WENO schemes applied to the quasi-relativistic Vlasov--Maxwell model for laser-plasma interaction

In this paper we focus on WENO-based methods for the simulation of the 1D Quasi-Relativistic Vlasov--Maxwell (QRVM) model used to describe how a laser wave interacts with and heats a plasma by penetrating into it. We propose several non-oscillatory methods based on either Runge--Kutta (explicit) or Time-Splitting (implicit) time discretizations. We then show preliminary numerical experiments

A Discontinuous Galerkin semi-Lagrangian solver for the guiding-center problem

Marseille, France, 19 Juillet - 27 Août 2010In this paper, we test an innovative numerical scheme for the simulation of the guiding-center model, of interest in the domain of plasma physics, namely for fusion devices. We propose a 1D Discontinuous Galerkin (DG) discretization, whose basis are the Lagrange polynomials interpolating the Gauss points inside each cell, coupled to a conservative semi-Lagrangian (SL) strategy. Then, we pass to the 2D setting by means of a second-order Strangsplitting strategy. In order to solve the 2D Poisson equation on the DG discretization, we adapt the spectral strategy used for equally-spaced meshes to our Gauss-point-based basis. The 1D solver is validated on a standard benchmark for the nonlinear advection; then, the 2D solver is tested against the swirling deformation ow test case; nally, we pass to the simulation of the guiding-center model, and compare our numerical results to those given by the Backward Semi-Lagrangian method

A Contribution to the simulation of Vlasov-based models

Consultable des del TDXTítol obtingut de la portada digitalitzadaEsta tesis está dedicada al desarrollo, aplicación y test de métodos para la simulación numérica de problemas procedentes de la física y de la ingeniería electrónica. La principal herramienta aplicada a lo largo de todo el trabajo es la ecuación de Vlasov (transporte) en la forma de la Boltzmann Transport Equation (BTE) para la descripción del transporte de partículas cargadas en plasmas y dispositivos electrónicos: las cargas se mueven bajo el efecto de un campo de fuerza y sufren scattering debido a otras cargas o fonones (pseudo-partículas que describen de manera efectiva las vibraciones de los iones del retículo cristalino). La BTE ha de ser acoplada con una ecuación o sistema de ecuaciones para calcular el campo de fuerza: para estructuras simples se usa la ecuación de Poisson; para plasmas, donde los efectos magnéticos no se pueden despreciar debido a las altas velocidades de las partículas, se usa la fuerza de Lorentz, por lo cual se han de resolver las ecuaciones de Maxwell; en nanoestructuras, por ejemplo transistores con dimensiones confinadas, la ecuación de Poisson necesita ser acoplada con la ecuación de Schrödinger para la descripción de las dimensiones cuánticas y para la descomposición en sub-bandas, o niveles de energía. Las colisiones son el scattering que las cargas padecen debido a las interacciones con otras cargas o con el retículo cristalino fijo, representado en forma de fonones. En la tesis se emplean diversos operadores de scattering: los más simples son operadores lineales de relajación; se estudia un modelo para la simulación de semiconductores donde se tienen en cuenta colisiones con fonones acústicos, en aproximación elástica, y fonones ópticos. Tras la introducción, en el primer capítulo se desarrollan los métodos numéricos más importantes: primero un método de interpolación no oscilante (PWENO), necesario para evitar las oscilaciones producidas por la reconstrucción por polinomios de Lagrange, que incrementa la variación total cuando aparecen choques: las oscilaciones en el espacio de fases son características del problema, pero si el método añade oscilaciones espúreas (es decir, debidas al método en sí), entonces el resultado numérico no tiene sentido, o simplemente explota. El segundo método numérico fundamental es la técnica de splitting: cuando se resuelve un problema complicado, si se puede dividir en sub-problemas y resolverlos por separado, entonces se puede reconstruir una aproximación para el problema completo; esta técnica se usa para el time splitting (separación de la parte de transporte y de colisión) y el splitting dimensional (dividir el espacio de fases en posición y velocidad). La tercera herramienta fundamental es un sólver para advección lineal: se usan dos métodos, uno basado en trazar hacia atrás las características a nivel puntual y otro basado en reconstruir valores integrales en segmentos en lugar de puntos; el primero controla mejor las oscilaciones, el segundo fuerza la conservación de masa. En el capítulo 2 estos métodos se aplican a algunos tests conocidos para averiguar su solidez. En el capítulo 3 estos métodos se aplican a la simulación de un diodo, y los resultados se comparan con resultados anteriores obtenidos por esquemas Runge-Kutta basados en diferencias finitas para aproximar las derivadas parciales. El capítulo 4 está dedicado a la construcción y simulación de modelos intermedios entre una ecuación cinética, con operador de colisión de tipo relajación, y su aproximación más grosera, ésta última siendo la ecuación del calor. Para obtener modelos intermedios, se busca un cierre de las ecuaciones de los momentos de orden cero y uno. Se proponen esquemas «asymptotic-preserving» para la ecuación cinética, que evitan la stiffness de la parte de advección a través de una descomposición de la función de distribución en su media más fluctuaciones. En cuanto a las clausuras de las ecuaciones de los momentos, se proponen esquemas de relajación para aislar las no-linealidades. Estos métodos son aplicados a un test conocido, el Su-Olson test. El último capítulo está dedicado a la simulación de un MOSFET (Metal Oxide Semiconductor Field Effect Transistor) 2D de dimensión nanométrica en el que los electrones se comportan como partículas en una dimensión y como ondas en las dimensiones confinadas. La descomposición en sub-bandas se realiza a través de una ecuación de Schrödinger 1D en estado estacionario. Las dimensiones, así como las sub-bandas, están acopladas por la ecuación de Poisson en la expresión de la densidad, y por el operador de colisión. Se propone un sólver microscópico para estados transitorios, basado en técnicas de splitting para las BTEs (una para cada nivel de energía), métodos de características para el transporte y una iteración de tipo Newton para resolver el problema acoplado Schrödinger-Poisson para el cálculo del campo de fuerza.This thesis is dedicated to the development, application and test of numerical methods for the numerical simulation of problems arising from physics and electronic engineering. The main tool which is used all along the work is the Vlasov (transport) equation in the form of the Boltzmann Transport Equation (BTE) for the description of the transport and collisions of charged particles in plasmas and electronic devices: charge carriers are driven by a force field and scattered by other carriers or phonons (pseudo-particles giving an effective representation of the oscillating field produced by the vibrating ions). The BTE must be coupled to an equation or a system of equations for the computation of the force field: for simple structures the Poisson equation is used; for plasmas, where the magnetic phenomena cannot be neglected due to the high velocities of the particles, the Lorentz force is used, so the Maxwell equations have to be solved; for nanostructures, e.g. transistors with confined dimensions, the Poisson equation needs coupling with Schrödinger equation for the description of the quantum dimensions and the decomposition into subbands, or energy levels. Collisions mean the scattering the carriers suffer due to the interactions with other carriers or the fixed lattice, in form of phonons. All along the thesis several scattering operator are used: the simplest ones are linear relaxation-time operators; a model for the simulation of a semiconductor is studied in which collisions are taken into account with acoustic phonons, in the elastic approximation, and optical phonons. After the introduction, in the first chapter the most important numerical methods are developed: first of all a pointwise non-oscillatory interpolation method (PWENO) needed to avoid the simple Lagrange polynomial reconstruction, which increases the total variation when shocks appear: oscillations are part of the physics of the problem, but if the method adds spurious, non-physical oscillations, then the numerical result is meaningless, or it simply blows up. The second fundamental numerical method is the splitting technique: when solving a complicated problem, if we are able to subdivide it into sub-problem and solve them for separate, then we can reconstruct an approximation for the complete problem; this technique is used for both time splitting (separate transport from collisions) and dimensional splitting (split the phase space into either dimensions). The third fundamental instrument is the solver for linear advections: two methods are used, one based on pointwise following backwards the characteristics and another one based on reconstructing integral values along segments instead of point values; the first one controls better oscillations, the second one forces mass conservation. These methods are applied in chapter 2 to some well-known benchmark tests to control their robustness. In chapter 3 these methods are applied to the simulation of a diode, and the results compared to previous results obtained by Runge-Kutta schemes based on finite differences schemes for the approximation of the partial derivatives. Chapter 4 is dedicated to the construction and simulation of intermediate models between a kinetic equation, with relaxation-time collision operator, and its coarsest approximation, this one being the heat equations. In order to obtain intermediate models, the moment equations are closed at zeroth and first order. Asymptotic-preserving schemes are proposed for the kinetic equation, which avoid the stiffness of the advection part by decomposing the distribution function into its average plus fluctuations. As for the moment closures, relaxation schemes are proposed in order to confine the non-linearities in the right hand side. These methods are then applied to a known benchmark, the Su-Olson test. The last chapter is dedicated to the simulation of a nanoscaled 2D MOSFET (Metal Oxide Field Effect Transistor) in which electrons behave as particles in one dimension and as waves in the confined dimensions. The subband decomposition is realized through a stationary-state 1D Schrödinger equation. The dimensions as well as the subbands are coupled by the Poisson equation in the expression of the density and by the collision operator. A transient-state microscopic solver is proposed, based on splitting techniques for the BTE's (one for each energy level), characteristics methods for the transport and a Newton iteration for the solution of the coupled Schrödinger-Poisson system for computing the force field

Efficient GPU implementation of a Boltzmann‑Schrödinger‑Poisson solver for the simulation of nanoscale DG MOSFETs

81–102, 2019) describes an efficient and accurate solver for nanoscale DG MOSFETs through a deterministic Boltzmann-Schrödinger-Poisson model with seven electron–phonon scattering mechanisms on a hybrid parallel CPU/GPU platform. The transport computational phase, i.e. the time integration of the Boltzmann equations, was ported to the GPU using CUDA extensions, but the computation of the system’s eigenstates, i.e. the solution of the Schrödinger-Poisson block, was parallelized only using OpenMP due to its complexity. This work fills the gap by describing a port to GPU for the solver of the Schrödinger-Poisson block. This new proposal implements on GPU a Scheduled Relaxation Jacobi method to solve the sparse linear systems which arise in the 2D Poisson equation. The 1D Schrödinger equation is solved on GPU by adapting a multi-section iteration and the Newton-Raphson algorithm to approximate the energy levels, and the Inverse Power Iterative Method is used to approximate the wave vectors. We want to stress that this solver for the Schrödinger-Poisson block can be thought as a module independent of the transport phase (Boltzmann) and can be used for solvers using different levels of description for the electrons; therefore, it is of particular interest because it can be adapted to other macroscopic, hence faster, solvers for confined devices exploited at industrial level.Project PID2020-117846GB-I00 funded by the Spanish Ministerio de Ciencia e InnovaciónProject A-TIC-344-UGR20 funded by European Regional Development Fund

Discontinuous Galerkin semi-Lagrangian method for Vlasov-Poisson

International audienceWe present a discontinuous Galerkin scheme for the numerical approximation of the one-dimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semi-Lagrangian method to emphasize the good behavior of this scheme when applied to Vlasov-Poisson test cases

Numerical schemes of diffusion asymptotics and moment closures for kinetic equations

We investigate different models that are intended to describe the small mean free path regime of a kinetic equation, a particular attention being paid to the moment closure by entropy minimization. We introduce a specific asymptotic-induced numerical strategy which is able to treat the stiff terms of the asymptotic diffusive regime. We evaluate on numerics the performances of the method and the abilities of the reduced models to capture the main features of the full kinetic equation

Hybrid CUDA-OpenMP parallel implementation of a deterministic solver for ultra-short DG-MOSFETs

International audienceThe simulation of nanoscale 2D DG-MOSFETs and similar semiconductor devices through a de-terministic and accurate model can be very useful for the industry but is particularly costly from the computational point of view. In this paper, we develop a hybrid parallel solver: the computing phases which corresponds to the simulation of the Boltzmann transport equation in the longitudinal dimension are performed on the GPU, while that phases related to the modelling of the electrons as waves in the transversal dimension are computed on the multi-core CPUs by using OpenMP. We have adapted the most costly computing phases to GPU in an efficient manner, achieving high performance and reducing drastically the simulation time. We give details about the parallel-design strategy and show the performance results

A numerical study of attraction/repulsion collective behavior models: 3D particle analyses and 1D kinetic simulations

39pInternational audienceWe study at particle and kinetic level a collective behavior model based on three phenomena: self-propulsion, friction (Rayleigh effect) and an attractive/repulsive (Morse) potential rescaled so that the total mass of the system remains constant independently of the number of particles N . In the first part of the paper, we introduce the particle model: the agents are numbered and described by their position and velocity. We iden- tify five parameters that govern the possible asymptotic states for this system (clumps, spheres, dispersion, mills, rigid-body rotation, flocks) and perform a numerical analysis on the 3D setting. Then, in the second part of the paper, we describe the kinetic system derived as the limit from the particle model as N tends to infinity; we propose, in 1D, a numerical scheme for the simulations, and perform a numerical analysis devoted to trying to recover asymptotically patterns similar to those emerging for the equivalent particle systems, when particles originally evolved on a circle

Numerical schemes of diffusion asymptotics and moment closures for kinetic equations

We investigate different models that are intended to describe the small mean free path regime of a kinetic equation, a particular attention being paid to the moment closure by entropy minimization. We introduce a specific asymptotic-induced numerical strategy which is able to treat the stiff terms of the asymptotic diffusive regime. We evaluate on numerics the performances of the method and the abilities of the reduced models to capture the main features of the full kinetic equation

Numerical Schemes of Diffusion Asymptotics and Moment Closures for Kinetic Equations

International audienceWe investigate different models that are intended to describe the small mean free path regime of a kinetic equation, a particular attention being paid to the moment closure by entropy minimization. We introduce a specific asymptotic-induced numerical strategy which is able to treat the stiff terms of the asymptotic diffusive regime. We evaluate on numerics the performances of the method and the abilities of the reduced models to capture the main features of the full kinetic equation