An Asymptotic-Preserving and Energy-Conserving Particle-In-Cell Method for Vlasov-Maxwell Equations

In this paper, we develop an asymptotic-preserving and energy-conserving (APEC) Particle-In-Cell (PIC) algorithm for the Vlasov-Maxwell system. This algorithm not only guarantees that the asymptotic limiting of the discrete scheme is a consistent and stable discretization of the quasi-neutral limit of the continuous model, but also preserves Gauss's law and energy conservation at the same time, thus it is promising to provide stable simulations of complex plasma systems even in the quasi-neutral regime. The key ingredients for achieving these properties include the generalized Ohm's law for electric field such that the asymptotic-preserving discretization can be achieved, and a proper decomposition of the effects of the electromagnetic fields such that a Lagrange multiplier method can be appropriately employed for correcting the kinetic energy. We investigate the performance of the APEC method with three benchmark tests in one dimension, including the linear Landau damping, the bump-on-tail problem and the two-stream instability. Detailed comparisons are conducted by including the results from the classical explicit leapfrog and the previously developed asymptotic-preserving PIC schemes. Our numerical experiments show that the proposed APEC scheme can give accurate and stable simulations both kinetic and quasi-neutral regimes, demonstrating the attractive properties of the method crossing scales.Comment: 21 pages, 30 figure

Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living Rev. Rel. 5 (2002)

Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well

Asymptotic-Preserving methods and multiscale models for plasma physics

The purpose of the present paper is to provide an overview of As ymptotic- Preserving methods for multiscale plasma simulations by ad dressing three sin- gular perturbation problems. First, the quasi-neutral lim it of fluid and kinetic models is investigated in the framework of non magnetized as well as magne- tized plasmas. Second, the drift limit for fluid description s of thermal plasmas under large magnetic fields is addressed. Finally efficient nu merical resolutions of anisotropic elliptic or diffusion equations arising in ma gnetized plasma simu- lation are reviewed

The Vlasov-Poisson system with radiation damping

We set up and analyze a model of radiation damping within the framework of continuum mechanics, inspired by a model of post-Newtonian hydrodynamics due to Blanchet, Damour and Schaefer. In order to simplify the problem as much as possible we replace the gravitational field by the electromagnetic field and the fluid by kinetic theory. We prove that the resulting system has a well-posed Cauchy problem globally in time for general initial data and in all solutions the fields decay to zero at late times. In particular, this means that the model is free from the runaway solutions which frequently occur in descriptions of radiation reaction

Hamiltonian formulation for the classical EM radiation-reaction problem: application to the kinetic theory for relativistic collisionless plasmas

A notorious difficulty in the covariant dynamics of classical charged particles subject to non-local electromagnetic (EM) interactions arising in the EM radiation-reaction (RR) phenomena is due to the definition of the related non-local Lagrangian and Hamiltonian systems. The lack of a standard Lagrangian/Hamiltonian formulation in the customary asymptotic approximation for the RR equation may inhibit the construction of consistent kinetic and fluid theories. In this paper the issue is investigated in the framework of Special Relativity. It is shown that, for finite-size spherically-symmetric classical charged particles, non-perturbative Lagrangian and Hamiltonian formulations in standard form can be obtained, which describe particle dynamics in the presence of the exact EM RR self-force. As a remarkable consequence, based on axiomatic formulation of classical statistical mechanics, the covariant kinetic theory for systems of charged particles subject to the EM RR self-force is formulated in Hamiltonian form. A fundamental feature is that the non-local effects enter the kinetic equation only through the retarded particle 4-position, which permits the construction of the related non-local fluid equations. In particular, the moment equations obtained in this way do not contain higher-order moments, allowing as a consequence the adoption of standard closure conditions. A remarkable aspect of the theory concerns the short delay-time asymptotic expansions. Here it is shown that two possible expansions are permitted. Both can be implemented for the single-particle dynamics as well as for the corresponding kinetic and fluid treatments. In the last case, they are performed a posteriori on the relevant moment equations obtained after integration of the kinetic equation over the velocity space. Comparisons with literature are pointed out