Spectral-Lagrangian solvers for non-linear
Boltzmann 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.
Motivations from statistical physics or interactive ‘particle’ systems
1.
Initial Motivation: rarefied ideal gases: conservative 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, nonconservative 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
‘v
v
v*
‘v*
C = number of particle in the box
a = diameter of the spheres
N=space dimension
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 (selfsimilar 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
Time irreversibility is expressed in this inequality
stability
In addition:
The Boltzmann Theorem:
there are only N+2 collision invariants
→yields the compressible Euler eqs → Small perturbations of Mawellians yield CNS eqs.
Hydrodynamic limits: evolution models of a ‘few’ statistical moments
(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-similarity?
Non-Gaussian (or Maxwellian) statistics!
Reviewing inelastic properties
e
e
e
e
e
Non-Equilibrium Stationary Statistical States
Elastic case
Inelastic case
A new deterministic approach to compute numerical solution for non-linear Boltzmann
equation: Spectral-Lagrangian constrained solvers
(Filbet, Pareschi & Russo)
(With H. Tharkabhushanam JCP’08)
In preparation, 08
:observing ‘purely kinetic phenomena’
•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 problem
The homogeneous dissipative BTE in Fourier space
(CMP’08)
(CMP’08)
(CMP’08)
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-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.
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
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
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)).
Example 2 : Purely kinetic phenomena:
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’
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:
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
Plots of v1- marginals at the wall and up to 1.5 mfp from the wall
Comparisons with K.Aoki, Y. Sone, K. Nijino, H. Sugimoto, 1991
t/t0= 0.12
Jump in pdf
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)