Spectral-Lagrangian solvers for non-Spectral-Lagrangian solvers for non-linear linear

non-conservative Boltzmann Transport non-conservative Boltzmann Transport Equations Equations

Irene M. GambaDepartment 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:Goals:•Understanding of analytical properties: large energy tailsUnderstanding of analytical properties: large energy tails •Long time asymptotics and characterization of Long time asymptotics and characterization of asymptotics statesasymptotics states

•Deterministic numerical approximations – observing ‘purely kinetic phenomena’Deterministic numerical approximations – observing ‘purely kinetic phenomena’

Statistical transport from collisional kinetic modelsStatistical transport from collisional kinetic models

• Rarefied ideal gases-elastic:Rarefied ideal gases-elastic: classical conservativeclassical conservative Boltzmann Transport eq.Boltzmann 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…).

‘v

‘v*

v

v*

C = number of particle in the box a = diameter of the spheresN=space dimension η

elastic collisioninelastic collisionη 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 termmodels 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*

η inelastic collision

Non-Equilibrium Stationary Statistical States

Elastic caseElastic case

Inelastic Inelastic casecase

A new deterministic approach to compute A new deterministic approach to compute numerical solution for non-linear non-conservative numerical solution for non-linear non-conservative

Boltzmann equations: Spectral-Lagrangian Boltzmann equations: Spectral-Lagrangian constrained solversconstrained solvers

(Filbet, Pareschi & Russo)

•observing ‘purely kinetic phenomena’

(With H. Tharkabhushanam JCP’08)

In preparation, 08

• Resolution of boundary layers discontinuitiesIn preparation, 08

Collision Integral AlgorithmCollision Integral Algorithm

‘conserve’ algorithm

Stabilization property

Discrete Conservation operator

A good test problemA good test problem The homogeneous dissipative BTE in Fourier spaceBTE in Fourier space

(CMP’08)

tr = reference time = mft Δt= 0.25 mft.

Bobylev, Cercignani, I.G (CMP’08)

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

Soft condensed matter Soft condensed matter phenomenaphenomena

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.

Maxwell Molecules modelRescaling of spectral modes exponentially by the continuous spectrum with λ(1)=-2/3

Testing: BTE with Thermostatexplicit solution problem of colored particles

Moments calculations:Moments calculations:Testing: BTE with Thermostat

Space inhomogeneous simulationsSpace inhomogeneous simulationsmean free time := the average time between collisionsmean 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 Elastic space inhomogeneous problem Shock tube simulations with a wall boundaryShock tube simulations with a wall boundary

Example 1: Shock propagation phenomena: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)).

Example 2 : Shock dissipationShock dissipation phenomenaphenomena:: Jump in wall kinetic temperature with diffusive boundary conditionsJump 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 l0 = 1. Number of Fourier modes N = 243, Spatial mesh size Δx = 0.01 l0 . 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 T0 to 2T0.

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

t0 the mean free time dx = t0/2

For Kn << 0.01 the method becomes too slow -> 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, 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

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 ( www.ma.utexas.edu/users/gamba/research 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 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 of regularity bounds-elastic d-dim VHP)

I.M.Gamba and Harsha Tarskabhushanam JCP’08(spectral-lagrangian solvers-computation of singulatities)

t/t0= 0.12

Jump in pdf

Comparisons with K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991

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

(CMP’08)