Irene M. Gamba

University of Texas at Austin | UT · Department of Mathematics and Institute for Computational Engineering and Sciences (ICES)

This review manuscript reports on two recent results for a time dynamics model for a non-linear system of Boltzmann equations in a space homogeneous setting that describes multi-component monatomic gas mixtures for binary interactions. This model describes the evolution of an arbitrary finite set of probability density functions, depending on time...

This work focus on the construction of weak solutions to a kinetic-fluid system of partial differential–integral equations modeling the evolution of particles droplets in a compressible fluid. The system is given by a coupling between the standard isentropic compressible Navier–Stokes equations for the macroscopic description of a gas fluid flow, a...

The ab initio model for heat propagation is the phonon transport equation, a Boltzmann-like kinetic equation. When two materials are put side by side, the heat that propagates from one material to the other experiences thermal boundary resistance. Mathematically, it is represented by the reflection coefficient of the phonon transport equation on th...

Error estimates are rigorously derived for a semi-discrete version of a conservative spectral method for approximating the space-homogeneous Fokker-Planck-Landau (FPL) equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a...

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 probabili...

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 t...

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 the existence and uniqueness of the vector value solution by means of an existence theorem for ODE systems in Banach spaces under the tra...

This work develops entropy-stable positivity-preserving DG methods as a computational scheme for Boltzmann-Poisson systems modeling the pdf of electronic transport along energy bands in semiconductor crystal lattices. We pose, using spherical or energy-angular variables as momentum coordinates, the corresponding Vlasov Boltzmann eq. with a linear c...

In this paper we prove global well-posedness for small initial data for the binary-ternary Boltzmann equation. The binary-ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and could possibly serve as a more accurate description model for...

In this paper, we conduct uniform error estimates of the bi-fidelity method for multi-scale kinetic equations. We take the Boltzmann and the linear transport equations as important examples. The main analytic tool is the hypocoercivity analysis for kinetic equations, considering solutions in a perturbative setting close to the global equilibrium. T...

Expanding upon the conservative spectral method for solving the Vlasov-Poisson Landau system, developed by Zhang and Gamba for Coulomb interactions, the deterministic scheme has been extended to model Vlasov-Poisson Fokker-Planck-Landau type equations with Maxwell type and hard sphere interactions. The original case, corresponding to the classical...

Expanding upon the conservative spectral method for solving the Landau equation, developed by Zhang and Gamba, a deterministic scheme has been developed for modeling Fokker-Planck-Landau type equations with Maxwell molecules and hard sphere interactions. The original case, corresponding to the classical physical problem of Coulomb interactions, is...

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 quadrati...

We develop efficient asymptotic-preserving time discretization to solve the disparate mass kinetic system of a binary gas or plasma in the "relaxation time scale" relevant to the epochal relaxation phenomenon. Both the Boltzmann and Fokker-Planck-Landau (FPL) binary collision operators will be considered. Other than utilizing several AP strategies...

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 quadrati...

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 operat...

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 transit...

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 equatio...

Immediately following the commentary below, this previously published article is reprinted in its entirety: Cathleen Synge Morawetz, “The mathematical approach to the sonic barrier”, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 127–145.

We consider in this paper the mathematical and numerical modeling 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 modeling at nano scales, and their implementation in Discontinuous Galerkin (DG) sch...

In the present work, we propose a deterministic numerical solver for the space homogeneous Boltzmann equation based on discontinuous Galerkin (DG) methods. Such an application has been rarely studied. The main goal of this manuscript is to generate a conservative solver for the collisional operator. As the key part, the weak form of the collision o...

We present in this document the Lebesgue and Sobolev propagation of exponential tails for solutions of the homogeneous Boltzmann equation for hard and Maxwell interactions. In addition, we show the $L^{p}$-integrability creation of such tails in the case of hard interactions. The document also presents a result on exponentially-fast convergence to...

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 w...

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 polynomi...

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 firs...

We study the rate of relaxation to equilibrium for Landau kinetic equation and some related models by considering the relatively simple case of radial solutions of the linear Landau-type equations. The well-known difficulty is that the evolution operator has no spectral gap, i.e. its spectrum is not separated from zero. Hence we do not expect purel...

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 r...

We present an overview of deterministic solvers for the Boltzmann and Landau equations inspired by their Fourier space representation as weighted convolutional forms, where the later can be obtained as a grazing collision limit of the former. This presentation offers an introduction to the area and elaborates on recent results for conservative spec...

We consider solutions to the initial value problem for the spatially homogeneous Boltzmann equation for pseudo-Maxwell molecules and show uniform in time propagation of upper Maxwellians bounds if the initial distribution function is bounded by a given Maxwellian. First we prove the corresponding integral estimate and then transform it to the desir...

We solve the Cauchy problem associated to the space homogeneous Boltzmann equation with an angle-potential singular concentration modeling the collision kernel, proposed in 2013 by Bobylev and Potapenko. The potential under consideration ranges from Coulomb to hard spheres cases. However, the motivation of such a collision kernel is to treat the ca...

We propose a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equation coupled with Poisson equation. Through time-splitting scheme, a Vlasov-Poisson (collisionless) problem and a homogeneous Landau (collisional) problem are obtained. These two subproblems can be treated separately. We use operator splitting where the tr...

In this paper we study the approximation properties of the spectral conservative method for the elastic and inelastic Boltzmann problem introduced by the authors in \cite{GT09}. The method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction used for conservation of mass, momentum and energy. We pre...

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 discret...

We prove the global existence of weak solutions to kinetic Kolmogorov-Vicsek models that can be considered a non-local non-linear Fokker-Planck type equation describing the dynamics of individuals with orientational interaction. This model is derived from the discrete Couzin-Vicsek algorithm as mean-field limit \cite{B-C-C,D-M}, which governs the i...

We solve the Cauchy problem for a kinetic quantum Boltzmann model that approximates the evolution of a radial distribution of quasiparticles in a dilute gas of bosons at very low temperature with a cubic kinetic transition probability kernel. We classify some relevant qualitative properties of such solutions which include the propagation and creati...

We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equation coupled with Poisson equation, which is a rather realistic and primary model for collisional plasmas. Two subproblems, i.e Vlasov-Poisson problem and homogeneous Landau problem, are obtained through time-splitting methods, and treated separatel...

This work concerns the numerical solution of a coupled system of self-consistent reaction-drift-diffusion-Poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical (PEC) solar cells with reactive semiconductor and electrolyte interfaces. We present three numerical algorithms, mainly based on a mixed finit...

We shall discuss the use of Discontinuous Galerkin (DG) Finite Element Methods to solve Boltzmann - Poisson (BP) models of electron transport in semiconductor devices at nano scales. We consider the mathematical and numerical modeling of Reflective Boundary Conditions in 2D devices and their implementation in DG-BP schemes. We study the specular, d...

We study generation and propagation properties of Mittag-Leffler moments for solutions of the spatially homogeneous Boltzmann equation for scattering collision kernels corresponding to hard potentials without angular Grad's cutoff assumption, i.e. the angular part of the scattering kernel is non-integrable with prescribed singularity rate. These ki...

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 c...

The mathematical modeling and numerical simulation of semiconductor-electrolyte systems play important roles in the design of high-performance semiconductor-liquid junction solar cells. In this work, we propose a macroscopic mathematical model, a system of nonlinear partial differential equations, for the complete description of charge transfer dyn...

We discuss some general properties of the Landau kinetic equation. In particular, the difference between the “true” Landau equation, which formally follows from classical mechanics, and the “generalized” Landau equation, which is just an interesting mathematical object, is stressed. We show how to approximate solutions to the Landau equation by the...

In this paper we present the first numerical method for a kinetic description of the Vicsek swarming model. The kinetic model poses a unique challenge, as there is a distribution dependent collision invariant to satisfy when computing the interaction term. We use a spectral representation linked with a discrete constrained optimization to compute t...

We present a conservative spectral method for the fully nonlinear Boltzmann collision operator based on the weighted convolution structure in Fourier space developed by Gamba and Tharkabhushnanam. This method can simulate a broad class of collisions, including both elastic and inelastic collisions as well as angularly dependent cross sections in w...

In the present work, we propose a deterministic numerical solver for the homogeneous Boltzmann equation based on Discontinuous Galerkin (DG) methods. The weak form of the collision operator is approximated by a quadratic form in linear algebra setting. We employ the property of “shifting symmetry” in the weight matrix to reduce the computing comple...

In this paper we perform, by means of Discontinuous Galerkin (DG) Finite Element Method (FEM) based numerical solvers for Boltzmann-Poisson (BP) semiclassical models of hot electronic transport in semiconductors, a numerical study of reflective boundary conditions in the BP system, such as specular reflection, diffusive reflection, and a mixed conv...

We study the dynamics defined by the Boltzmann equation set in the Euclidean space $\mathbb{R}^D$ in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches th...

We will discuss recent development in the simulation of Boltzmann-Poisson systems and Wigner transport by deterministic numerical solvers. We have proposed to solve linear transport problems using a Discontinuous Galerkin (DG) Finite Element Method (FEM) approach that allows adaptivity and accuracy by a flexible choice of basis functions, as well a...

In this paper a spectral-Lagrangian method is proposed for the full, non-linear Boltzmann equation for a multi-energy level gas typical of a hypersonic re-entry flow. Internal energy levels are treated as separate species and inelastic collisions (leading to internal energy excitation and relaxation) are accounted for. The formulation developed can...

We present the formulation of a conservative spectral scheme for Boltzmann collision operators with anisotropic scattering mechanisms to model grazing collision limit regimes approximating the solution to the Landau equation in space homogeneous setting. The scheme is based on the conservative spectral method of Gamba and Tharkabhushanam [17, 18]....

form only given. We present recent work extending the conservative spectral method for the Boltzmann transport equation developed by Gamba and Tharkabhushanam. This formulation is derived from the weak form of the Boltzmann equation, which represents the collisional term as a weighted convolution in Fourier space. We have extended the method to the...

Discontinuous Galerkin methods are developed for solving the Vlasov-Maxwell system, methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Maxwell system. Th...

A spectral-Lagrangian deterministic solver for the Boltzmann equation for rarefied gas flows is proposed. Numerical solutions are obtained for the flow across normal shock waves of pure gases and mixtures by means of a time-marching method. Operator splitting is used. The solution update is obtained as a combination of the operators for the advecti...

We present new results building on the conservative deterministic spectral method for the space homogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into...

We present new results building on the conservative deterministic spectral method for the space inhomogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral int...

We propose to extend an existing spectral-Lagrangian numerical method for the Boltzmann equation for a pure gas (without internal energy) to a multi energy level gas. The numerical method is based on the weak form of the collision operator and can be used with any type of cross-section model. The formulation is developed in order to account for bot...

We describe the Runge-Kutta discontinuous Galerkin (RKDG) schemefootnotetextR. E. Heath, I. M. Gamba, P. J. Morrison, and C. Michler, J. Comp. Phys. 231, 1140 (2012). for the Vlasov-Poisson system that models collisionless plasmas. One-dimensional systems are emphasized. This numerical method used is seen to have excellent conservation properties,...

In this paper we consider Runge-Kutta discontinuous Galerkin (RKDG) schemes for Vlasov-Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and c...

One of the main applications in plasma physics concerns the energy production through thermo-nuclear fusion. The controlled fusion is achieved by magnetic confinement i.e., the plasma is confined into a toroidal domain (tokamak) under the action of huge magnetic fields. Several models exist for describing the evolution of strongly magnetized plasma...

We consider the gravitational Vlasov–Poisson (VP), or the so-called collisionless Boltzmann–Poisson equations for the self-gravitating collisionless stellar systems. We compute the solutions using a high-order discontinuous Galerkin method for the Vlasov equation, and the classical representation by Green’s function for the Poisson equation in the...

The present work is motivated by the development of a fast DG based deterministic solver for the extension of the BTE to a system of transport Boltzmann equations for full electronic multiband transport with intraband scattering mechanisms. Our proposed method allows to find scattering effects of high complexity, such as anisotropic electronic band...

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous $d$-dimensional Boltzmann equation. In particular, when the collision kernel is of the form $|v-v_*|^\beta b(\cos(\theta))$ for $\beta \in (0,2]$ with $\cos(\theta)= |v-v_*|^{-1}(v-v_*)\cdot \sigma$ and $\sigma \in \mathbb{S}^{d-1}$, and assuming...

A discontinuous Galerkin (DG) method for integrating the Vlasov-Poisson system [1] is described and generalized. Higher order polynomials on basis elements are used in recent calculations and an extensive error analysis has been performed. In particular, recurrence properties have been examined in detail. The method is conservative and a limiter is...

It is shown that a broad class of generalized Dirichlet series (includ-ing the polylogarithm, related to the Riemann zeta-function) can be presented as a class of solutions of the Fourier transformed spatially homogeneous linear Boltz-mann equation with a special Maxwell-type collision kernel. The result is based on an explicit integral representat...

We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials. The solutions of such BTEs are positive probability distribution functions and it is very challenging to have a mass-conservative, high-ord...

The study of the convergence to equilibrium of solutions to Fokker-Planck type equations with linear diffusion and super-linear drift leads in a natural way to a minimization problem for an energy functional (entropy) which relies on a sub-linear convex function. In many cases, conditions linked both to the non-linearity of the drift and to the spa...

The discontinuous Galerkin (DG) method developed by some of us for integrating the Vlasov-Poisson systemootnotetextR.E. Heath, I.M. Gamba, P.J. Morrison, and C. Michler, arXiv:1009.3046v1 [physics.plasm-ph]. is described and generalized. Higher order polynomials on basis elements are used and extensive error analyses, including recurrence propertie...

We investigate numerically an inverse problem related to the Boltzmann–Poisson system of equations for transport of electrons in semiconductor devices. The objective of the (ill-posed) inverse problem is to recover the doping profile of a device, presented as a source function in the mathematical model, from its current–voltage characteristics. To...

This short note complements the recent paper of the authors (Alonso, Gamba in J. Stat. Phys. 137(5–6):1147–1165, 2009). We revisit the results on propagation of regularity and stability using L p estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator. We show here the correct range of expon...

We are interested in the deterministic computation of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The main difficulty of such computation arises from the very high dimensions of the model, making it necessary to use relatively coarse meshes and hence requiring the numerical solver to be st...

We are interested in the deterministic computation of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The main difficulty of such computation arises from the very high dimensions of the model, making it necessary to use relatively coarse meshes and hence requiring the numerical solver to be st...

We revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the W k l,r regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of previous approaches by...

We present the effectiveness and competitiveness of a discontinuous Galerkin (DG) scheme applied to deterministic computations of the transients for the Boltzmann-Poisson (BP) system describing electron transport in semiconductor devices. In particular, we show that the scheme can maintain reasonable accuracy even with rather coarse meshes, hence p...

We consider the linear Wigner-Fokker-Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of s...

A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enf...

The numerical approximation of the Spectral-Lagrangian scheme developed by the au-thors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock a...

We investigate the well posedness of the stationary linear Boltz-mann equation with space periodic electric field. We discuss the different behaviors that occur depending if the average electric field vanishes or not. The existence follows by perturbation techniques and stability arguments un-der uniform a priori estimates. The uniqueness of the we...

We investigate the well posedness of stationary Vlasov-Boltzmann equations both in the simpler case of linear problem with a space varying force field and a collisional integral satisfy-ing the detailed balanced principle with a non-singular scattering function, and, the non-linear Vlasov-Poisson-Boltzmann system. For the former we obtain existence...

The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker–Planck interaction ter...