Spectral-Lagrangian solvers for non-linear Boltzmann type equations: numerics and analysis

1. Spectral-Lagrangian solvers for non-linearBoltzmann type equations: numerics and analysis Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin Austin, Ausust 2008 RTG Workshop In collaboration with: Harsha Tharskabhushanam, ICES, UT Austin; currently at P.R.O.S.

2. Motivations from statistical physics or interactive ‘particle’ systems • Initial Motivation: rarefied ideal gases: 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, 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 • Goals: Search for common features that • characterizes the statistical flow • A unified approach for Maxwell type interactions and • generalizations. • Analytical properties - long time asymptotics and characterization of asymptotics states: high energy tails and singularity formation

3. ‘v v C = number of particle in the box a = diameter of the spheres N=space dimension 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). u’= (1-β) u + β |u| σ , with σthe direction of elastic post-collisional relative velocity Inelastic Collision

6. Review of Properties of the collisional integral and the equation: conservation of moments

7. Time irreversibility is expressed in this inequality stability In addition: The Boltzmann Theorem:there are only N+2 collision invariants

8. →yields the compressible Euler eqs→Small perturbations of Mawellians yield CNS eqs.

9. Hydrodynamic limits: evolution models of a ‘few’ statistical moments (mass, momentum and energy)

10. Exact energy identity for a Maxwell type interaction models Then f(v,t) → δ0ast → ∞ to a singular concentrated measure (unless exits a heat source) Current issues of interest regarding energy dissipation: Can one tell the shape or classify possible stationary states and their asymptotics, such as self-similarity? Non-Gaussian (or Maxwellian) statistics!

11. Reviewing inelastic properties e e e e e

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

13. A new deterministic approach to compute numerical solution for non-linear Boltzmann equation: Spectral-Lagrangian constrained solvers (Filbet, Pareschi & Russo) (With H. TharkabhushanamJCP’08) In preparation, 08 :observing ‘purely kinetic phenomena’ • This scheme is an alternative to the well known stochastically basedDSMC (Discrete Simulation by Monte Carlo) • for particle methods or alternatively called theBird scheme.

16. in σ

17. such that: Given find

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

21. (CMP’08)

24. (CMP’08)

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

28. 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.

29. reference time = mean free time Δt= 0.25 * reference time step.

32. 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

33. mean free time= the average time between collisions mean free path = average speed X mft= average distance travelled between collisions mfp= 1 (mean free path ) Spatial mesh size Δx = r mfp Time step Δt = mean free time With N= Number of Fourier modes in each k-v-direction

35. 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 = mean free time, 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)).

36. Example 2 : Purely kineticphenomena: 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’

37. 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 = 0.15 l0. Time step Δt = mean free time

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

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

42. 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)

43. 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: • adaptive hybrid – methods: coupling of kinetic/fluid interfaces (use • hydrodynamic limit equations for statistical equilibrium) • Implementation of parallel solvers. • inverse problems: determination of transition probabilities (collision kernels from BTE) Thank you very much for your attention! References ( www.ma.utexas.edu/users/gamba/research and references therein)