling
An asymptotic-preserving micro-macro scheme for Vlasov-BGK-like equations in the diffusion scaling Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3 Saint-Malo 13 December 2016 1 Université
More informationUne décomposition micro-macro particulaire pour des équations de type Boltzmann-BGK en régime de diffusion
Une décomposition micro-macro particulaire pour des équations de type Boltzmann-BGK en régime de diffusion Anaïs Crestetto 1, Nicolas Crouseilles 2 et Mohammed Lemou 3 La Tremblade, Congrès SMAI 2017 5
More informationModèles hybrides préservant l asymptotique pour les plasmas
Modèles hybrides préservant l asymptotique pour les plasmas Anaïs Crestetto 1, Nicolas Crouseilles 2, Fabrice Deluzet 1, Mohammed Lemou 3, Jacek Narski 1 et Claudia Negulescu 1. Groupe de Travail Méthodes
More informationUne décomposition micro-macro particulaire pour des équations de type Boltzmann-BGK en régime de diffusion
Une décomposition micro-macro particulaire pour des équations de type Boltzmann-BGK en régime de diffusion Anaïs Crestetto 1, Nicolas Crouseilles 2 et Mohammed Lemou 3 Rennes, 14ème Journée de l équipe
More informationMicro-macro methods for Boltzmann-BGK-like equations in the diffusion scaling
Micro-macro methods for Boltzmann-BGK-like equations in the diffusion scaling Anaïs Crestetto 1, Nicolas Crouseilles 2, Giacomo Dimarco 3 et Mohammed Lemou 4 Saint-Malo, 14 décembre 2017 1 Université de
More informationKinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles
Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou To cite this version: Anaïs Crestetto, Nicolas Crouseilles,
More informationAsymptotic-Preserving scheme based on a Finite Volume/Particle-In-Cell coupling for Boltzmann- BGK-like equations in the diffusion scaling
Asymptotic-Preserving scheme based on a Finite Volume/Particle-In-Cell coupling for Boltzmann- BGK-like equations in the diffusion scaling Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou To cite this
More informationHybrid and Moment Guided Monte Carlo Methods for Kinetic Equations
Hybrid and Moment Guided Monte Carlo Methods for Kinetic Equations Giacomo Dimarco Institut des Mathématiques de Toulouse Université de Toulouse France http://perso.math.univ-toulouse.fr/dimarco giacomo.dimarco@math.univ-toulouse.fr
More informationMonte Carlo methods for kinetic equations
Monte Carlo methods for kinetic equations Lecture 4: Hybrid methods and variance reduction Lorenzo Pareschi Department of Mathematics & CMCS University of Ferrara Italy http://utenti.unife.it/lorenzo.pareschi/
More informationNumerical methods for kinetic equations
Numerical methods for kinetic equations Lecture 6: fluid-kinetic coupling and hybrid methods Lorenzo Pareschi Department of Mathematics and Computer Science University of Ferrara, Italy http://www.lorenzopareschi.com
More informationFluid Equations for Rarefied Gases
1 Fluid Equations for Rarefied Gases Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 21 May 2001 with E. A. Spiegel
More informationKinetic relaxation models for reacting gas mixtures
Kinetic relaxation models for reacting gas mixtures M. Groppi Department of Mathematics and Computer Science University of Parma - ITALY Main collaborators: Giampiero Spiga, Giuseppe Stracquadanio, Univ.
More informationarxiv: v1 [math.na] 7 Nov 2018
A NUMERICAL METHOD FOR COUPLING THE BGK MODEL AND EULER EQUATION THROUGH THE LINEARIZED KNUDSEN LAYER HONGXU CHEN, QIN LI, AND JIANFENG LU arxiv:8.34v [math.na] 7 Nov 8 Abstract. The Bhatnagar-Gross-Krook
More informationExponential methods for kinetic equations
Exponential methods for kinetic equations Lorenzo Pareschi Department of Mathematics & CMCS University of Ferrara, Italy http://utenti.unife.it/lorenzo.pareschi/ lorenzo.pareschi@unife.it Joint research
More informationMicro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations
Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations Irene M. Gamba, Shi Jin, and Liu Liu Abstract In this
More informationDissertation. presented to obtain the. Université Paul Sabatier Toulouse 3. Mention: Applied Mathematics. Luc MIEUSSENS
Dissertation presented to obtain the HABILITATION À DIRIGER DES RECHERCHES Université Paul Sabatier Toulouse 3 Mention: Applied Mathematics by Luc MIEUSSENS Contributions to the numerical simulation in
More informationFluid Equations for Rarefied Gases
1 Fluid Equations for Rarefied Gases Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 23 March 2001 with E. A. Spiegel
More informationStochastic Particle Methods for Rarefied Gases
CCES Seminar WS 2/3 Stochastic Particle Methods for Rarefied Gases Julian Köllermeier RWTH Aachen University Supervisor: Prof. Dr. Manuel Torrilhon Center for Computational Engineering Science Mathematics
More informationanalysis for transport equations and applications
Multi-scale analysis for transport equations and applications Mihaï BOSTAN, Aurélie FINOT University of Aix-Marseille, FRANCE mihai.bostan@univ-amu.fr Numerical methods for kinetic equations Strasbourg
More informationHigh Order Semi-Lagrangian WENO scheme for Vlasov Equations
High Order WENO scheme for Equations Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Andrew Christlieb Supported by AFOSR. Computational Mathematics Seminar, UC Boulder
More informationUncertainty Quantification for multiscale kinetic equations with random inputs. Shi Jin. University of Wisconsin-Madison, USA
Uncertainty Quantification for multiscale kinetic equations with random inputs Shi Jin University of Wisconsin-Madison, USA Where do kinetic equations sit in physics Kinetic equations with applications
More informationLecture 4: The particle equations (1)
Lecture 4: The particle equations (1) Presenter: Mark Eric Dieckmann Department of Science and Technology (ITN), Linköping University, Sweden July 17, 2014 Overview We have previously discussed the leapfrog
More informationA particle-in-cell method with adaptive phase-space remapping for kinetic plasmas
A particle-in-cell method with adaptive phase-space remapping for kinetic plasmas Bei Wang 1 Greg Miller 2 Phil Colella 3 1 Princeton Institute of Computational Science and Engineering Princeton University
More informationControlling numerical dissipation and time stepping in some multi-scale kinetic/fluid simulations
Controlling numerical dissipation and time stepping in some multi-scale kinetic/fluid simulations Jian-Guo Liu Department of Physics and Department of Mathematics, Duke University Collaborators: Pierre
More informationUncertainty Quantification and hypocoercivity based sensitivity analysis for multiscale kinetic equations with random inputs.
Uncertainty Quantification and hypocoercivity based sensitivity analysis for multiscale kinetic equations with random inputs Shi Jin University of Wisconsin-Madison, USA Shanghai Jiao Tong University,
More informationUncertainty Quantification for multiscale kinetic equations with high dimensional random inputs with sparse grids
Uncertainty Quantification for multiscale kinetic equations with high dimensional random inputs with sparse grids Shi Jin University of Wisconsin-Madison, USA Kinetic equations Different Q Boltmann Landau
More informationA hybrid kinetic-fluid model for solving the gas dynamics Boltzmann-BGK equation.
A hybrid kinetic-fluid model for solving the gas dynamics Boltzmann-BGK equation. N. Crouseilles a, b, P. Degond a, and M. Lemou a a MIP, UMR CNRS 564, UFR MIG, Université Paul Sabatier 8, route de Narbonne,
More informationAn improved unified gas-kinetic scheme and the study of shock structures
IMA Journal of Applied Mathematics (2011) 76, 698 711 doi:10.1093/imamat/hxr002 Advance Access publication on March 16, 2011 An improved unified gas-kinetic scheme and the study of shock structures KUN
More informationImplicit kinetic relaxation schemes. Application to the plasma physic
Implicit kinetic relaxation schemes. Application to the plasma physic D. Coulette 5, E. Franck 12, P. Helluy 12, C. Courtes 2, L. Navoret 2, L. Mendoza 2, F. Drui 2 ABPDE II, Lille, August 2018 1 Inria
More informationExponential Runge-Kutta for inhomogeneous Boltzmann equations with high order of accuracy
Exponential Runge-Kutta for inhomogeneous Boltzmann equations with high order of accuracy Qin Li, Lorenzo Pareschi Abstract We consider the development of exponential methods for the robust time discretization
More informationA hybrid method for hydrodynamic-kinetic flow - Part II - Coupling of hydrodynamic and kinetic models
A hybrid method for hydrodynamic-kinetic flow - Part II - Coupling of hydrodynamic and kinetic models Alessandro Alaia, Gabriella Puppo May 31, 2011 Abstract In this work we present a non stationary domain
More informationarxiv: v1 [math.na] 25 Oct 2018
Multi-scale control variate methods for uncertainty quantification in kinetic equations arxiv:80.0844v [math.na] 25 Oct 208 Giacomo Dimarco and Lorenzo Pareschi October 26, 208 Abstract Kinetic equations
More informationAsymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality
Asymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality Pierre Degond 1, Fabrice Deluzet 2, Laurent Navoret 3, An-Bang Sun 4, Marie-Hélène Vignal 5 1 Université
More informationMicroscopically Implicit-Macroscopically Explicit schemes for the BGK equation
Microscopically Implicit-Macroscopically Explicit schemes for the BGK equation Sandra Pieraccini, Gabriella Puppo July 25, 2 Abstract In this work a new class of numerical methods for the BGK model of
More informationAnomalous transport of particles in Plasma physics
Anomalous transport of particles in Plasma physics L. Cesbron a, A. Mellet b,1, K. Trivisa b, a École Normale Supérieure de Cachan Campus de Ker Lann 35170 Bruz rance. b Department of Mathematics, University
More informationAdaptive semi-lagrangian schemes for transport
for transport (how to predict accurate grids?) Martin Campos Pinto CNRS & University of Strasbourg, France joint work Albert Cohen (Paris 6), Michel Mehrenberger and Eric Sonnendrücker (Strasbourg) MAMCDP
More informationMoments conservation in adaptive Vlasov solver
12 Moments conservation in adaptive Vlasov solver M. Gutnic a,c, M. Haefele b,c and E. Sonnendrücker a,c a IRMA, Université Louis Pasteur, Strasbourg, France. b LSIIT, Université Louis Pasteur, Strasbourg,
More informationSemi-Lagrangian Formulations for Linear Advection Equations and Applications to Kinetic Equations
Semi-Lagrangian Formulations for Linear Advection and Applications to Kinetic Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Chi-Wang Shu Supported by NSF and AFOSR.
More informationAccurate representation of velocity space using truncated Hermite expansions.
Accurate representation of velocity space using truncated Hermite expansions. Joseph Parker Oxford Centre for Collaborative Applied Mathematics Mathematical Institute, University of Oxford Wolfgang Pauli
More informationModelling and numerical methods for the diffusion of impurities in a gas
INERNAIONAL JOURNAL FOR NUMERICAL MEHODS IN FLUIDS Int. J. Numer. Meth. Fluids 6; : 6 [Version: /9/8 v.] Modelling and numerical methods for the diffusion of impurities in a gas E. Ferrari, L. Pareschi
More informationThe Moment Guided Monte Carlo Method
The Moment Guided Monte Carlo Method Pierre Degond,, Giacomo Dimarco,,3 and Lorenzo Pareschi Université de Toulouse; UPS, INSA, UT, UTM ; Institut de Mathématiques de Toulouse ; F-3 Toulouse, France. CNRS;
More informationOn a class of implicit-explicit Runge-Kutta schemes for stiff kinetic equations preserving the Navier-Stokes limit
On a class of implicit-eplicit Runge-Kutta schemes for stiff kinetic equations preserving the Navier-Stokes limit Jingwei Hu Xiangiong Zhang June 8, 17 Abstract Implicit-eplicit (IMEX) Runge-Kutta (RK)
More informationCORBIS: Code Raréfié Bidimensionnel Implicite Stationnaire
CORBIS: Code Raréfié Bidimensionnel Implicite Stationnaire main ingredients: [LM (M3AS 00, JCP 00)] plane flow: D BGK Model conservative and entropic velocity discretization space discretization: finite
More informationASYMPTOTIC PRESERVING (AP) SCHEMES FOR MULTISCALE KINETIC AND HYPERBOLIC EQUATIONS: A REVIEW
ASYMPTOTIC PRESERVING (AP) SCHEMES FOR MULTISCALE KINETIC AND HYPERBOLIC EQUATIONS: A REVIEW SHI JIN Contents 1. Introduction 1 2. Hyperbolic systems with stiff relaxations 3 3. Kinetic equations: the
More informationOn the Boltzmann equation: global solutions in one spatial dimension
On the Boltzmann equation: global solutions in one spatial dimension Department of Mathematics & Statistics Colloque de mathématiques de Montréal Centre de Recherches Mathématiques November 11, 2005 Collaborators
More informationAn Asymptotic-Preserving Monte Carlo Method for the Boltzmann Equation
An Asymptotic-Preserving Monte Carlo Method for the Boltzmann Equation Wei Ren a, Hong Liu a,, Shi Jin b,c a J C Wu Center for Aerodynamics, School of Aeronautics and Aerospace, Shanghai Jiao Tong University,
More informationBoltzmann Equation and Hydrodynamics beyond Navier-Stokes
Boltzmann Equation and Hydrodynamics beyond Navier-Stokes Alexander Bobylev Keldysh Institute for Applied Mathematics, RAS, Moscow Chapman-Enskog method and Burnett equations Notation: f (x, v, t) - distribution
More informationHypocoercivity and Sensitivity Analysis in Kinetic Equations and Uncertainty Quantification October 2 nd 5 th
Hypocoercivity and Sensitivity Analysis in Kinetic Equations and Uncertainty Quantification October 2 nd 5 th Department of Mathematics, University of Wisconsin Madison Venue: van Vleck Hall 911 Monday,
More informationUne approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck
Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck Jean Dolbeault dolbeaul@ceremade.dauphine.fr CEREMADE CNRS & Université Paris-Dauphine http://www.ceremade.dauphine.fr/ dolbeaul (EN
More informationFluid-Particles Interaction Models Asymptotics, Theory and Numerics I
Fluid-Particles Interaction Models Asymptotics, Theory and Numerics I J. A. Carrillo collaborators: T. Goudon (Lille), P. Lafitte (Lille) and F. Vecil (UAB) (CPDE 2005), (JCP, 2008), (JSC, 2008) ICREA
More informationHypocoercivity for kinetic equations with linear relaxation terms
Hypocoercivity for kinetic equations with linear relaxation terms Jean Dolbeault dolbeaul@ceremade.dauphine.fr CEREMADE CNRS & Université Paris-Dauphine http://www.ceremade.dauphine.fr/ dolbeaul (A JOINT
More informationA successive penalty based Asymptotic-Preserving scheme for kinetic equations
A successive penalty based Asymptotic-Preserving scheme for kinetic equations Bokai Yan Shi Jin September 30, 202 Abstract We propose an asymptotic-preserving AP) scheme for kinetic equations that is efficient
More informationMulti-water-bag model and method of moments for Vlasov
and method of moments for the Vlasov equation 1, Philippe Helluy 2, Nicolas Besse 3 FVCA 2011, Praha, June 6. 1 INRIA Nancy - Grand Est & University of Strasbourg - IRMA crestetto@math.unistra.fr (PhD
More informationKinetic Models and Gas-Kinetic Schemes with Rotational Degrees of Freedom for Hybrid Continuum/Kinetic Boltzmann Methods
Kinetic Models and Gas-Kinetic Schemes with Rotational Degrees of Freedom for Hybrid Continuum/Kinetic Boltzmann Methods Simone Colonia, René Steijl and George N. Barakos CFD Laboratory - School of Engineering
More informationBoundary Value Problems and Multiscale Coupling Methods for Kinetic Equations SCHEDULE
Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations April 21-24, 2016 Department of Mathematics University of Wisconsin-Madison SCHEDULE Thursday, April 21 Friday, April 22 Saturday,
More informationA CLASS OF ASYMPTOTIC PRESERVING SCHEMES FOR KINETIC EQUATIONS AND RELATED PROBLEMS WITH STIFF SOURCES
A CLASS OF ASYMPTOTIC PRESERVING SCHEMES FOR KINETIC EQUATIONS AND RELATED PROBLEMS WITH STIFF SOURCES FRANCIS FILBET AND SHI JIN Abstract. In this paper, we propose a general framework to design asymptotic
More informationA Unified Gas-kinetic Scheme for Continuum and Rarefied Flows
A Unified Gas-kinetic Scheme for Continuum and Rarefied Flows Kun Xu a,, Juan-Chen Huang b a Mathematics Department Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong b
More informationA class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation
A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation Shi Jin Bokai Yan October 2, 200 Abstract We present a class of asymptotic-preserving AP) schemes for the nonhomogeneous Fokker-Planck-Landau
More informationFrom Boltzmann Equations to Gas Dynamics: From DiPerna-Lions to Leray
From Boltzmann Equations to Gas Dynamics: From DiPerna-Lions to Leray C. David Levermore Department of Mathematics and Institute for Physical Science and Technology University of Maryland, College Park
More informationNumerical methods for plasma physics in collisional regimes
J. Plasma Physics (15), vol. 81, 358116 c Cambridge University Press 1 doi:1.117/s3778176 1 Numerical methods for plasma physics in collisional regimes G. Dimarco 1,Q.Li,L.Pareschi 1 and B. Yan 3 1 Department
More informationWhat place for mathematicians in plasma physics
What place for mathematicians in plasma physics Eric Sonnendrücker IRMA Université Louis Pasteur, Strasbourg projet CALVI INRIA Nancy Grand Est 15-19 September 2008 Eric Sonnendrücker (U. Strasbourg) Math
More informationUniformly accurate averaging numerical schemes for oscillatory evolution equations
Uniformly accurate averaging numerical schemes for oscillatory evolution equations Philippe Chartier University of Rennes, INRIA Joint work with M. Lemou (University of Rennes-CNRS), F. Méhats (University
More informationQuasi-periodic solutions of the 2D Euler equation
Quasi-periodic solutions of the 2D Euler equation Nicolas Crouseilles, Erwan Faou To cite this version: Nicolas Crouseilles, Erwan Faou. Quasi-periodic solutions of the 2D Euler equation. Asymptotic Analysis,
More informationModels of collective displacements: from microscopic to macroscopic description
Models of collective displacements: from microscopic to macroscopic description Sébastien Motsch CSCAMM, University of Maryland joint work with : P. Degond, L. Navoret (IMT, Toulouse) SIAM Analysis of
More informationLecture 5: Kinetic theory of fluids
Lecture 5: Kinetic theory of fluids September 21, 2015 1 Goal 2 From atoms to probabilities Fluid dynamics descrines fluids as continnum media (fields); however under conditions of strong inhomogeneities
More informationLandau Damping Simulation Models
Landau Damping Simulation Models Hua-sheng XIE (u) huashengxie@gmail.com) Department of Physics, Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, P.R.China Oct. 9, 2013
More informationFinite volumes schemes preserving the low Mach number limit for the Euler system
Finite volumes schemes preserving the low Mach number limit for the Euler system M.-H. Vignal Low Velocity Flows, Paris, Nov. 205 Giacomo Dimarco, Univ. de Ferrara, Italie Raphael Loubere, IMT, CNRS, France
More informationBerk-Breizman and diocotron instability testcases
Berk-Breizman and diocotron instability testcases M. Mehrenberger, N. Crouseilles, V. Grandgirard, S. Hirstoaga, E. Madaule, J. Petri, E. Sonnendrücker Université de Strasbourg, IRMA (France); INRIA Grand
More informationMichel Mehrenberger 1 & Eric Sonnendrücker 2 ECCOMAS 2016
for Vlasov type for Vlasov type 1 2 ECCOMAS 2016 In collaboration with : Bedros Afeyan, Aurore Back, Fernando Casas, Nicolas Crouseilles, Adila Dodhy, Erwan Faou, Yaman Güclü, Adnane Hamiaz, Guillaume
More informationA HIERARCHY OF HYBRID NUMERICAL METHODS FOR MULTI-SCALE KINETIC EQUATIONS
A HIERARCHY OF HYBRID NUMERICAL METHODS FOR MULTI-SCALE KINETIC EQUATIONS FRANCIS FILBET AND THOMAS REY Abstract. In this paper, we construct a hierarchy of hybrid numerical methods for multi-scale kinetic
More informationDirect Modeling for Computational Fluid Dynamics
Direct Modeling for Computational Fluid Dynamics Kun Xu February 20, 2013 Computational fluid dynamics (CFD) is new emerging scientific discipline, and targets to simulate fluid motion in different scales.
More informationAnisotropic fluid dynamics. Thomas Schaefer, North Carolina State University
Anisotropic fluid dynamics Thomas Schaefer, North Carolina State University Outline We wish to extract the properties of nearly perfect (low viscosity) fluids from experiments with trapped gases, colliding
More informationOverview of Accelerated Simulation Methods for Plasma Kinetics
Overview of Accelerated Simulation Methods for Plasma Kinetics R.E. Caflisch 1 In collaboration with: J.L. Cambier 2, B.I. Cohen 3, A.M. Dimits 3, L.F. Ricketson 1,4, M.S. Rosin 1,5, B. Yann 1 1 UCLA Math
More informationLattice Bhatnagar Gross Krook model for the Lorenz attractor
Physica D 154 (2001) 43 50 Lattice Bhatnagar Gross Krook model for the Lorenz attractor Guangwu Yan a,b,,liyuan a a LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences,
More informationPHYSICS OF HOT DENSE PLASMAS
Chapter 6 PHYSICS OF HOT DENSE PLASMAS 10 26 10 24 Solar Center Electron density (e/cm 3 ) 10 22 10 20 10 18 10 16 10 14 10 12 High pressure arcs Chromosphere Discharge plasmas Solar interior Nd (nω) laserproduced
More informationAn asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations
An asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations Institute of Applied Physics and Computational Mathematics, Beijing NUS, Singapore, March 2-6, 2015 (joint
More informationBlock-Structured Adaptive Mesh Refinement
Block-Structured Adaptive Mesh Refinement Lecture 2 Incompressible Navier-Stokes Equations Fractional Step Scheme 1-D AMR for classical PDE s hyperbolic elliptic parabolic Accuracy considerations Bell
More informationSummer College on Plasma Physics. 30 July - 24 August, The particle-in-cell simulation method: Concept and limitations
1856-30 2007 Summer College on Plasma Physics 30 July - 24 August, 2007 The particle-in-cell M. E. Dieckmann Institut fuer Theoretische Physik IV, Ruhr-Universitaet, Bochum, Germany The particle-in-cell
More informationA Unified Gas-kinetic Scheme for Continuum and Rarefied Flows
A Unified Gas-kinetic Scheme for Continuum and Rarefied Flows K. Xu and J.C. Huang Mathematics Department, Hong Kong University of Science and Technology, Hong Kong Department of Merchant Marine, National
More informationComparison 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,
More informationPostprint.
http://www.diva-portal.org Postprint This is the accepted version of a chapter published in Domain Decomposition Methods in Science and Engineering XXI. Citation for the original published chapter: Gander,
More informationMACROSCOPIC FLUID MODELS WITH LOCALIZED KINETIC UPSCALING EFFECTS
MACROSCOPIC FLUID MODELS WITH LOCALIZED KINETIC UPSCALING EFFECTS Pierre Degond, Jian-Guo Liu 2, Luc Mieussens Abstract. This paper presents a general methodology to design macroscopic fluid models that
More informationComputer simulation of multiscale problems
Progress in the SSF project CutFEM, Geometry, and Optimal design Computer simulation of multiscale problems Axel Målqvist and Daniel Elfverson University of Gothenburg and Uppsala University Umeå 2015-05-20
More informationConservative semi-lagrangian schemes for Vlasov equations
Conservative semi-lagrangian schemes for Vlasov equations Nicolas Crouseilles Michel Mehrenberger Eric Sonnendrücker October 3, 9 Abstract Conservative methods for the numerical solution of the Vlasov
More informationComputational plasma physics
Computational plasma physics Eric Sonnendrücker Max-Planck-Institut für Plasmaphysik und Zentrum Mathematik, TU München Lecture notes Sommersemester 015 July 13, 015 Contents 1 Introduction 3 Plasma models
More informationAn asymptotic preserving scheme in the drift limit for the Euler-Lorentz system. Stéphane Brull, Pierre Degond, Fabrice Deluzet, Marie-Hélène Vignal
1 An asymptotic preserving scheme in the drift limit for the Euler-Lorentz system. Stéphane Brull, Pierre Degond, Fabrice Deluzet, Marie-Hélène Vignal IMT: Institut de Mathématiques de Toulouse 1. Introduction.
More informationLattice Boltzmann Method for Moving Boundaries
Lattice Boltzmann Method for Moving Boundaries Hans Groot March 18, 2009 Outline 1 Introduction 2 Moving Boundary Conditions 3 Cylinder in Transient Couette Flow 4 Collision-Advection Process for Moving
More informationNumerical 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
More informationPalindromic Discontinuous Galerkin Method
Palindromic Discontinuous Galerkin Method David Coulette, Emmanuel Franck, Philippe Helluy, Michel Mehrenberger, Laurent Navoret To cite this version: David Coulette, Emmanuel Franck, Philippe Helluy,
More informationUn schéma volumes finis well-balanced pour un modèle hyperbolique de chimiotactisme
Un schéma volumes finis well-balanced pour un modèle hyperbolique de chimiotactisme Christophe Berthon, Anaïs Crestetto et Françoise Foucher LMJL, Université de Nantes ANR GEONUM Séminaire de l équipe
More informationWeighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods
Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu School of Mathematical Science Xiamen University jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu
More informationResearch Article On a Quasi-Neutral Approximation to the Incompressible Euler Equations
Applied Mathematics Volume 2012, Article ID 957185, 8 pages doi:10.1155/2012/957185 Research Article On a Quasi-Neutral Approximation to the Incompressible Euler Equations Jianwei Yang and Zhitao Zhuang
More informationHybrid Kinetic-MHD simulations with NIMROD
simulations with NIMROD Charlson C. Kim and the NIMROD Team Plasma Science and Innovation Center University of Washington CSCADS Workshop July 18, 2011 NIMROD C.R. Sovinec, JCP, 195, 2004 parallel, 3-D,
More informationRecent Innovative Numerical Methods
Recent Innovative Numerical Methods Russel Caflisch IPAM Mathematics Department, UCLA 1 Overview Multi Level Monte Carlo (MLMC) Hybrid Direct Simulation Monte Carlo (DSMC) Accelerated Molecular Dynamics
More informationKinetic Solvers with Adaptive Mesh in Phase Space for Low- Temperature Plasmas
Kinetic Solvers with Adaptive Mesh in Phase Space for Low- Temperature Plasmas Vladimir Kolobov, a,b,1 Robert Arslanbekov a and Dmitry Levko a a CFD Research Corporation, Huntsville, AL 35806, USA b The
More informationKinetic damping in gyro-kinetic simulation and the role in multi-scale turbulence
2013 US-Japan JIFT workshop on New Aspects of Plasmas Kinetic Simulation NIFS, November 22-23, 2013 Kinetic damping in gyro-kinetic simulation and the role in multi-scale turbulence cf. Revisit for Landau
More informationLattice Boltzmann Method
Chapter 5 Lattice Boltzmann Method 5.1 Pseudo-kinetic computational methods Understanding the flow properties of complex fluids in complex topologies is of importance to technological applications and
More informationA Discontinuous Galerkin Method for Vlasov Systems
A Discontinuous Galerkin Method for Vlasov Systems P. J. Morrison Department of Physics and Institute for Fusion Studies The University of Texas at Austin morrison@physics.utexas.edu http://www.ph.utexas.edu/
More informationCHAPTER 9. Microscopic Approach: from Boltzmann to Navier-Stokes. In the previous chapter we derived the closed Boltzmann equation:
CHAPTER 9 Microscopic Approach: from Boltzmann to Navier-Stokes In the previous chapter we derived the closed Boltzmann equation: df dt = f +{f,h} = I[f] where I[f] is the collision integral, and we have
More information