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

Goals: Understanding of analytical properties: large energy tailsUnderstanding of analytical properties: large energy tails Long time asymptotics and characterization ofLong time asymptotics and characterization of asymptotics states Deterministic numerical approximations – observing purely kinetic phenomenaDeterministic numerical approximations – observing purely kinetic phenomena Statistical transport from collisional kinetic models Rarefied ideal gases-elastic:classical conservativeBoltzmann Transport eq. Rarefied ideal gases-elastic: classical conservative Boltzmann Transport eq. Energy dissipative phenomena: Gas of elastic or inelastic interacting systems in the presence of a thermostat with a fixed background temperature ө b 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 …).

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

A general form statistical 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). Inelastic Collision u= (1-β) u + β |u| σ, with σ the direction of elastic post-collisional relative velocity v v * v v*v* η inelastic collision

Non-Equilibrium Stationary Statistical States Elastic case Inelastic case

A new deterministic approach to compute numerical solution for non-linear non-conservative Boltzmann equations: Spectral-Lagrangian constrained solvers (Filbet, Pareschi & Russo) observing purely kinetic phenomena ( With H. Tharkabhushanam JCP08) In preparation, 08 Resolution of boundary layers discontinuities In preparation, 08

Collision Integral Algorithm

conserve algorithm Stabilization property Discrete Conservation operator

A good test problem BTE in Fourier space The homogeneous dissipative BTE in Fourier space (CMP08)

t r = reference time = mft Δt= 0.25 mft.

Bobylev, Cercignani, I.G (CMP08)

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

Soft condensed matter phenomena Remark: The numerical algorithm is based on the evolution of the continuous spectrum of the solution as in Greengard-Lin00 spectral calculation of the free space heat kernel, i.e. self-similar of the heat equation in all space.

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

Moments calculations: Testing: BTE with Thermostat

Space inhomogeneous simulations mean free time := the average time between collisions mean free path := average speed x mft (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: N= Number of Fourier modes in each j-direction in 3D Spatial mesh size Δx = O.O1 mfp Time step Δt = r mft, mft= reference time

Elastic space inhomogeneous problem Shock tube simulations with a wall boundary Shock propagation phenomena: 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 = mft, mean free path l = 1, 700 time steps, CPU 55hs mesh points: phase velocity N v = 16^3 in [-5,5)^3 - Space: N x =50 mesh points in 30 mean free paths: Δx=3/5 Total number of operations : O(N x N v 2 log(N v )).

Shock dissipation phenomena: Example 2 : Shock dissipation phenomena: Jump in wall kinetic temperature with diffusive boundary conditions 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)

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 twice its magnitude : Calculations in the next four pages: Mean free path l 0 = 1. Number of Fourier modes N = 24 3, Spatial mesh size Δx = 0.01 l 0. Time step Δt = r mft Boundary Conditions for sudden heating:

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

Formation of a shock wave by an initial sudden change of wall temperature from T 0 to 2T 0. Sudden heating problem (BGK eq. with lattice Boltzmann solvers) K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991

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

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

The Riemann Problem: 1D-3D hard spheres elastic gas The macroscopic i.c. satisfy the Rankine-Hugoniot Kn=0.01 t 0 the mean free time dx = t 0 /2 For Kn use hydrodynamic solvers

Shannon Sampling theorem

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, Russo01, Filbet03 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 Deterministic spectral/Lagrangian Method

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 ( and references therein)

Recent related work related to the problem: Cercignani'95( inelastic BTE derivation ) ; Bobylev, JSP 97 ( elastic,hard spheres in 3 d: propagation of L 1 -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, CMP04 ( 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 (CMP08); (generalized multi-linear Maxwell type interactions-inelastic/elastic: global energy dissipation) I.M.G, V.Panferov, C.Villani, arXiv.org07, ARMA08 ( elastic n-dimensional variable hard potentials Grad cut-off:: propagation of L 1 and L - exponential estimates) C. Mouhot, CMP06 ( elastic, VHP, bounded angular cross section: creation of L 1 -exponential ) Ricardo Alonso and I.M.G., JMPA08 (Grad cut-off, propagation of regularity bounds-elastic d-dim VHP) I.M.Gamba and Harsha Tarskabhushanam JCP08 (spectral-lagrangian solvers-computation of singulatities)

t/t 0 = 0.12 Jump in pdf Comparisons with K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991

Plots of v 1 - marginals at the wall and up to 1.5 mfp from the wall

(CMP08)