Spectral-Lagrangian solvers for non-Spectral-Lagrangian solvers for non-linearlinear

Boltzmann type equations: numerics Boltzmann type equations: numerics and analysisand analysis

Irene M. GambaDepartment 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.

Motivations from statistical physics or interactive ‘particle’ systems

1. Initial Motivation: rarefied ideal gases: conservativeconservative Boltzmann Transport eq.Boltzmann Transport eq.

2. 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.

3. Soft condensed matter at nano scale: Bose-Einstein condensates models, charge transport in solids: current/voltage transport modeling semiconductor.

4. 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 A unified approach for Maxwell type interactions and •generalizations.generalizations. • Analytical properties - long time asymptotics and characterization of Analytical properties - long time asymptotics and characterization of asymptotics states: high energy tails and singularity formation asymptotics states: high energy tails and singularity formation

May be extended to multi-linear interactions

‘v

‘v*

v

v*

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

i.e. enough intersitial space

A general form statistical transport :The space-homogenous BTE with external heating sourcesImportant 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

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

The Boltzmann Theorem:The Boltzmann Theorem: there are only N+2 collision invariants

Time irreversibility is expressed in this inequality stability

In addition:

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

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

Exact energy identity for a Maxwell type interaction models

Then f(v,t) → δ0 as t → ∞ 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-similarityself-similarity? Non-Gaussian (or Maxwellian) statistics!

Reviewing Reviewing inelastic inelastic propertiesproperties

e

ee

e e

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 Boltzmann numerical solution for non-linear Boltzmann

equation: Spectral-Lagrangian constrained solversequation: Spectral-Lagrangian constrained solvers

(Filbet, Pareschi & Russo)

:observing ‘purely kinetic phenomena’

(With H. Tharkabhushanam JCP’08)

In preparation, 08

•This scheme is an alternative to the well known stochastically based DSMC (Discrete Simulation by Monte Carlo) for particle methods or alternatively called the Bird scheme.

in σ

Given find such that:

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

(CMP’08)

(CMP’08)

(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.

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

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

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

mean free path = average speed X mft = average distance travelled between collisions

mean free time = the average time between collisions

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 = 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)).

Example 2 : Purely kineticPurely kinetic 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’

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 = 0.15 l0 . Time step Δt = mean free time

Boundary Conditions for sudden heating:

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

t/t0= 0.12

Jump in pdf

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

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)

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)