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 Degond, Jeff Haack, Shi Jin, Luc Mieussens
Outline of talk Two numerical difficulties in multi-scale fluids/kinetic simulations Stiffness in time step due to fast waves In the Eulerian description, numerical dissipation is amplified due to the dynamics in small scale motions such as fast waves and small mean free path. I will focus on three examples All speed scheme for compressible flows (with Jaff Haack and Shi Jin) Effective local kinetic upscaling (with Pierre Degond and Luc Mieussens) Linear kinetic equation in diffusion region (with Luc Mieussens) In connection with neutron transport equation, I will also show some PDE estimates on short path statics (with Pierre Degond)
Asymptotic Preserving (AP) schemes definition: a scheme uniformly stable and accurate w.r.t ε ( t and x independent of ε) consequence: in the fluid regime (ε 1): scheme consistent with the fluid equation with only slight implicit time discretization to ensure stability and gain efficiency (remove fast variables) numerical dissipation is controlled in the leading variables (remove 1/ɛ factor in numerical dissipation, or appear only in the space orthogonal to the leading variables) preserving important conservation laws key: decomposition equations
Euler equation for compressible flows, 1757 Leonhard Euler 1707-1783 ρ t + (ρu) = 0 conservation of mass (ρu) t + (ρu u) + p = f conservation of momentum ρ density, u velocity, f long range force Pre-thermodynamics era (1st and 2nd law of thermal dynamics not known yet), equation of state not well defined When Mach number is not small, conservation of energy becomes significant (Ma 2 = ρ 0U 2 p 0 = 2 kinetic energy γ 1 internal energy ). Scaling invariant under t Tt, x Lx, u Uu, L = UT Universal equation: from system as small as cold Fermi gas ( 100 microns) to galaxy.
Euler equation for compressible flows, 1757 Leonhard Euler 1707-1783 ρ t + (ρu) = 0 conservation of mass (ρu) t + (ρu u) + p = f conservation of momentum ρ density, u velocity, f long range force Pre-thermodynamics era (1st and 2nd law of thermal dynamics not known yet), equation of state not well defined When Mach number is not small, conservation of energy becomes significant (Ma 2 = ρ 0U 2 p 0 = 2 kinetic energy γ 1 internal energy ). Scaling invariant under t Tt, x Lx, u Uu, L = UT Universal equation: from system as small as cold Fermi gas ( 100 microns) to galaxy.
Euler eq for incompressible flows, 1752, 1761 Leonhard Euler 1707-1783 Dimensionless rescaling of compressible Euler eq: ρ t + (ρu) = 0 (ρu) t + (ρu u) + 1 Ma 2 p = f Mach number: Ma = U/c 0, c 0 = p0 ρ 0, ρ 0 and p 0 : dimensional scaling factors for pressure and density. As Ma 0, we eliminate the unimportant acoustic waves asymptotic expansion ρ = ρ 0 + Ma 2 ρ +, p = p 0 + Ma 2 p + ρ 0 ( u t u = 0 + u u) + p = f incompressibility momentum eq, Newton s 2nd law
1827, Navier-Stokes equations, 1845 C.-L. Navier 1785-1836 Sir G.G. Stokes 1819-1903 The dimensionless isentropic compressible Navier-Stokes equations are t ρ + m = 0 t m + (ρu u + 1ɛ ) 2 p(ρ) = (µσ), σ = u + u T 2 d u The incompressible Navier-Stokes equations are ρ( u t + u u) + p = µ u + f in Ω momentum u = 0 in Ω incompressibility The only dimensionless quantity: Reynolds number, Re = ρul/µ.
Hyperbolic decomposition t U + F (U) + QU = 0: Sub-systems U t + F (U) = 0, U t + QU = 0 are hyperbolic. i.e., they are dynamically stable sub-systems. F (U) is nonlinear, but it is non-stiff, any stable hyperbolic solver shall work, there is no excessive numerical dissipation. QU is stiff, but it is a linear system with constant coefficients! High order discretization with implicit non-dissipative scheme is used to stiffness in time stepping and introduce NO numerical dissipation. Automatically under-resolve small-scale details when it has no effect on bulk behavior. Asymptotic preserving: physical asymptotic behavior realized in numerical discretization, enable to do all speed computation.
Semi-implicit Scheme: Un+1 U n t + AU n + BU n+1 = 0 If both schemes U n+1 U n + AU n = 0, t U n+1 U n + BU n+1 = 0 t are stable, then the original scheme is also stable. Scheme U n+1 = (I ta)u n is stable iff I ta 1 + c t Scheme U n+1 = (I + tb) 1 U n is stable iff (I + tb) 1 1 + c t Combined scheme U n+1 = (I + tb) 1 (I ta)u n is stable iff (I + tb) 1 (I ta) 1 + C t
Hyperbolic decomposition t U + x F (U) + Q x U = 0: U = ( ρ m t ρ + α x m + (1 α) x m = 0 (1) ( t m + x ρu 2 + p ɛ (ρ) ) + a(t) ɛ 2 xρ = 0 (2) p(ρ) a(t)ρ p ɛ (ρ) = ɛ 2, a(t) = min p (ρ(x, t)) x ) ( ) ( 0 1 α, F (U) =, Q = Jacobian and Eigenvaues ( F 0 α (U) = p ɛ(ρ) u 2 2u αm ρu 2 + p ɛ (ρ) a(t) ɛ 2 0 ) x ), λ ± = u ± (1 α)u 2 + α p ɛ(ρ) p ɛ(ρ) 0 implies hyperbolicity, p ɛ(ρ) is bounded implies no stiffness.
A stable scheme Denote D u = (u j u j 1 )/h, D + u = (u j+1 u j )/h and ( ) 0 (1 α)d Q = a n D ɛ 2 + 0 (3) F ± = 1 2 ( F (U) ± λ(u) U), { } λ(u) = max u ± (1 α)u 2 + α p(ρ) U n+1 U n + D F + (U n ) + D + F (U n ) + QU n+1 = 0 t This has a natural extension to a second order MUSCL scheme.
A stable scheme U n+1 U n + D F + (U n ) + D + F (U n ) + QU n+1 = 0 t ( ) 0 (1 α)d Q = a n D ɛ 2 + 0 We can reformulate the density equation as (4) ρ n+1 2ρ n + ρ n 1 t 2 (1 α)an ɛ 2 D D + ρ n+1 = Ex(U n, U n 1 ) We can use a Fast Helmholtz Solver! After finding ρ n+1, update m n+1 explicitly.
Asymptotic preserving: discrete incompressible limit The incompressible limit scheme becomes u n+1 u n t α h u n + (1 α) h un+1 = 0 + h (u n u n ) + 1 ρ c + h pn+1 2 = µ h u n Note: The incompressible pressure 1 ρ c p 2 is obtained by p 2 = ρ c p (ρ c ) ρ.
1D Riemann Problem, Mach = 1, α =.25 %,-./012131$"!!%)(!-!!# $"' $ #"' #!!"#!"$!"%!"&!"'!"(!")!"*!"+ # #"' 4012131$"!!%)(!-!!# #!"'!!!"#!"$!"%!"&!"'!"(!")!"*!"+ #
2 D Riemann Problem, Mach = 1, α =.75! =.75 2 1.5 1 0.5 0 100 80 60 40 20 0 0 20 40 60 80 100
Driven Cavity flow, Re=4000, Mach =.1 and 10 4 1 eps =.1, T=20, Re=4000, alpha =.01, beta = 1 #,-./0/#!!& 1/2/0/$!1/3,/0/&!!!1/45-64/0/,-. $ 1/7,84/0/# 0.9!"+ 0.8!"* 0.7!") 0.6!"( 0.5!"' 0.4!"& 0.3!"% 0.2!"$ 0.1!"# 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Top velocity U = 16x 2 (1 x 2 )!!!"#!"$!"%!"&!"'!"(!")!"*!"+ #
Backward Facing Step Flow, Re=100, Mach =.01 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0 1 2 3 4 5 6 0.5 0 1 2 3 4 5 AP Scheme captures correct reattachment point for low Reynolds number
Efficient localized kinetic upscaling [Degond-Liu-Mieussens, MMS 06] the micro-macro model (for Boltzmann): t U + x F (U) + x vmg = 0, t g + (I Π M(U) )(v x g) = Lg + Q(g, g) the localized micro-macro model t U + x F (U) + x vmg K = 0, (I Π M(U) )(v x M(U)) t g K + h(i Π M(U) )(v x g K ) = hlg K + hq(g K, g K ) h(i Π M(U) )(v x M(U)) simple idea: localization of g use the cutoff function h: determine Fluid zones : set h = 0 determine Kinetic zones : set h = 1 define buffer zones between (F) and (K) zones: set 0 h 1
Numerical tests: hydrodynamical scale [Degond-Liu-Mieussens, MMS 06] Sod test (density and h function) for the BGK equation with dynamic localization. 6 x 10 6 Density Sod Test for t=0.018s 1.2 Interface position, Local Knudsen number and equilibrium parameter for t=0.018 s transition function Knudsen x 10 Q density(kg/m 3 ) 5 4 3 1.4 x 10 6 1.3 1.2 1.1 1 0.9 0.8 7.4 7.6 7.8 8 8.2 8.4 h, Kn x 10, Q 1 0.8 0.6 0.4 2 0.2 1 kinetic model macroscopic model micmac 0 20 15 10 5 0 5 10 15 20 x(m) 15 10 5 0 5 10 15 x(m)
Numerical tests: diffusion scale [Degond-Liu-Mieussens, MMS 06] Temperature for the radiative-heat transfer model, static localization. 1 Kinetic!=1 Fluid!=100 0.8 buffer kinetic model fluid model (diff micro-macro mo temperature 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 x
Neutron transport equation in diffusion time scale ε t f + v f = 1 ε Lf εσ af + εs, Lf (v) = s(v, v )(f (v ) f (v))dv S 2 where v S 2 angular direction, σ total cross section, σ a absorption cross section, S internal source
Micro-macro decomposition. slab geometry t f + 1 ε v xf = σ ε 2 Lf σ af + S v [ 1, 1] is the x cosine direction. micro-macro decomposition: f = ρ + εg, ρ = [f ] = 1 1 2 1 f (v) dv, [g] = 0 decomposition of the kinetic equation: t ρ + x [vg] = σ a ρ + S, t g + 1 ε (I [.])(v xg) = σ ε 2 Lg 1 ε 2 v xρ this formulation is well suited to derive the diffusion limit
Discretization and diffusion limit. the scheme: ρ n+1 i ρ n i t g n+1 i+ 2 1 g n i+ 1 2 t 2 + 4v σ i+ 1 = 2 g n+1 i+ 1 2 g n+1 i 1 2 x + 1 (I [.]) ε ε 2 Lg n+1 i+ 1 2 3 5 = σ ai ρ n+1 i + S i, 0 g n @v + i+ 1 2 g n i 1 2 x 1 ε 2 v ρn i+1 ρn i x g n + v i+ 2 3 g n 1 i+ 1 2 A x limit ε = 0: the scheme gives ρ n+1 i ρ n i t κ ρn i+1 2ρn i + ρ n i 1 x 2 = σ ai ρ n+1 i + S i, time explicit scheme with 3-point stencil
Analysis (Mieussens + L) the scheme is uniformly stable: under the CFL constraint: ρ n + ε g n C ( ρ 0 + ε g 0 ) t 2 3 ( 1 x 2 ) 3 2κ + ε x small ε: t 2 x 2 9 2κ (CFL for diffusion) large ε: t 2 3ε x (CFL for convection) the scheme is uniformly accurate: ρ(t n ) ρ n + ε g(t n ) g n C( t + x 2 + ε x)
Mean chord length for convex domain l = [ ] S dv 2 Γ,v n 0 l(s, v)v nds [ ] S dv Ω = 2 Γ ds S 2,v n 0 v ndv Γ dsπ = 4 Ω Γ Uniform and isotropic flux of particles entering the domain Ω across the surface Γ (neutron transport, Dirac, Fuchs, 1943) Integral geometry, geometric probability, Cauchy 1841, Czuber 1884)
Related references on mean chord length A.-L. Cauchy 1789-1857 Paul Dirac 1902-84 Fuchs 1911-88 1841, Cauchy, mean projected area S = Γ /4 1884, Czuber, mean chord length l = Ω / S = 4 Ω / Γ 1943, Dirac, Fuchs etc, Declassified British Report MS-D-5 Atomic bomb: fissile material goes super-critical only depending on the mean chord of the piece of uranium (not mass as normally thought). Probability of reaction = l σ, σ: cross section 1958, A.M. Weinberg, E.P. Wigner, The Physical Theory of Neutron Chain Reactors. 1981, J N Bardsley and A Dubi, SIAM Appl. Math. 2003, S. Blanco and R. Fournier, Europhys. Lett.
orresponding to the first way of reasoning. that corresponding to the first way of r viously, these two images are both meaningful, but the corresponding processesobviously, are these two images are bo aneously under way and compensate each other exactly. The first image concerns simultaneously long under way and compens Heuristic arguments, Blanco-Fournier, 2003 A! << R 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R(m) R! 1 ~ R For small mean free path: B A Fig. 3 B2! 2!1 Monte Carlo simulations. Simulated average trajectory length inside a circle of radius Fig. R 2 for Monte Carlo simulations. Simulat sects with distinct random-walk characteristics: mean free path λ =5 10 3 m (measured two insects with distinct random-walk cha ant species Messor sancta L.); mean free path λ =5 10 2 m (roughly estimated for for the the ant species Messor sancta L.); m frequent direction changes ach species Blattella germanica L.). In both simulations, the asymmetry parameter cockroach of the species Blattella germanica L.) scattering phase function is g = 0.5. single scattering phase function is g = 0.5. Trajectory homothety. Illustration of the quasi-homothety of short trajectories. The Fig. second 3 Trajectory homothety. Illustration ory corresponds to a mean free path half as long as that of the first trajectory (λ 2 = trajectory λ corresponds to a mean free pat 1/2), for same statistical choices, and neglecting circle curvature, leads to a trajectory length which half as for same statistical choices, and negl For any isotropic uniform incidence, the average 2 L long (L 1/2). 2 length L 1/2). of B1 <L> 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R(m) Fig. 2 Once ant entered domain, it travels a longer distance due to On the other hand, most ants have an exit point very close to their entry point trajectories through the domain is independent of the characteristics of the diffusion process
Heuristic arguments Blanco-Fournier 2003, Bardsley-Dubi 1981 Steady neutron transport equation v f = 1 s(v, v )(f (v ) f (v))dv ε S 2 with BC: f = φ 0 for x Γ, v n 0. Constant solution (statistical equilibrium): f (x, v) φ 0. Rate at which reactions occur R = σ f (x, v)dxdv = σφ 0 4π Ω Ω S 2 R = σ l in-flux = σ l φ 0 Γ S 2,v n 0 v ndsdv = σ l φ 0 π Γ l = 4 Ω / Γ for convex domain, any mean free path
A. Mazzolo: Properties of diffusive random walks in bounded dom Heuristic arguments Mazzolo 2004, Blanco-Fournier 2006 1 0.5 0 l n α n τ n+1 for small τ n l r n 1 = l n, n 2-0.5-1 -1-0.5 0 0.5 1 path lengths l of first exit trajectories starting uniformly and isotropically at the boundary Fig. 1 Example of two-dimensional random walks generated with a mean free path a non-convex body (non-concentric hollow disk): solid lines represent random walk domain, while dotted lines represent random walks starting on the surface. path lengths r of first exit trajectories starting uniformly and isotropically within domain Random-walks properties. In the following, we consider constant-speed whose probabilistic law for the independent random jumps has an exponentia diffusion approximation t ρ = τ 3 ρ with BC ρ + 2τ p(l) =1/λ exp[ l/λ]. 3 nρ = 0 and uniform initial data ρ t=0 = ρ 0. where λ is the mean free path. In reactor physics, this corresponds to the monocinetic neutrons in a purely diffusing medium. First, we select a point tributed inside K, a bounded body in R 3 of volume V and surface S. Then allowed to move according to eq. (1) in a direction that is uniformly distrib
Some estimates (Degond + L) Theorem u τ = 1, in Ω, u τ + τ n u τ = 0, on Γ Ω 2 Γ τ + u0 2 u τ Ω 2 Ω Γ τ + u 2 where u 0 and v 0 are solutions of u 0 = 1, in Ω, u 0 = 0, on Γ u = 1, in Ω, n u = Ω / Γ, on Γ, Proposition inf u H 1, R H(u) inf H(u), Γ u=0 u H0 1 H(u) The equality holds if and only if Ω is a sphere. Γ u = 0 ( ) 1 2 u 2 u Ω
Expansion for τ K 0 u τ = Ω Γ τ + u + τ k v k k=1 u = 1, in Ω, n u = Ω / Γ, on Γ, v k = 0, in Ω, n v k = v k 1, on Γ, Γ Γ u = 0 v k = 0 Convergence analysis: v k Γ C (I n n) v k Γ K 0 n v k Γ = K 0 v k 1 Γ v k 2 Ω = Γ (u τ u ) v k n v k K 0 v k 1 2 Γ K0 τ K 0 u Γ
A technique lemma Lemma d dτ Ω u τ = Ω 2 / Γ + n u τ + Ω / Γ (u τ u 0 ) 2 + τ n u τ 2 2 Γ τ n u 0 2 2 Γ 0 u 0 2 + n u τ + Ω / Γ 2 = u 2 Γ 0 2 Γ