Comparison of two Eulerian solvers for the four-dimensional Vlasov equation: Part I

This paper presents two methods for solving the four-dimensional Vlasov equation on a grid of the phase space. The two methods are based on the semi-Lagrangian method which consists in computing the distribution function at each grid point by following the characteristic curves ending there. The first method reconstructs the distribution function using local splines which are well suited for a parallel implementation. The second method is adaptive using wavelets interpolation: only a subset of the grid points are conserved to manage data locality. Numerical results are presented in the second part.     

Comparison of two Eulerian solvers for the four-dimensional Vlasov equation: Part II

In this second part, we carry out a numerical comparison between two Vlasov solvers, which solve directly the Vlasov equation on a grid of the phase space. The two methods are based on the semi-Lagrangian method as presented in Part I: the first one (LOSS, local splines simulator) uses a uniform mesh of the phase space whereas the second one (OBI, ondelets based interpolation) is an adaptive method. The numerical comparisons are performed by solving the four-dimensional Vlasov equation for some classical problems of plasma and beam physics. We shall also investigate the speedup and the CPU time as well as the compression rate of the adaptive method which are important features because of the size of the problems.     

Global existence for the Vlasov–Darwin system in ℝ3 for small initial data

We prove the global existence of weak solutions to the Vlasov–Darwin system in R3 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 L2‐bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov–Darwin system towards the Vlasov–Poisson system. Copyright © 2003 John Wiley & Sons, Ltd.     

On the dynamical reduction of the Vlasov equation

The elimination of a fast-time scale from the Vlasov equation by Lie-transform methods is an important step in deriving a reduced Vlasov equation such as the drift-kinetic Vlasov equation or the gyrokinetic Vlasov equation. It is shown here that this dynamical reduction also leads to the introduction of polarization and magnetization effects in the reduced Maxwell equations, which ensure that the reduced Vlasov–Maxwell equations possess an exact energy–momentum conservation law.     

Equilibrium of stellar dynamical systems in the context of the Vlasov–Poisson model

This short review is devoted to the problem of the equilibrium of stellar dynamical systems in the context of the Vlasov–Poisson model. In a first part we will review some classical problems posed by the application of the Vlasov–Poisson model to the astrophysical systems like globular clusters or galaxies. In a second part we will recall some recent numerical results which may give us some quantitative hints about the equilibrium state associated to those systems.     

A hybrid level-set/moving-mesh interface tracking method

An approach for combining Arbitrary–Lagrangian–Eulerian (ALE) moving-mesh and level-set interface tracking methods is presented that allows the two methods to be used in different spatial regions and coupled across the region boundaries. The coupling allows interface shapes to be convected from the ALE method to the level-set method and vice-versa across the ALE/level-set boundary. The motivation for this is to allow high-order ALE methods to represent interface motion in regions where there is no topology change, and the level-set function to be used in regions where topology change occurs. The coupling method is based on the characteristic directions of information propagation and can be implemented in any geometrical configuration. In addition, an iterative method for the hybrid formulation has been developed that can be combined with pre-existing solution methods. Tests of a propagating interface in a uniform flow show that the hybrid approach provides accuracy equivalent to what one is able to obtain with either of the methods individually.     

Eulerian polynomials and B-splines

An interrelationship between Eulerian polynomials, Eulerian fractions and Euler–Frobe nius polynomials, Euler–Frobenius fractions, and B-splines is presented. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed.     

Fischer decomposition of the space of entire functions for the convolution operator

It is known that any function in a Hilbert Bargmann–Fock space can be represented as the sum of a solution of a given homogeneous differential equation with constant coefficients and a function being a multiple of the characteristic function of this equation with conjugate coefficients. In the paper, a decomposition of the space of entire functions of one complex variable with the topology of uniform convergence on compact sets for the convolution operator is presented. As a corollary, a solution of the de la Vallée Poussin interpolation problem for the convolution operator with interpolation points at the zeros of the characteristic function with conjugate coefficient is obtained.     

Runge–Kutta characteristic methods for first-order linear hyperbolic equations

We develop two Runge–Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian–Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov–Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge–Kutta methods developed in this article. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 617–661, 1997     

On the Eulerian–Lagrangian formulation of balance equations in porous media

The article presents a general approach to modeling the transport of extensive quantities in the case of flow of multiple multicomponent fluid phases in a deformable porous medium domain under nonisothermal conditions. The models are written in a modified Eulerian–Lagrangian formulation. In this modified formulation, the material derivatives are written in terms of modified velocities. These are the velocities at which the various phase and component variables propagate in the domain, along their respective characteristic curves. It is shown that these velocities depend on the heterogeneity of various solid matrix and fluid properties. The advantage of this formulation, with respect to the usually employed Eulerian one, is that numerical dispersion, associated with the advective fluxes of extensive quantities, are eliminated. The methodology presented in the article shows how the Eulerian–Lagrangian formulation is written in terms of the relatively small number of primary variables of a transport problem. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 505–530, 1997     

A class of methods for diffusion-convection equations

A class of method with a free parameter(s) is considered for the solution of the diffusion–convection equation. This class is obtained using interpolation function approach and so, some classic methods appears when s take specific values. In order to obtain a non–oscillatory numerical solution some restrictions will be placed on s. Choosing two suitable methods of that class, an algorithm will be established and computational requirements will be considered     

A CFL‐free explicit characteristic interior penalty scheme for linear advection‐reaction equations

We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L² norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The classical limit of the relativistic Vlasov–Maxwell system in two space dimensions

The motion of a collisionless plasma is modelled by the Vlasov–Maxwell system. In this paper, solutions of the relativistic Vlasov–Maxwell system are considered in two space dimensions. The speed of light, c, appears as a parameter in the system. With representations of the electric and magnetic fields, conditions are established under which solutions of the relativistic Vlasov–Maxwell system converge pointwise to solutions of the non‐relativistic Vlasov–Poisson system as c tends to infinity, at the asymptotic rate of 1/c. Copyright © 2004 John Wiley & Sons, Ltd.     

Vlasov equilibria: Varying the temperature or the density distributions

Stationary selfconsistent solutions of the Vlasov–Maxwell system in a magnetized inhomogeneous plasma (so called Vlasov equilibria) provide the natural starting point for the investigation of plasma stability and of the nonlinear development of plasma instabilities in collisionless or weakly collisional regimes. In view of the different mechanisms that drive these instabilities, we discuss Vlasov equilibria with both density and temperature gradients.     

An Eulerian‐Lagrangian control volume scheme for two‐dimensional unsteady advection‐diffusion problems

We develop a mass conservative Eulerian‐Lagrangian control volume scheme (ELCVS) for the solution of the transient advection‐diffusion equations in two space dimensions. This method uses finite volume test functions over the space‐time domain defined by the characteristics within the framework of the class of Eulerian‐Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight‐node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well‐received classes of numerical methods for the equations considered. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Cauchy problem on a characteristic cone for the Einstein–Vlasov system: (I) The initial data constraints

In this paper, one considers a Cauchy problem with data on a characteristic cone for the Einstein–Vlasov system in temporal gauge. One highlights gauge-dependent constraints that, supplemented by the standard constraints i.e. the Hamiltonian and the momentum constraints, define the full set of constraints for the considered setting. One studies their global resolution from a suitable choice of some free data, the behavior of the deduced initial data at the vertex of the cone, and the preservation of the gauge.     

Convergence of Solutions of the Landau Equations in the Collisionless Limit

This paper addresses the question: when the frequency of collisions vanishes, will the solutions of the Landau system converge to the solutions of the Vlasov system? We give a positive answer to the question for solutions satisfying certain regularity conditions. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 679–688 (1997).     

A locally conservative Eulerian‐Lagrangian control‐volume method for transient advection‐diffusion equations

Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection‐diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian‐Lagrangian control‐volume method (ELCVM) for transient advection‐diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well‐regarded Eulerian‐Lagrangian methods, which were previously shown to be very competitive with many well‐perceived methods. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Discrete Orthogonality of the Malmquist Takenaka System of the Upper Half Plane and Rational Interpolation

In the representation of non periodic signals the use of special rational orthogonal systems is more efficient. One of these bases is the Malmquist–Takenaka system for the upper half plane. We will prove the discrete orthogonality of this system. Based on the discretization we introduce a new rational interpolation operator and we will study the properties of this operator. A finite sampling theorem for a special subset of non periodic analytic signals will be presented.