Spectral-Lagrangian solvers for non-linear non-conservative Boltzmann Transport Equations

1. Spectral-Lagrangian solvers for non-linear non-conservative Boltzmann Transport Equations Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin BIRS, September 2008 In collaboration with: Harsha Tharskabhushanam, ICES, UT Austin, currently P.R.O.S

2. Statistical transport from collisional kinetic models • Rarefied ideal gases-elastic:classical conservativeBoltzmann Transport eq. • Energy dissipative phenomena: Gas of elastic or inelastic interacting systems in the presence of a thermostat with a fixed background temperature өb or Rapid granular flow dynamics: (inelastic hard sphere interactions): homogeneous cooling states, randomly heated states, shear flows, shockwaves past wedges, etc. • (Soft) condensed matter at nano scale: Bose-Einstein condensates models and charge transport in solids: current/voltage transport modeling semiconductor. • Emerging applications from stochastic dynamics for multi-linear Maxwell type interactions : Multiplicatively Interactive Stochastic Processes: • Pareto tails for wealth distribution, non-conservative dynamics: opinion dynamic models, particle swarms in population dynamics, etc (Fujihara, Ohtsuki, Yamamoto’ 06,Toscani, Pareschi, Caceres 05-06…). • Goals: • Understanding of analytical properties: large energy tails • Long time asymptotics and characterization of • asymptotics states • Deterministic numerical approximations – observing ‘purely kinetic phenomena’

3. elastic collision ‘v v inelastic collision C = number of particle in the box a = diameter of the spheres N=space dimension ηthe impact direction η v* ‘v* i.e. enough intersitial space May be extended to multi-linear interactions

4. A general formstatistical transport : The space-homogenous BTE with external heating sources Important examples from mathematical physics and social sciences: The term models external heating sources: • background thermostat (linear collisions), • thermal bath (diffusion) • shear flow (friction), • dynamically scaled long time limits (self-similar solutions). ‘v v inelastic collision η v* ‘v* u’= (1-β) u + β |u| σ , with σthe direction of elastic post-collisional relativevelocity Inelastic Collision

6. Non-Equilibrium Stationary Statistical States Elastic case Inelastic case

7. A new deterministic approach to compute numerical solution for non-linear non-conservative Boltzmann equations: Spectral-Lagrangian constrained solvers (Filbet, Pareschi & Russo) (With H. TharkabhushanamJCP’08) In preparation, 08 In preparation, 08 • Resolution of boundary layers discontinuities • observing ‘purely kinetic phenomena’

11. Collision Integral Algorithm

14. Discrete Conservation operator ‘conserve’ algorithm Stabilization property

15. A good test problem The homogeneous dissipative BTE in Fourier space (CMP’08)

16. tr = reference time = mft Δt= 0.25mft.

19. Bobylev, Cercignani, I.G (CMP’08)

21. Self-similar asymptotics for a for a slowdown process given by elastic BTE with a thermostat A benchmark case:

23. Soft condensed matter phenomena Remark: The numerical algorithm is based on the evolution of the continuous spectrum of the solution as in Greengard-Lin’00 spectral calculation of the free space heat kernel, i.e. self-similar of the heat equation in all space.

24. Testing: BTE with Thermostat explicit solution problem of colored particles Maxwell Molecules model Rescaling of spectral modes exponentially by the continuous spectrum with λ(1)=-2/3

25. Testing: BTE with Thermostat Moments calculations:

26. Space inhomogeneous simulations mean free time := the average time between collisions mean free path := average speed xmft (average distance traveled between collisions)  Set the scaled equation for 1= Kn := mfp/geometry of length scale Spectral-Lagrangian methods in 3D-velocity space and 1D physical space discretization in the simplest setting: Spatial mesh size Δx = O.O1mfp Time step Δt = r mft , mft=reference time N= Number of Fourier modes in each j-direction in 3D

28. Elastic space inhomogeneous problem Shock tube simulations with a wall boundary Example 1: Shock propagation phenomena:Traveling shock with specular reflection boundary conditions at the left wall and a wall shock initial state. Time step:Δt = 0.005 mft, mean free path l = 1, 700 time steps, CPU ≈ 55hs mesh points:phase velocity Nv = 16^3 in [-5,5)^3 - Space: Nx=50 mesh points in 30 mean free paths: Δx=3/5 Total number of operations :O(Nx Nv2 log(Nv)).

29. Example 2 : Shock dissipationphenomena: Jump in wall kinetic temperature with diffusive boundary conditions. Constant moments initial state with a discontinuous pdf at the boundary, with wall kinetic temperature decreased by half its magnitude= `sudden cooling’ Sudden cooling problem K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991 (Lattice Boltzmann on BGK)

30. Resolution of discontinuity ’near the wall’ for diffusive boundary conditions: (K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991) Sudden heating: Constant moments initial state with a discontinuous pdf at the boundary wall, with wall kinetic temperature increased by twiceits magnitude: Boundary Conditions for sudden heating: Calculations in the next four pages: Mean free path l0= 1. Number of Fourier modes N = 243, Spatial mesh size Δx = 0.01 l0. Time step Δt = r mft

31. Comparisons withK.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991 (Lattice Boltzmann on BGK) Jump in pdf Sudden heating problem

32. Formation of a shock wave by an initial sudden change of wall temperature from T0 to 2T0. Sudden heating problem (BGK eq. with lattice Boltzmann solvers) K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991

33. Sudden heating problem K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991

34. Sudden cooling problem K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991

35. The Riemann Problem: 1D-3D hard spheres elastic gas The macroscopic i.c. satisfy the Rankine-Hugoniot Kn=0.01 dx = t0/2 t0 the mean free time For Kn << 0.01 the method becomes too slow -> use hydrodynamic solvers

37. Shannon Sampling theorem

38. Deterministic spectral/Lagrangian Method • The method is designed to capture the distribution behavior for elastic and inelastic collisions. • Conservation is achieved by a constrained Lagrange multiplier technique wherein • the conservation properties are the constraints in the optimization problem. • The resulting scalar objective function is optimized. • Other deterministic methods based on Fourier Series (Pareschi, Russo’01, Filbet’03 Rjasanov and Ibrahimov-02) are only for elastic/conservative interactions and conserves only the density and not higher moments. • Required moments can be conserved →computation of very accurate kinetic energy dissipative problems, independent of micro reversibility properties. • The method produces no oscillatory behavior, even at lower order time discretization: Homogeneous Boltzmann collision Integral is a strong smoothing operator

39. In the works and future plans • Spectral – Lagrangian solvers for non-linear Boltzmann transport eqs. • Space inhomogeneous calculations: temperature gradient induced flows like a • Cylindrical Taylor-Couette flow and the Benard convective problem. • Chemical gas mixture implementation. Correction to hydrodynamics closures • Challenge problems: • The proposed deterministic method does not guarantee the positivity of the pdf. • This problem may be solved by primal-dual interior point method from linear programming algorithms for solutions of discrete inequations, but it may not be worth the effort. • adaptive hybrid – methods: coupling of kinetic/fluid interfaces (use • hydrodynamic limit equations for statistical equilibrium) • Implementation of parallel solvers. Thank you very much for your attention! References ( www.ma.utexas.edu/users/gamba/research and references therein)

40. Recent related work related to the problem: Cercignani'95(inelastic BTE derivation); Bobylev, JSP 97 (elastic,hard spheres in 3 d: propagation of L1-exponential estimates); Bobylev, Carrillo and I.M.G., JSP'00 (inelastic Maxwell type interactions); Bobylev, Cercignani , and with Toscani, JSP '02 &'03(inelastic Maxwell type interactions); Bobylev, I.M.G, V.Panferov, C.Villani, JSP'04, CMP’04(inelastic + heat sources); Mischler and Mouhout, Rodriguez Ricart JSP '06 (inelastic + self-similar hard spheres); Bobylev and I.M.G. JSP'06 (Maxwell type interactions-inelastic/elastic + thermostat), Bobylev, Cercignani and I.M.G arXiv.org,06 (CMP’08);(generalized multi-linear Maxwell type interactions-inelastic/elastic: global energy dissipation) I.M.G, V.Panferov, C.Villani, arXiv.org’07, ARMA’08 (elastic n-dimensional variable hard potentials Grad cut-off:: propagation of L1 and L∞-exponential estimates) C. Mouhot, CMP’06 (elastic, VHP, bounded angular cross section: creation of L1-exponential ) Ricardo Alonso and I.M.G., JMPA’08(Grad cut-off, propagation ofregularity bounds-elastic d-dim VHP) I.M.Gamba and Harsha Tarskabhushanam JCP’08(spectral-lagrangian solvers-computation of singulatities)

41. Comparisons with K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991 t/t0= 0.12 Jump in pdf

42. Plots of v1- marginals at the wall and up to 1.5mfp from the wall

43. (CMP’08)