Eric Sonnendrucker | Technische Universität München (original) (raw)

Papers by Eric Sonnendrucker

Computer Physics Communications, 2004

We introduce here a new method for the numerical resolution of the Vlasov equation on a phase spa... more We introduce here a new method for the numerical resolution of the Vlasov equation on a phase space grid using an adaptive semi-Lagrangian method. The adaptivity is obtained through a multiresolution analysis which enables to keep or remove grid points from the simulation depending on the size of their associated coefficients in a multiresolution expansion. The adaptive algorithm consists in three steps : prediction of the active grid points at the next time step, the usual semi-Lagrangian algorithm, and a compression allowing to eliminate unnecessary grid points.

Fluid-Acoustic Coupling and Wave Propagation

Different strategies regarding the simulation of sound generation and propagation are explored. A... more Different strategies regarding the simulation of sound generation and propagation are explored. A hydrodynamic/acoustic splitting method for computational aeroacoustics in low Mach number flows with variable density, temperature gradients and heat conduction is described. The resulting equations can be formulated as linearized Euler equations plus source terms and reduce to the linear acoustic wave equation, if convection speeds can be

Numerische Mathematik, 2007

We study the two-scale asymptotics for a charged beam under the action of a rapidly oscil- lating... more We study the two-scale asymptotics for a charged beam under the action of a rapidly oscil- lating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution. can be performed either in

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000

On considère le mouvement d'un gaz de particules chargées soumisesà un chamṕ electrique donné et ... more On considère le mouvement d'un gaz de particules chargées soumisesà un chamṕ electrique donné et un champ magnétique uniforme très grand. Onétudie la limite faible de la fonction de distribution des particules dans le cas où l'échelle d'observation est de l'ordre du rayon de Larmor.

An exponential integrator for a highly oscillatory vlasov equation

Discrete and Continuous Dynamical Systems - Series S, 2014

ABSTRACT In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system dependin... more ABSTRACT In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an exponential time differencing approach, the scheme is able to use large time steps with respect to the typical size of the fast oscillations of the solution.

Journal de Mathématiques Pures et Appliquées, 2001

In magnetic fusion, a plasma is constrained by a very large magnetic field, which introduces a ne... more In magnetic fusion, a plasma is constrained by a very large magnetic field, which introduces a new time scale, namely the period of rotation of the particles around the magnetic field lines. This new time scale is very restrictive for numerical simulation, which makes it important to find approximate models of the Vlasov-Poisson equation where it is removed. The gyrokinetic models aim at exactly this. Such models have been derived in the physics literature for several decades now, but only in the last few years there have been rigorous mathematical derivations. Those have only addressed the limit when the magnetic field becomes infinite. We consider here the Vlasov equation in different physical regimes for which small parameters are identified, and cast the obtained dimensionless equations into the abstract framework of a singularly perturbed convection equation. In this framework we derive an asymptotic expansion with respect to the small parameter of its solution, and characterize the terms of the expansion. The proofs make use of Allaire's two-scale convergence.  2001 Éditions scientifiques et médicales Elsevier SAS

Motivated by the difficulty arising in the numerical simulation of the movement of charged partic... more Motivated by the difficulty arising in the numerical simulation of the movement of charged particles in presence of a large external magnetic field, which adds an additional time scale and thus imposes to use a much smaller time step, we perform in this paper a homogenization of the Vlasov equation and the Vlasov-Poisson system which yield approximate equations describing the mean behavior of the particles. The convergence proof is based on the two scale convergence tools introduced by N'Guetseng and Allaire. We also consider the case where, in addition to the magnetic field, a large external electric field orthogonal to the magnetic field and of the same magnitude is applied.

A new scheme for solving the Vlasov equation using a phase space grid is proposed. The algorithm ... more A new scheme for solving the Vlasov equation using a phase space grid is proposed. The algorithm is based on the conservation of the flux of particles, and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity. Several numerical results are presented in two-and four-dimensional phase space and the scheme is compared with the semiLagrangian method. This method is almost as accurate as the semi-Lagrangian one, and the local reconstruction technique is well suited for parallel computation.

We prove the global existence of weak solutions to the Vlasov-Darwin system in R 3 for small init... more We prove the global existence of weak solutions to the Vlasov-Darwin system in R 3 for small initial data. The Vlasov-Darwin system is an approximation of the Vlasov-Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov-Maxwell system, the total energy conservation does not provide an L 2 -bound on the transverse part of the electric ÿeld. This di culty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We ÿnally investigate the convergence of the Vlasov-Darwin system towards the Vlasov-Poisson system.

A European Infrastructure for Fusion Simulations

2010 18th Euromicro Conference on Parallel, Distributed and Network-based Processing, 2010

Abstract The Integrated Tokamak Modelling Task Force (ITM-TF) is developing an infrastructure whe... more Abstract The Integrated Tokamak Modelling Task Force (ITM-TF) is developing an infrastructure where the validation needs, as being formulated in terms of multi-device data access and detailed physics comparisons aiming for inclusion of synthetic diagnostics in the simulation chain, are key components. A device independent approach to data transport and a standardized approach to data management (data structures, naming, and access) is being developed in order to allow cross validation between different fusion devices using ...

Gysela 5D, a gyrokinetic semilagrangian parallel code

We are interested in solving the Vlasov equation used to describe collective eects in plasmas. Th... more We are interested in solving the Vlasov equation used to describe collective eects in plasmas. This non- linear partial dierential equation coupled with a field equation describes the time evolution of the particle dis- tribution in phase space. In this paper, we focuses on a recently developed 5D parallel numerical application dedicated to gyrokinetic simulation of Takamak sys- tems. We got a multi-level parallelized application that achieves a good scalability up to hundreds of processors on a cluster of SMP nodes.

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2001

The methods and codes employed in the US Heavy Ion Fusion program to simulate the beams in an Int... more The methods and codes employed in the US Heavy Ion Fusion program to simulate the beams in an Integrated Research Experiments (IRE) facility and a fusion driver are presented in overview. A new family of models incorporating accelerating module impedance, multi-beam, and self-magnetic effects is described, and initial WARP3D particle simulations of beams using these models are presented. Finally, plans for streamlining the machine-design simulation sequence, and for simulating beam dynamics from the source to the target in a consistent and comprehensive manner, are described. #

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2001

We introduce the semi-Lagrangian Vlasov method, which computes the distribution function of the p... more We introduce the semi-Lagrangian Vlasov method, which computes the distribution function of the particles on a grid in phase space, to beam propagation in a uniform focusing channel. With this new tool, we study halo formation in a mismatched thermal beam, and the evolution of an initial semi-Gaussian beam. For the latter problem comparisons are made with the Particle-In-Cell code WARP. #

Journal of Computational Physics, 1999

The numerical resolution of kinetic equations and in particular of Vlasov type equations is most ... more The numerical resolution of kinetic equations and in particular of Vlasov type equations is most of the time performed using PIC (Particle In Cell) methods which consist in describing the time evolution of the equation through a finite number of particles which follow the characteristic curves of the equation, the interaction with the external and self consistent fields being resolved using a grid. Another approach consists in computing directly the distribution function on a grid by following the characteristics backward in time for one time step and interpolating the value at the feet of the characteristics using the grid points values of the distribution function at the previous time step. In this report we introduce this last method and its use for different types of Vlasov equations.

Computer Physics Communications, 2004

Computer Physics Communications, 2004

We introduce here a new method for the numerical resolution of the Vlasov equation on a phase spa... more We introduce here a new method for the numerical resolution of the Vlasov equation on a phase space grid using an adaptive semi-Lagrangian method. The adaptivity is obtained through a multiresolution analysis which enables to keep or remove grid points from the simulation depending on the size of their associated coefficients in a multiresolution expansion. The adaptive algorithm consists in three steps : prediction of the active grid points at the next time step, the usual semi-Lagrangian algorithm, and a compression allowing to eliminate unnecessary grid points.

Fluid-Acoustic Coupling and Wave Propagation

Different strategies regarding the simulation of sound generation and propagation are explored. A... more Different strategies regarding the simulation of sound generation and propagation are explored. A hydrodynamic/acoustic splitting method for computational aeroacoustics in low Mach number flows with variable density, temperature gradients and heat conduction is described. The resulting equations can be formulated as linearized Euler equations plus source terms and reduce to the linear acoustic wave equation, if convection speeds can be

Numerische Mathematik, 2007

We study the two-scale asymptotics for a charged beam under the action of a rapidly oscil- lating... more We study the two-scale asymptotics for a charged beam under the action of a rapidly oscil- lating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution. can be performed either in

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000

On considère le mouvement d'un gaz de particules chargées soumisesà un chamṕ electrique donné et ... more On considère le mouvement d'un gaz de particules chargées soumisesà un chamṕ electrique donné et un champ magnétique uniforme très grand. Onétudie la limite faible de la fonction de distribution des particules dans le cas où l'échelle d'observation est de l'ordre du rayon de Larmor.

An exponential integrator for a highly oscillatory vlasov equation

Discrete and Continuous Dynamical Systems - Series S, 2014

ABSTRACT In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system dependin... more ABSTRACT In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an exponential time differencing approach, the scheme is able to use large time steps with respect to the typical size of the fast oscillations of the solution.

Journal de Mathématiques Pures et Appliquées, 2001

In magnetic fusion, a plasma is constrained by a very large magnetic field, which introduces a ne... more In magnetic fusion, a plasma is constrained by a very large magnetic field, which introduces a new time scale, namely the period of rotation of the particles around the magnetic field lines. This new time scale is very restrictive for numerical simulation, which makes it important to find approximate models of the Vlasov-Poisson equation where it is removed. The gyrokinetic models aim at exactly this. Such models have been derived in the physics literature for several decades now, but only in the last few years there have been rigorous mathematical derivations. Those have only addressed the limit when the magnetic field becomes infinite. We consider here the Vlasov equation in different physical regimes for which small parameters are identified, and cast the obtained dimensionless equations into the abstract framework of a singularly perturbed convection equation. In this framework we derive an asymptotic expansion with respect to the small parameter of its solution, and characterize the terms of the expansion. The proofs make use of Allaire's two-scale convergence.  2001 Éditions scientifiques et médicales Elsevier SAS

Motivated by the difficulty arising in the numerical simulation of the movement of charged partic... more Motivated by the difficulty arising in the numerical simulation of the movement of charged particles in presence of a large external magnetic field, which adds an additional time scale and thus imposes to use a much smaller time step, we perform in this paper a homogenization of the Vlasov equation and the Vlasov-Poisson system which yield approximate equations describing the mean behavior of the particles. The convergence proof is based on the two scale convergence tools introduced by N'Guetseng and Allaire. We also consider the case where, in addition to the magnetic field, a large external electric field orthogonal to the magnetic field and of the same magnitude is applied.

A new scheme for solving the Vlasov equation using a phase space grid is proposed. The algorithm ... more A new scheme for solving the Vlasov equation using a phase space grid is proposed. The algorithm is based on the conservation of the flux of particles, and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity. Several numerical results are presented in two-and four-dimensional phase space and the scheme is compared with the semiLagrangian method. This method is almost as accurate as the semi-Lagrangian one, and the local reconstruction technique is well suited for parallel computation.

We prove the global existence of weak solutions to the Vlasov-Darwin system in R 3 for small init... more We prove the global existence of weak solutions to the Vlasov-Darwin system in R 3 for small initial data. The Vlasov-Darwin system is an approximation of the Vlasov-Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov-Maxwell system, the total energy conservation does not provide an L 2 -bound on the transverse part of the electric ÿeld. This di culty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We ÿnally investigate the convergence of the Vlasov-Darwin system towards the Vlasov-Poisson system.

A European Infrastructure for Fusion Simulations

2010 18th Euromicro Conference on Parallel, Distributed and Network-based Processing, 2010

Abstract The Integrated Tokamak Modelling Task Force (ITM-TF) is developing an infrastructure whe... more Abstract The Integrated Tokamak Modelling Task Force (ITM-TF) is developing an infrastructure where the validation needs, as being formulated in terms of multi-device data access and detailed physics comparisons aiming for inclusion of synthetic diagnostics in the simulation chain, are key components. A device independent approach to data transport and a standardized approach to data management (data structures, naming, and access) is being developed in order to allow cross validation between different fusion devices using ...

Gysela 5D, a gyrokinetic semilagrangian parallel code

We are interested in solving the Vlasov equation used to describe collective eects in plasmas. Th... more We are interested in solving the Vlasov equation used to describe collective eects in plasmas. This non- linear partial dierential equation coupled with a field equation describes the time evolution of the particle dis- tribution in phase space. In this paper, we focuses on a recently developed 5D parallel numerical application dedicated to gyrokinetic simulation of Takamak sys- tems. We got a multi-level parallelized application that achieves a good scalability up to hundreds of processors on a cluster of SMP nodes.

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2001

The methods and codes employed in the US Heavy Ion Fusion program to simulate the beams in an Int... more The methods and codes employed in the US Heavy Ion Fusion program to simulate the beams in an Integrated Research Experiments (IRE) facility and a fusion driver are presented in overview. A new family of models incorporating accelerating module impedance, multi-beam, and self-magnetic effects is described, and initial WARP3D particle simulations of beams using these models are presented. Finally, plans for streamlining the machine-design simulation sequence, and for simulating beam dynamics from the source to the target in a consistent and comprehensive manner, are described. #

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2001

We introduce the semi-Lagrangian Vlasov method, which computes the distribution function of the p... more We introduce the semi-Lagrangian Vlasov method, which computes the distribution function of the particles on a grid in phase space, to beam propagation in a uniform focusing channel. With this new tool, we study halo formation in a mismatched thermal beam, and the evolution of an initial semi-Gaussian beam. For the latter problem comparisons are made with the Particle-In-Cell code WARP. #

Journal of Computational Physics, 1999

The numerical resolution of kinetic equations and in particular of Vlasov type equations is most ... more The numerical resolution of kinetic equations and in particular of Vlasov type equations is most of the time performed using PIC (Particle In Cell) methods which consist in describing the time evolution of the equation through a finite number of particles which follow the characteristic curves of the equation, the interaction with the external and self consistent fields being resolved using a grid. Another approach consists in computing directly the distribution function on a grid by following the characteristics backward in time for one time step and interpolating the value at the feet of the characteristics using the grid points values of the distribution function at the previous time step. In this report we introduce this last method and its use for different types of Vlasov equations.

Computer Physics Communications, 2004