The Lattice Boltzmann method for hyperbolic systems. Benjamin Graille. October 19, 2016

Transcription

1 The Lattice Boltzmann method for hyperbolic systems Benjamin Graille October 19, 2016

2 Framework The Lattice Boltzmann method 1 Description of the lattice Boltzmann method Link with the kinetic theory Classical schemes Analysis methods Boundary conditions 2 pylbm (collaboration with Loïc Gouarin) Presentation Examples 3 The vectorial schemes Presentation Numerical tests 4 Conclusion 2 / 36

3 Description of the lattice Boltzmann method The Lattice Boltzmann method 1 Description of the lattice Boltzmann method Link with the kinetic theory Classical schemes Analysis methods Boundary conditions 2 pylbm (collaboration with Loïc Gouarin) 3 The vectorial schemes 4 Conclusion 3 / 36

4 Description of the lattice Boltzmann method Link with the kinetic theory Different scales for modeling Taylor expansion t 0 microscopic models lattice Boltzmann macroscopic models mesoscopic models Chapman-Enskog ɛ 0 particles positions, velocities statistical descriptions mean distribution functions observable quantities density, velocity, temperature 4 / 36

5 Description of the lattice Boltzmann method Link with the kinetic theory From mesoscopic to macroscopic Mesoscopic scale: the Boltzmann equation t f (t, x, c) + c x f (t, x, c) = 1 Q(f ), ɛ where ɛ is the Knudsen number. Chapman-Enskog method: formal asymptotic expansion according to ɛ the macroscopic equations on the moments. ρ = f (t, x, c) dc, (mass) q = cf (t, x, c) dc, (momentum) 1 E = 2 c2 f (t, x, c) dc, (energy) 5 / 36

6 Description of the lattice Boltzmann method Link with the kinetic theory Mimic the kinetic to simulate the macroscopic Example in 2 dimensions a uniform cartesian mesh 6 / 36

7 Description of the lattice Boltzmann method Link with the kinetic theory Mimic the kinetic to simulate the macroscopic Example in 2 dimensions a uniform cartesian mesh on each spot, particles with adapted discret velocities 6 / 36

8 Description of the lattice Boltzmann method Link with the kinetic theory Mimic the kinetic to simulate the macroscopic Example in 2 dimensions a uniform cartesian mesh on each spot, particles with adapted discret velocities the transport phase 6 / 36

9 Description of the lattice Boltzmann method Link with the kinetic theory Mimic the kinetic to simulate the macroscopic Example in 2 dimensions a uniform cartesian mesh on each spot, particles with adapted discret velocities the transport phase the relaxation phase 6 / 36

10 Description of the lattice Boltzmann method Link with the kinetic theory Mimic the kinetic to simulate the macroscopic Example in 2 dimensions a uniform cartesian mesh on each spot, particles with adapted discret velocities the transport phase the relaxation phase do it again! 6 / 36

11 Description of the lattice Boltzmann method Link with the kinetic theory Sketch of the collision phase conserved moments The collision operator is a linear relaxation of the vector f toward an equilibrium value f eq f = f + M 1 SM (f eq f )). s > 1 f f s < 1 f eq The value of f eq is function of the conserved moments. odd non conserved moments even non conserved moments 7 / 36

12 Description of the lattice Boltzmann method Link with the kinetic theory Sketch of the collision phase conserved moments The collision operator is a linear relaxation of the vector f toward an equilibrium value f eq f = f + M 1 SM (f eq f )). s > 1 f f s < 1 f eq The value of f eq is function of the conserved moments. odd non conserved moments even non conserved moments 7 / 36

13 Description of the lattice Boltzmann method Link with the kinetic theory Define a lattice Boltzmann scheme A lattice Boltzmann scheme is given by a set of q velocities adapted to the mesh, c 0,..., c q 1, an invertible matrix M that transforms the densities into the moments: q 1 q 1 m k = M kj f j = P k (c j )f j, P k R[X ], j =0 j =0 functions defining the equilibrium m eq k, relaxation parameters s k. The nth first moments are necessary the unknowns of the PDEs: these moments are conserved, that is m eq k = m k, 0 k n 1. The next moments are not conserved and their equilibrium value depends on the conserved moments: m eq k = m eq k (m 0,..., m n 1 ), n k q 1. 8 / 36

14 Description of the lattice Boltzmann method Classical schemes Examples D 1 Q 2 advection, heat D 1 Q 3 advection, wave, heat D 1 Q 5 Euler D 2 Q 4 advection, heat D 2 Q 5 advection, wave, heat D 2 Q 9 Navier-Stokes / 36

15 Description of the lattice Boltzmann method Analysis methods Example of the D 1 Q 2 The D 1 Q 2 can be used to simulate the mono-dimensional hyperbolic conservative equation t u(t, x) + x ϕ(u)(t, x) = 0, t > 0, x R, where u : R R is the unknown. The D 1 Q 2 is given by two velocities { λ, λ} with λ = x/ t and the associated densities (f, f + ) the matrix M and it s inverse ( ) 1 1 M = λ λ M 1 = ( 1/2 ) 1/(2λ) 1/2 1/(2λ) the conserved moment u and the non conserved moment v, where u = f + f + and v = λf + λf +. the equilibrium value v eq = v eq (u) and the relaxation parameter s. 10 / 36

16 Description of the lattice Boltzmann method Analysis methods One time step of the linear D 1 Q 2 In the linear case (v eq = αu), one time step of the scheme reads v (x, t) = (1 s)v(x, t) + sαu(x, t), f (x, t + t) = f (x + x, t), f + (x, t + t) = f + In terms of densities, it reads relaxation, transport to the left, (x x, t), transport to the right. f (x, t + t) = 1 (2 s sα )f 2 λ (x + x, t) + 1 (s sα )f 2 λ + (x + x, t), f + (x, t + t) = 1 (s + sα )f 2 λ (x + x, t). + 1 (2 s + sα )f 2 λ + (x + x, t) In terms of moments, it reads u n+1 j v n+1 j = 1 (1 s α 2 λ )un j (1 + s α 2 λ )un j 1 1 s 2λ (vn j +1 v n j 1), = 1 s 2 (vn j +1 + v n j 1) λ (1 s α 2 λ )un j +1 + λ (1 + s α 2 λ )un j / 36

17 Description of the lattice Boltzmann method Analysis methods Equivalent equations (0) Assuming that the densities are regular functions, the Taylor expansion method for small t and x yields Zeroth order At the order 0, the distribution are at equilibrium f j = f eq j + O( t ), f j = f eq j + O( t ), j {, +}. We have f j (t + t, x) = f j (t, x v j t), v j = j λ, j {, +}, f j + O( t ) = f j + O( t ), v = v + O( t ) = (1 s)v + sv eq + O( t ), v = v eq + O( t ) and v = v eq + O( t ). 12 / 36

18 Description of the lattice Boltzmann method Analysis methods Equivalent equations (1) First order The first moment u satisfies the partial differential equation t u + x v eq = O( t ). The choice v eq = ϕ(u) is then done so that u satisfies the conservative equation at order 1. Transition lemma The second moment v satisfies v = v eq t s θ + O( t2 ), v = v eq + t ( 1 1 s ) θ + O( t 2 ), with θ = t v eq + λ 2 x u. We have f j (t + t, x) = f j (t, x v j t), f j + t t f j = f j v j t x f j + O( t 2 ), f j + t t f eq j Taking the first moment: = f j v j t x f eq j + O( t 2 ). u + t t u = u t x v eq + O( t 2 ). Taking the second moment: v + t t v eq = v λ 2 t x u + O( t 2 ) = (1 s)v + sv eq λ 2 t x u + O( t 2 ) 13 / 36

19 Description of the lattice Boltzmann method Analysis methods Equivalent equations (2) Second order The first moment u satisfies the second-order partial differential equation t u + x ϕ(u) = t σ x [( λ 2 ϕ (u) 2) x u ] + O( t 2 ), with σ = 1/s 1/2. Taking the first moment at second order: u + t t u + 1 t 2 2 tt u = u t x v λ2 t 2 xx u + O( t 3 ) t u + x ϕ(u) = x (v eq v ) t(λ2 xx u tt u) + O( t 2 ) v eq v = t( 1 1)θ θ = s ϕ (u) t u + λ 2 xx u = (λ 2 ϕ (u) 2 ) xx u 14 / 36

20 Description of the lattice Boltzmann method Analysis methods Maximum principle Theorem We assume that s [0, 1], u 0 j [α, β], v 0 j = ϕ(u 0 j ), j, and λ max α u β ϕ (u). Then n 0 j u n j [α, β]. The functions h ± (u) = λu±ϕ(u) 2λ f 0 ±,j are increasing. We then have [h ± (α), h ± (β)]. The relaxation phase reads as a linear convex combination: And f n ±,j = λun j ± v n j = λun j ± v n j ± vn j v n j = f n ±,j 2λ 2λ 2λ = (1 s)f n ±,j + sh ± (u n j ) [h ± (α), h ± (β)]. u n j = f n,j + f n +,j ± s ϕ(un j ) v n j 2λ [h (α) + h + (α), h (β) + h + (β)] = [α, β]. 15 / 36

21 Description of the lattice Boltzmann method Analysis methods L2 stability Theorem We assume that veq = αu, λ > α, and s [0, 2]. Then the D1 Q2 scheme is stable for the L2 -norm. The amplification matrix of the linear D1 Q2 scheme is given by ξ s 1 2s (1 + α ) e i x ξ (1 α )e i x λ 2 λ G( x, ξ) =. i x ξ i x ξ s (1 + α )e 1 2s (1 α ) e 2 λ λ Eigenvalues for α = Eigenvalues for α = real 0.0 part Eigenvalues for α = s 1.0 s 1.0 s rt 0.0 ry pa a 0.5 agin im rt 0.0 ry pa a 0.5 agin im rt 0.0 ry pa a 0.5 agin im real 0.0 part real 0.0 part N 16 / 36

22 Description of the lattice Boltzmann method Boundary conditions Bounce Back and anti Bounce Back The black point has outside neighbors. For each one, we have to specify the distribution functions with incoming velocities. 17 / 36

23 Description of the lattice Boltzmann method Boundary conditions Bounce Back and anti Bounce Back When a fluid particle (discrete distribution function) reaches a boundary node, the particle will scatter back to the fluid along with its incoming direction. For anti Bounce Back condition, the sign is changed. The bound of the domain is then pixelized. 18 / 36

24 Description of the lattice Boltzmann method Boundary conditions Bouzidi type boundary conditions In order to improve the accuracy of the boundary conditions, the interface need to be localized more precisely. x i 1 x i x i+1 1 s s s < 1/2 x i 1 x i x i+1 s 1 s s > 1/2 19 / 36

25 Description of the lattice Boltzmann method Boundary conditions Bouzidi type boundary conditions Case s < 1/2 (explicit interpolation). x x i 1 i 1 x i x i+1 2 s 1 s 1 s s < 1/2 We have Then x i 1 2 = (1 2s)x i 1 + 2sx i. f i+1 = f i = f + = (1 2s)f + i 1 i 1 + 2sf + i / 36

26 Description of the lattice Boltzmann method Boundary conditions Bouzidi type boundary conditions Case s > 1/2 (implicit interpolation). x i 1 x x i i+ 1 x i+1 2 s 1 s 1 s s > 1/2 We have Then x i = 1 2s x i+ 1 2 f i+1 = f i = 1 2s f i s 1 x i 1. 2s + 2s 1 f i 1 = 1 2s 2s f + i + 2s 1 2s f i 21 / 36

27 pylbm (collaboration with Loïc Gouarin) The Lattice Boltzmann method 1 Description of the lattice Boltzmann method 2 pylbm (collaboration with Loïc Gouarin) Presentation Examples 3 The vectorial schemes 4 Conclusion 22 / 36

28 pylbm (collaboration with Loïc Gouarin) Presentation Motivations pylbm academic and flexible code to test and to compare the schemes: simple and brief implementation can be used by non specialists or students optimisations: mpi, openmp, cython (cuda and opencl in test) python: compromise efficiency / simplicity fine management of the geometry built from simple objects The user defines 2D: circle, ellipse, triangle, parallelogram 3D: sphere, ellipsoid, parallelepiped, cylindrer the geometry of the domain and the scheme through a dictionary, initial conditions and boundary conditions through functions, parameters for the optimization (optional), the code is then generated and executed. 23 / 36

29 pylbm (collaboration with Loïc Gouarin) Presentation Structure of the code Circle Dictionnary Ellipse Parallelogram Simulation Triangle Geometry Stencil Velocity Sphere Ellipsoid Domain Scheme Generator Cylinder Parallelepiped Array : f, m Fonctions : transport relaxation m2f f2m Initialization Boundary conditions Result 24 / 36

30 pylbm (collaboration with Loïc Gouarin) Examples D 1 Q 2 for the advection Geometry import pylbm xmin, xmax = 0., 1. # domain Geometry informations spatial dimension: 1 bounds of the box: [[ 0. 1.]] dic = { ' box ' :{ ' x ' : [xmin, xmax ]}, } geom = pylbm. Geometry ( dic ) print ( geom ) geom. visualize () 25 / 36

31 pylbm (collaboration with Loïc Gouarin) Examples D 1 Q 2 for the advection Stencil import pylbm dic = { ' dim ' : 1, ' schemes ' :[ { ' velocities ' : [1, 2], } ], } Stencil informations * spatial dimension: 1 * maximal velocity in each direction: [1] * minimal velocity in each direction: [-1] * Informations for each elementary stencil: stencil 0 - number of velocities: 2 - velocities: (1: 1), (2: -1), sten = pylbm. Stencil ( dic ) print ( sten ) sten. visualize () 26 / 36

32 pylbm (collaboration with Loïc Gouarin) Examples D 1 Q 2 for the advection Domain import pylbm Domain informations spatial dimension: 1 space step: dx= 1.000e-01 xmin, xmax = 0., 1. # domain N = 10 # number of points dic = { ' box ' :{ ' x ' : [xmin, xmax ]}, ' space_step ' : (xmax - xmin )/N, ' scheme_velocity ' : 1., ' schemes ' :[ { ' velocities ' : [1, 2], } ], } dom = pylbm. Domain ( dic ) print ( dom ) dom. visualize () 27 / 36

33 Scheme pylbm (collaboration with Loïc Gouarin) Examples import pylbm import sympy as sp D 1 Q 2 for the advection u, X = sp. symbols ( ' u, X ' ) c =.25 # velocity s = 1. # relaxation parameter dic = { ' dim ' : 1, ' scheme_velocity ' : 1., ' schemes ' :[ { ' velocities ' : [1, 2], ' conserved_moments ' : u, ' polynomials ' : [1, X], ' equilibrium ' : [u, c*u], ' relaxation_parameters ' : [0., s], ' init ' : {u: 0.} } ], } scheme = pylbm. Scheme ( dic ) print ( scheme ) Scheme informations spatial dimension: dim=1 number of schemes: nscheme=1 number of velocities: Stencil.nv[0]=2 velocities value: v[0]=(1: 1), (2: -1), polynomials: P[0]=Matrix([[1], [X]]) equilibria: EQ[0]=Matrix([[u], [0.25*u]]) relaxation parameters: s[0]=[0.0, 1.0] moments matrices M = [Matrix([ [1, 1], [1, -1]])] invm = [Matrix([ [1/2, 1/2], [1/2, -1/2]])] 28 / 36

34 pylbm (collaboration with Loïc Gouarin) Examples D 1 Q 2 for the advection (numerical results) Smooth solution Discontinuous solution k s e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-04 slope 2.000e e e e e e-01 k s e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-02 slope 3.374e e e e e e / 36

35 The vectorial schemes The Lattice Boltzmann method 1 Description of the lattice Boltzmann method 2 pylbm (collaboration with Loïc Gouarin) 3 The vectorial schemes Presentation Numerical tests 4 Conclusion 30 / 36

36 The vectorial schemes Presentation Motivations We focus on the numerical simulation of hyperbolic systems of pde s by the lattice Boltzmann method. In this presentation, we deal with mono dimensional conservation laws: t u(t, x) + x ϕ(u)(t, x) = 0, t > 0, x R, where u is the unknown vector of size n. Goals: generic scheme for all flux functions ϕ, identifiable stability conditions, straightforward treatment of the boundary conditions. 31 / 36

37 The vectorial schemes Presentation The vectorial D 1 Q 2,...,2 For a system of n equations, we duplicate n times the D 1 Q 2 : the velocity of the scheme: λ = x/ t the velocities of the particles: v 0 = λ, v 1 = λ the particles distributions: f 1,0, f 1,1,..., f n,0, f n,1 the moments: u k = f k,0 + f k,1, v k = λ( f k,0 + f k,1 ), 1 k n the relaxation phase: u k = u k, v k = v k + s k (v eq k v k ), 1 k n Theorem (Equivalent equation) Taking v eq k = ϕ k (u), the vector of the first moments u satisfy t u + x ϕ(u) = t S x ((λ 2 I n ( dϕ(u) ) ) ) 2 x u + O( t 2 ), with S = diag(σ 1,..., σ n), σ k = 1/s k 1/2, 1 k n. 32 / 36

38 The vectorial schemes Presentation The D 1 Q 2,...,2 as a relaxation scheme The relaxation system proposed by Jin and Xin reads { t u ɛ + x v ɛ = 0, t v ɛ + A x u ɛ = 1 ɛ (ϕ(uɛ ) v ɛ ). Denoting u n i = u(x i, t n ), v n i = v(x i, t n ), x i L, t n = n t, the D 1 Q 2,...,2 can be rewritten in the form (if all relaxation parameters are equal to s) v n i = v n i s(v n i ϕ(u n i )), u n+1 i Splitting between v n+1 i = 1 2 (un i+1 + u n t i 1) 2 x (vn i+1 vi 1), n = 1 2 (vn i+1 + v n i 1) λ 2 t 2 x (un i+1 u n i 1). the relaxation part treated by the explicit Euler method with ɛ = t/s, the hyperbolic part treated by the Lax-Friedrichs method with A = λ 2 I n. 33 / 36

39 The vectorial schemes Numerical tests Numerical tests done with pylbm dimension 1 advection with constant velocity advection-diffusion-reaction equation Burgers p-system shallow water system full compressible Euler system dimension 2 advection with constant velocity shallow water system incompressible thermo-hydrodynamic (Boussinesq approximation) magneto-hydro-dynamic system dimension 3 advection with constant velocity incompressible thermo-hydrodynamic (Boussinesq approximation) 34 / 36

40 Conclusion The Lattice Boltzmann method 1 Description of the lattice Boltzmann method 2 pylbm (collaboration with Loïc Gouarin) 3 The vectorial schemes 4 Conclusion 35 / 36

41 Conclusion The Lattice Boltzmann method The lattice Boltzmann method and the Boltzmann equation very rough discretization in term of velocities no claim to solve the Boltzmann equation a taylor expansion to mimic the Chapman-Enskog expansion The software pylbm flexible and simple interface to create a simulation performances of a compiled langage some features to add: optimization with grpahics cards, relative velocity schemes (Tony Février), mesh refinement... The vectorial schemes simple way to simulate conservative hyperbolic problems link with the relaxation methods to do: obtain a convergence theorem with this restrictive framework 36 / 36