arXiv subjects

Irene M. Gamba

Publications and source records attributed to Irene M. Gamba.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the Cauchy problem for Boltzmann equation modelling a polyatomic gas

In the present manuscript we consider the Boltzmann equation that models a polyatomic gas by introducing one additional continuous variable, referred to as microscopic internal energy. We establish existence and uniqueness theory in the space homogeneous setting for the full non-linear case, under an extended Grad assumption on transition probability rate, that comprises hard potentials for both the relative speed and internal energy with the rate in the interval $(0,2]$, which is multiplied by an integrable angular part and integrable partition functions. The Cauchy problem is resolved by means of an abstract ODE theory in Banach spaces, for an initial data with finite and strictly positive gas mass and energy, finite momentum, and additionally finite $k_*$ polynomial moment, with $k_*$ depending on the rate of the transition probability and the structure of a polyatomic molecule or its internal degrees of freedom. Moreover, we prove that polynomially and exponentially weighted Banach space norms associated to the solution are both generated and propagated uniformly in time.

math-ph↗

arXiv2020

On existence and uniqueness to homogeneous Boltzmann flows of monatomic gas mixtures

We solve the Cauchy problem for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions in three dimensions. More precisely, we show existence and uniqueness of the vector value solution by means of an existence theorem for ODE systems in Banach spaces under the transition probability rates assumption corresponding to hard potentials rates in the interval $(0,1]$, with an angular section modeled by an integrable function of the angular transition rates modeling binary scattering effects. The initial data for the vector valued solutions needs to be a vector of non-negative measures with finite total number density, momentum and strictly positive energy, as well as to have a finite $L^1_{k_*}(\mathbb{R}^3)$-integrability property corresponding to a sum across each species of $k_*$-polynomial weighted norms depending on the corresponding mass fraction parameter for each species as much as on the intermolecular potential rates, referred as to the scalar polynomial moment of order $k_*$. The existence and uniqueness rigorous results rely on a new angular averaging lemma adjusted to vector values solution that yield a Povzner estimate with constants that decay with the order of the corresponding dimensionless scalar polynomial moment. In addition, such initial data yields global generation of such scalar polynomial moments at any order as well as their summability of moments to obtain estimates for corresponding scalar exponentially decaying high energy tails, referred as to scalar exponential moments associated to the system solution. Such scalar polynomial and exponential moments propagate as well.

math-ph↗

arXiv2020

Propagation of $L^p_β$-norm, $ 1<p\le \infty$, for the system of Boltzmann equations for monatomic gas mixtures

With the existence and uniqueness of a vector value solution for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions proved [8], we present in this manuscript several properties for such a solution. We start by proving the gain of integrability of the gain term of the multispecies collision operator, extending the work done previously for the single species case [3]. In addition, we study the integrability properties of the multispecies collision operator as a bilinear form, revisiting and expanding the work done for a single gas [2]. With these estimates, together with a control by below for the loss term of the collision operator as in [8], we develop the propagation for the polynomially and exponentially $β$-weighted $L^p_β$-norms for the vector value solution. Finally, we extend such $L^p_β$-norms propagation property to $p=\infty$.

math.AP↗

arXiv2019

On the wave turbulence theory for stratified flows in the ocean

After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of quantum Boltzmann type. In this work, we provide the first rigorous mathematical study for the equation by showing the global existence and uniqueness of strong solutions.

math.AP↗

arXiv2019

Asymptotic-preserving schemes for two-species binary collisional kinetic system with disparate masses I: time discretization and asymptotic analysis

We develop efficient asymptotic-preserving time discretization schemes to solve the disparate mass kinetic system of a binary gas or plasma in the "relaxation time scale" relevant to the epochal relaxation phenomenon. Since the resulting model is associated to a parameter given by the square of the mass ratio between the light and heavy particles, we develop an asymptotic-preserving scheme in a novel decomposition strategy using the penalization method for multiscale collisional kinetic equations. Both the Boltzmann and Fokker-Planck-Landau (FPL) binary collision operators will be considered. Other than utilizing several AP strategies for single-species binary kinetic equations, we also introduce a novel splitting and a carefully designed explicit-implicit approximation, which are guided by the asymptotic analysis of the system. We also conduct asymptotic-preserving analysis for the time discretization, for both space homogenous and inhomogeneous systems.

math.NA↗

arXiv2019

Convergence and error estimates for the Lagrangian based Conservative Spectral method for Boltzmann Equations

We develop error estimates for the semi-discrete conservative spectral method for the approximation of the elastic and inelastic space homogeneous Boltzmann equation introduced by the authors in \cite{GT09}. In addition we study the long time convergence of such semi-discrete solution to equilibrium Maxwellian distribution that conserves the mass, momentum and energy associated to the initial data. The numerical method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction that enforces the collision invariants, namely conservation of mass, momentum and energy in the elastic case, and just mass and momentum in the inelastic one. We present a detailed semi-discrete analysis on convergence of the proposed numerical method which includes the $L^{1}-L^{2}$ theory for the scheme. This analysis allows us to present, additionally, convergence in Sobolev spaces and convergence to equilibrium for the numerical approximation. The results of this work answer a long standing open problem posed by Cercignani et al. in \cite[Chapter 12]{CIP} about finding error estimates for a numerical scheme associated to the Boltzmann equation, as well as showing the semi-discrete numerical solution converges to the equilibrium Maxwellian distribution associated to the initial value problem.

math.NA↗

arXiv2019

Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations

In this paper, we first extend the micro-macro decomposition method for multiscale kinetic equations from the BGK model to general collisional kinetic equations, including the Boltzmann and the Fokker-Planck Landau equations. The main idea is to use a relation between the (numerically stiff) linearized collision operator with the nonlinear quadratic ones, the later's stiffness can be overcome using the BGK penalization method of Filbet and Jin for the Boltzmann, or the linear Fokker-Planck penalization method of Jin and Yan for the Fokker-Planck Landau equations. Such a scheme allows the computation of multiscale collisional kinetic equations efficiently in all regimes, including the fluid regime in which the fluid dynamic behavior can be correctly computed even without resolving the small Knudsen number. A distinguished feature of these schemes is that although they contain implicit terms, they can be implemented explicitly. These schemes preserve the moments (mass, momentum and energy) exactly thanks to the use of the macroscopic system which is naturally in a conservative form. We further utilize this conservation property for more general kinetic systems, using the Vlasov-Ampére and Vlasov-Ampére-Boltzmann systems as examples. The main idea is to evolve both the kinetic equation for the probability density distribution and the moment system, the later naturally induces a scheme that conserves exactly the moments numerically if they are physically conserved.

math.NA↗

arXiv2018

On pointwise exponentially weighted estimates for the Boltzmann equation

In this paper we prove propagation in time of weighted $L^\infty$ bounds for solutions to the non-cutoff homogeneous Boltzmann equation that satisfy propagation in time of weighted $L^1$ bounds. To emphasize that the propagation in time of weighted $L^{\infty}$ bounds relies on the propagation in time of weighted $L^1$ bounds, we express our main result using certain general weights. Consequently we apply the main result to cases of exponential and Mittag-Leffler weights, for which propagation in time of weighted $L^1$ bounds holds.

math.AP↗

arXiv2018

Spectral Gap Computations for Linearized Boltzmann Operators

The quantitative information on the spectral gaps for the linearized Boltzmann operator is of primary importance on justifying the Boltzmann model and study of relaxation to equilibrium. This work, for the first time, provides numerical evidences on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a "collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function being the Rayleigh quotient of the "collision matrix" and with constraints being the conservation laws. A conservation correction then applies. We also showed the convergence of the approximate Rayleigh quotient to the real spectral gap for the case of integrable angular cross-sections. Some distributed eigen-solvers and hybrid OpenMP and MPI parallel computing are implemented. Numerical results on integrable as well as non-integrable angular cross-sections are provided.

math-ph↗

arXiv2018

Global weak solutions to compressible Navier-Stokes-Vlasov-Boltzmann systems for spray dynamics

This work concerns the global existence of the weak solutions to a system of partial differential equations modeling the evolution of particles in the fluid. That system is given by a coupling between the standard isentropic compressible Navier-Stokes equations for the macroscopic description of a gas fluid flow, and a Vlasov-Boltzmann type equation governing the evolution of spray droplets modeled as particles with varying radius. We establish the existence of global weak solutions with finite energy, whose density of gas satisfies the renormalized mass equation. The proof, is partially motivated by the work of Feireisl- Novotny-Petzeltov on the weak solutions of the compressible Navier-Stokes equations coupled to the kinetic problem for the spray droplets extending the techniques of Legger and Vasseur developed for the incompressible fluid-kinetic system.

math.AP↗

arXiv2018

The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation system for bosons at very low temperature

We study a quantum Boltzmann-Condensation system that describes the evolution of the interaction between a well formed Bose-Einstein condensate and the quasi-particles cloud. The kinetic model is valid for a dilute regime at which the temperature of the gas is very low compared to the Bose-Einstein condensation critical temperature. In particular, our system couples the density of the condensate from a Gross-Pitaevskii type equation to the kinetic equation through the dispersion relation in the kinetic model and the corresponding transition probability rate from pre to post collision momentum states. We rigorously show the following three properties (1) the well-posedness of the Cauchy problem for the system in the case of a radially symmetric initial configuration, (2) find qualitative properties of the solution such as instantaneous creation of exponential tails and, (3) prove the uniform condensate stability related to the initial mass ratio between condensed particles and quasi-particles. The stability result from (3) leads to global in time existence of the initial value problem for the quantum Boltzmann-Condensation system.

math.AP↗

arXiv2018

Galerkin Methods for Boltzmann-Poisson transport with reflection conditions on rough boundaries

We consider in this paper the mathematical and numerical modelling of reflective boundary conditions (BC) associated to Boltzmann - Poisson systems, including diffusive reflection in addition to specularity, in the context of electron transport in semiconductor device modelling at nano scales, and their implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the physical boundaries of the device and develop a numerical approximation to model an insulating boundary condition, or equivalently, a pointwise zero flux mathematical condition for the electron transport equation. Such condition balances the incident and reflective momentum flux at the microscopic level, pointwise at the boundary, in the case of a more general mixed reflection with momentum dependant specularity probability $p(\vec{k})$. We compare the computational prediction of physical observables given by the numerical implementation of these different reflection conditions in our DG scheme for BP models, and observe that the diffusive condition influences the kinetic moments over the whole domain in position space.

math.NA↗

arXiv2018

Discontinuous Galerkin Deterministic Solvers for a Boltzmann-Poisson Model of Hot Electron Transport by Averaged Empirical Pseudopotential Band Structures

The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver of a Boltzmann-Poisson model for hot electron transport, an electronic conduction band whose values are obtained by the spherical averaging of the full band structure given by a local empirical pseudopotential method (EPM) around a local minimum of the conduction band for silicon, as a midpoint between a radial band model and an anisotropic full band, in order to provide a more accurate physical description of the electron group velocity and conduction energy band structure in a semiconductor. This gives a better quantitative description of the transport and collision phenomena that fundamentally define the behaviour of the Boltzmann - Poisson model for electron transport used in this work. The numerical values of the derivatives of this conduction energy band, needed for the description of the electron group velocity, are obtained by means of a cubic spline interpolation. The EPM-Boltzmann-Poisson transport with this spherically averaged EPM calculated energy surface is numerically simulated and compared to the output of traditional analytic band models such as the parabolic and Kane bands, numerically implemented too, for the case of 1D $n^+-n-n^+$ silicon diodes with 400nm and 50nm channels. Quantitative differences are observed in the kinetic moments related to the conduction energy band used, such as mean velocity, average energy, and electric current (momentum).

cs.CE↗

arXiv2017

Positivity preserving DG schemes for a Boltzmann - Poisson model of electrons in semiconductors in curvilinear momentum coordinates

The work presented in this paper is related to the development of positivity preserving Discontinuous Galerkin (DG) methods for Boltzmann - Poisson (BP) computational models of electronic transport in semiconductors. We pose the Boltzmann Equation for electron transport in curvilinear coordinates for the momentum. We consider the 1D diode problem with azimuthal symmetry, which is a 3D plus time problem. We choose for this problem the spherical coordinate system $\vec{p}(|\vec{p}|,μ=cosθ,φ)$, slightly different to the choice in previous DG solvers for BP, because its DG formulation gives simpler integrals involving just piecewise polynomial functions for both transport and collision terms. Applying the strategy of Zhang \& Shu, \cite{ZhangShu1}, \cite{ZhangShu2}, Cheng, Gamba, Proft, \cite{CGP}, and Endeve et al. \cite{EECHXM-JCP}, we treat the collision operator as a source term, and find convex combinations of the transport and collision terms which guarantee the positivity of the cell average of our numerical probability density function at the next time step. The positivity of the numerical solution to the pdf in the whole domain is guaranteed by applying the limiters in \cite{ZhangShu1}, \cite{ZhangShu2} that preserve the cell average but modify the slope of the piecewise linear solutions in order to make the function non - negative. In addition of the proofs of positivity preservation in the DG scheme, we prove the stability of the semi-discrete DG scheme under an entropy norm, using the dissipative properties of our collisional operator given by its entropy inequalities. The entropy inequality we use depends on an exponential of the Hamiltonian rather than the Maxwellian associated just to the kinetic energy.

math.NA↗

arXiv2017

Galerkin-Petrov approach for the Boltzmann equation

In this work, we propose a new Galerkin-Petrov method for the numerical solution of the classical spatially homogeneous Boltzmann equation. This method is based on an approximation of the distribution function by associated Laguerre polynomials and spherical harmonics and test an a variational manner with globally defined three-dimensional polynomials. A numerical realization of the algorithm is presented. The algorithmic developments are illustrated with the help of several numerical tests.

math.NA↗

arXiv2017

A Conservative Scheme for Vlasov Poisson Landau modeling collisional plasmas

We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equation coupled with the Poisson equation, which is a {classical mean-field} primary model for collisional plasmas. Two subproblems, i.e. the Vlasov-Poisson problem and homogeneous Landau problem, are obtained through time-splitting methods, and treated separately by the Runge-Kutta Discontinuous Galerkin method and a conservative spectral method, respectively. To ensure conservation when projecting between the two different computing grids, a special conservation routine is designed to link the solutions of these two subproblems. This conservation routine accurately enforces conservation of moments in Fourier space. The entire numerical scheme is implemented with parallelization with hybrid MPI and OpenMP. Numerical experiments are provided to study linear and nonlinear Landau Damping problems and two-stream flow problem as well.

physics.comp-ph↗

arXiv2017

On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff

We establish the $L^1$ weighted propagation properties for solutions of the Boltzmann equation with hard potentials and non-integrable angular components in the collision kernel. Our method identifies null forms by angular averaging and deploys moment estimates of solutions to the Boltzmann equation whose summability is achieved by introducing the new concept of Mittag-Leffler moments - extensions of $L^1$ exponentially weighted norms. Such $L^1$ weighted norms of solutions to the Boltzmann equation are, both, generated and propagated in time and the characterization of their corresponding Mittag-Leffler weights depends on the angular singularity and potential rates in the collision kernel. These estimates are a fundamental step in order to obtain $L^\infty$ exponentially weighted estimates for solutions of the Boltzmann equation being developed in a follow up work.

math.AP↗

arXiv2016

A fast spectral method for the Boltzmann collision operator with general collision kernels

We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method \cite{PR00, GT09} which requires $O(N^6)$ memory to store precomputed weights and has $O(N^6)$ numerical complexity, the new method has complexity $O(MN^4\log N)$, where $N$ is the number of discretization points in each of the three velocity dimensions and $M$ is the total number of discretization points on the sphere and $M\ll N^2$. Furthermore, it requires no precomputation for the variable hard sphere (VHS) model and only $O(MN^4)$ memory to store precomputed functions for more general collision kernels. Although a faster spectral method is available \cite{MP06} (with complexity $O(MN^3\log N)$), it works only for hard sphere molecules, thus limiting its use for practical problems. Our new method, on the other hand, can apply to arbitrary collision kernels. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method.

math.NA↗