a new code for electrostatic simulation by numerical integration of the vlasov and ampere equations using maccormack s method abstract we present a new simulation code for electrostatic waves in one dimension which uses the vlasov equation to integrate the distribution function and ampere s equation to integrate the electric field forward in time previous vlasov codes used the vlasov and poisson equations using ampere s equation has two advantages first boundary conditions do not have to be set on the electric field second it forms a logical basis for an electromagnetic code since the time integration of the electric and magnetic fields is treated in a similar way maccormack s method is used to integrate the vlasov equation which was found to be easy to implement and reliable a stability analysis is presented for the maccormack scheme applied to the vlasov equation conditions for stability are more stringent than the simple courant friedrich s lewy cfl conditions for the spatial and velocity grids we provide a simple linear function which when combined with the cfl conditions should ensure stability simulation results for landau damping are in excellent agreement with numerical solutions of the linear dispersion relation for a wide range of wavelengths the code is also able to retain phase memory as demonstrated by the recurrence effect and reproduce the effects of particle trapping the use of ampere s equation enables standing and traveling waves to be produced depending on whether the current is zero or non zero respectively in simulations where the initial current is non zero and maxwell s equations are satisfied initially additional standing waves may be set up which could be important when the coupling of wave fields from a transmitter to a plasma is considered

high order maximum principle preserving semi lagrangian finite difference weno schemes for the vlasov equation in this paper we propose the parametrized maximum principle preserving mpp flux limiter originally developed in 37 to the semi lagrangian finite difference weighted essentially non oscillatory scheme for solving the vlasov equation the mpp flux limiter is proved to maintain up to fourth order accuracy for the semi lagrangian finite difference scheme without any time step restriction numerical studies on the vlasov poisson system demonstrate the performance of the proposed method and its ability in preserving the positivity of the probability distribution function while maintaining the high order accuracy

power law decay and the ergodic nonergodic transition in simple fluids it is well known that mode coupling theory mct leads to a two step power law time decay in dense simple fluids we show that much of the mathematical machinery used in the mct analysis can be taken over to the analysis of the systematic theory developed in the fundamental theory of statistical particle dynamics mazenko in phys rev e 81 6 061102 2010 we show how the power law exponents can be computed in the second order approximation where we treat hard sphere fluids with statics described by the percus yevick solution

semi lagrangian schemes for the vlasov equation on an unstructured mesh of phase space a new scheme for solving the vlasov equation using an unstructured mesh for the phase space is proposed the algorithm is based on the semi lagrangian method which exploits the fact that the distribution function is constant along the characteristic curves we use different local interpolation operators to reconstruct the distribution function f some of which need the knowledge of the gradient of f we can use limiter coefficients to maintain the positivity and the l bound of f and optimize these coefficients to ensure the conservation of the l1 norm that is to say the mass by solving a linear programming problem several numerical results are presented in two and three axisymmetric case dimensional phase space the local interpolation technique is well suited for parallel computation

conservative numerical schemes for the vlasov equation abstract 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 semilagrangian one and the local reconstruction technique is well suited for parallel computation

coldice a parallel vlasov poisson solver using moving adaptive simplicial tessellation abstract resolving numerically vlasov poisson equations for initially cold systems can be reduced to following the evolution of a three dimensional sheet evolving in six dimensional phase space we describe a public parallel numerical algorithm consisting in representing the phase space sheet with a conforming self adaptive simplicial tessellation of which the vertices follow the lagrangian equations of motion the algorithm is implemented both in six and four dimensional phase space refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase space sheet at second order relying on additional tracers created when needed at runtime in order to preserve in the best way the hamiltonian nature of the system refinement is anisotropic and constrained by measurements of local poincare invariants resolution of poisson equation is performed using the fast fourier method on a regular rectangular grid similarly to particle in cells codes to compute the density projected onto this grid the intersection of the tessellation and the grid is calculated using the method of franklin and kankanhalli 65 66 67 generalised to linear order as preliminary tests of the code we study in four dimensional phase space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves we also perform a warm dark matter simulation in six dimensional phase space that we use to check the parallel scaling of the code

nested iteration and first order system least squares for incompressible resistive magnetohydrodynamics this paper develops a nested iteration algorithm to solve time dependent nonlinear systems of partial differential equations for each time step newton s method is used to form approximate solutions from a sequence of nested spaces where the resolution of the approximations increases as the algorithm progresses nested iteration results in most of the iterations being performed on coarser grids where minimal work is needed to reduce error to the level of discretization error the approximate solution on a given coarse grid is interpolated to a refined grid and is used as an initial guess for the problem posed there the approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem due to the rapid convergence of newton s method near a solution the paper develops an algorithm that attempts to optimize accuracy per computational cost on each grid so that essentially no unnecessary work is done on any grid the nested iteration algorithm is then applied to a reduced two dimensional model of the incompressible resistive magnetohydrodynamic mhd equations using this algorithm on the mhd equations in the context of a first order system least squares finite element discretization and algebraic multigrid to solve the linearized systems instabilities in a model tokamak fusion reactor are simulated numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30 80 fine grid relaxation sweeps per time step

block structured adaptive mesh refinement algorithms for vlasov simulation direct discretization of continuum kinetic equations like the vlasov equation are under utilized because the distribution function generally exists in a high dimensional 3d space and computational cost increases geometrically with dimension we propose to use high order finite volume techniques with block structured adaptive mesh refinement amr to reduce the computational cost the primary complication comes from a solution state comprised of variables of different dimensions we develop the algorithms required to extend standard single dimension block structured amr to the multi dimension case specifically algorithms for reduction and injection operations that transfer data between mesh hierarchies of different dimensions are explained in detail in addition modifications to the basic amr algorithm that enable the use of high order spatial and temporal discretizations are discussed preliminary results for a standard 1d 1v vlasov poisson test problem are presented results indicate that there is potential for significant savings for some classes of vlasov problems

umbral calculus and the theory of multispecies non ideal gases abstract relations in the cluster expansion theory of non ideal gases are presented using the recently developed formalism of umbral calculus this paper provides an overview of the mathematical theory and explicitly derives the virial expansion and relations between the partition function and reducible and irreducible cluster coefficients for single and multispecies non ideal gases it is shown that these results are obtained remarkably easily using the methods of umbral calculus

kinetic equations governing smoluchowski dynamics in equilibrium we continue our study of the statistical properties of particles in equilibrium obeying smoluchowski dynamics we show that the system is governed by a kinetic equation of the memory function form and that the memory function is given by one of the self energies available via perturbation theory as introduced in previous work we determine the memory function explicitly to second order in an expansion in a pseudopotential the method we use allows for a straightforward computation of corrections via a formal expansion and we therefore view it as an improvement over the conventional mode coupling theory mct formalism where it is not clear how to make systematic corrections in addition the formalism we have introduced is flexible enough to allow for a wide array of different approximation schemes including density expansions the convergence criteria for our formal series are not worked out here but the second order equation that we derive is promising in the sense that it leads to analytic and numerical results consistent with expectations from computer simulations of the hard sphere system in addition to replicating the desired features from conventional mct e g a two step decay these particular solutions will be discussed in forthcoming work

arbitrarily high order convected scheme solution of the vlasov poisson system abstract the convected scheme cs is a forward trajectory semi lagrangian method for solution of transport equations which has been most often applied to the kinetic description of plasmas and rarefied neutral gases in its simplest form the cs propagates the solution forward in time by advecting the so called moving cells along their characteristic trajectories and by remapping them on the mesh at the end of the time step the cs is conservative positivity preserving simple to implement and it is not subject to time step restriction to maintain stability recently guclu and hitchon 2012 1 a new methodology was introduced for reducing numerical diffusion based on a modified equation analysis the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping while the spatial accuracy was increased from 2nd to 4th order the new scheme retained the important properties of the original method and was shown to be extremely simple and efficient for constant advection problems here the cs is applied to the solution of the vlasov poisson system which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal coulomb interactions the vlasov equation is split into two constant advection equations one in configuration space and one in velocity space and high order time accuracy is achieved by proper composition of the operators the splitting procedure enables us to use the constant advection solver which we extend to arbitrarily high order of accuracy in time and space a new improved procedure is given which makes the calculation of the corrections straightforward focusing on periodic domains we describe a spectrally accurate scheme based on the fast fourier transform the proposed implementation is strictly conservative and positivity preserving the ability to correctly reproduce the system dynamics as well as resolving small scale features in the solution is shown in classical 1d 1v test cases both in the linear and the non linear regimes

a new conservative unsplit method for the solution of the vlasov equation we have developed a new conservative method for solving the vlasov equation without using any splitting technique our goal is to maintain the positivity of the distribution function and to avoid un physical oscillations which might lead to numerical instabilities based on a finite volume conservative discretization of the conservative form of the vlasov equation we implemented a highly accurate second order upwind scheme in order to avoid un physical oscillations and their possible numerical instability we apply a flux limiter in the second order we validate our new vlasov solver by considering standard cases of one dimensional current driven ion acoustic instabilities solving a vlasov ampere set of equations

arbitrary order time accurate semi lagrangian spectral approximations of the vlasov poisson system abstract the vlasov poisson system modeling the evolution of non collisional plasmas in the electrostatic limit is approximated by a semi lagrangian technique spectral methods of periodic type are implemented through a collocation approach groups of particles are represented by the fourier lagrangian basis and evolve for a single timestep along a high order accurate representation of the local characteristic lines the time advancing technique is based on truncated taylor series that can be in principle of any order of accuracy a variant is obtained by coupling the phase space discretization with high order accurate backward differentiation formulas bdf at each timestep particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs thus these techniques combine the excellent features of spectral approximations with high order time integration the resulting method has excellent conservation properties indeed it can be proven that the total number of particles proportional to the total mass and charge is conserved up to the machine precision series of numerical experiments are performed in order to assess the real performance in particular comparisons with standard benchmarks are examined

degenerate poly cauchy polynomials in this paper we study several properties for the degenerate poly cauchy polynomials we present several explicit formulas and recurrence relations for these polynomials also we establish a connection between our polynomials and several known families of polynomials

novel identities involving genocchi numbers and polynomials arising from applications of umbral calculus abstract the aim of this paper is to deal with applications of umbral calculus on fermionic p adic integral on z p from those applications we derive some new identities on genocchi numbers and polynomials moreover a systemic study of the class of sheffer sequences on the generating function of genocchi polynomials are given

asymptotic preserving particle in cell method for the vlasov poisson system near quasineutrality this paper deals with the numerical resolution of the vlasov poisson system in a nearly quasineutral regime by particle in cell pic methods in this regime classical pic methods are subject to stability constraints on the time and space steps related to the small debye length and large plasma frequency here we propose an asymptotic preserving pic scheme which is not subjected to these limitations additionally when the plasma period and debye length are small compared to the time and space steps this method provides a consistent pic discretization of the quasineutral vlasov equation we perform several one dimensional numerical experiments which provide a solid validation of the method and its underlying concepts and compare the method with classical pic and direct implicit methods

field theoretic formulation of kinetic theory basic development we show how kinetic theory the statistics of classical particles obeying newtonian dynamics can be formulated as a field theory the field theory can be organized to produce a self consistent perturbation theory expansion in an effective interaction potential the need for a self consistent approach is suggested by our interest in investigating ergodic nonergodic transitions in dense fluids the formal structure we develop has been implemented in detail for the simpler case of smoluchowski dynamics one aspect of the approach is the identification of a core problem spanned by the variables the number density and b a response density in this paper we set up the perturbation theory expansion with explicit development at zeroth and first order we also determine all of the cumulants in the noninteracting limit among the core variables and b

a high order semi lagrangian discontinuous galerkin method for vlasov poisson simulations without operator splitting abstract in this paper we develop a high order semi lagrangian sl discontinuous galerkin dg method for nonlinear vlasov poisson vp simulations without operator splitting in particular we combine two recently developed novel techniques one is the high order non splitting sldg transport method cai et al 2017 4 and the other is the high order characteristics tracing technique proposed in qiu and russo 2017 29 the proposed method with up to third order accuracy in both space and time is locally mass conservative free of splitting error positivity preserving stable and robust for large time stepping size the sldg vp solver is applied to classic benchmark test problems such as landau damping and two stream instabilities for vp simulations efficiency and effectiveness of the proposed scheme is extensively tested tremendous cpu savings are shown by comparisons between the proposed sl dg scheme and the classical runge kutta dg method

an umbral calculus for polynomials characterizing u n tensor operators after the change of variables i i i and xi i 1 i i 1 we show that the invariant polynomials g n q i xi i 1 characterizing u n tensor operators p q q 0 0 become an integral linear combination of schur functions s in the symbol where denotes the difference of the two sets of variables 1 n and 1 n we obtain a similar result for the yet more general bisymmetric polynomials m g n q 1 n 1 m making use of properties of skew schur functions s and s we put together an umbral calculus for m g n q that is working entirely with polynomials we uniquely determine m g n q from m g n q 1 and combinatorial rules involving ferrers diagrams i e partitions provided that n 1 q this restriction does not interfere with writing the general case of m g n q as a linear combination of s as an application we deduce conjugation symmetry for n g n q from transposition symmetry by showing that these two symmetries are equivalent

a performance comparison of semi lagrangian discontinuous galerkin and spline based vlasov solvers in four dimensions abstract the purpose of the present paper is to compare two semi lagrangian methods in the context of the four dimensional vlasov poisson equation more specifically our goal is to compare the performance of the more recently developed semi lagrangian discontinuous galerkin scheme to cubic spline interpolation which is widely used in eulerian vlasov simulation to that end we perform simulations for nonlinear landau damping and a two stream instability and provide benchmarks for the selalib and sldg codes both on a workstation and using mpi on a cluster in addition we will present results for the graphic processing unit gpu implementation contained in sldg we find that the semi lagrangian discontinuous galerkin scheme shows a moderate improvement in run time for nonlinear landau damping and a substantial improvement for the two stream instability it should be emphasized that these results are markedly different from results obtained in the asymptotic regime which favor spline interpolation thus we conclude that the traditional approach of evaluating numerical methods is misleading even for short time simulations in addition the absence of any global communication in the semi lagrangian discontinuous galerkin method gives it a decisive advantage for scaling to more than 256 cores

fundamental theory of statistical particle dynamics we present a fundamental theory for the kinetics of systems of classical particles the theory represents a unification of kinetic theory brownian motion and field theory it is self consistent and is the dynamic generalization of the functional theory of fluids in equilibrium this gives one a powerful tool for investigating the existence of ergodic nonergodic transitions near the liquid glass transition

more on the umbral calculus with emphasis on the q umbral calculus abstract the interrelationship between distinct umbral calculi is studied these ideas are applied in particular to q umbral calculus which shows how andrews q theory relates to the q theory discussed in the previous paper by the author the q hermite polynomials and basic hypergeometric series are briefly discussed

a particle in cell method for the simulation of plasmas based on an unconditionally stable field solver abstract we propose a new particle in cell pic method for the simulation of plasmas based on a recently developed unconditionally stable solver for the wave equation this method is not subject to a cfl restriction limiting the ratio of the time step size to the spatial step size typical of explicit methods while maintaining computational cost and code complexity comparable to such explicit schemes we describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases and present the results of several standard test problems showing good agreement with theory with time step sizes much larger than allowed by typical cfl restrictions

a splitting fourier pseudospectral method for vlasov poisson fokker planck system abstract in this paper we propose a splitting fourier pseudospectral method for vlasov poisson fokker planck vpfp system which describes the motion of charged particles in plasma the numerical integration for the system is performed by employing the splitting method in time fourier galerkin method in space direction and fourier collocation method in phase direction respectively the algorithm has spectral accuracy in both space and phase directions and can be implemented efficiently with the fast fourier transform and technique of diagonalization respectively extensive numerical results in one dimensional phase space or 1 x 1 v from the proposed numerical method are shown and have proven the good agreement with the theory and previous studies numerical algorithm for the vpfp system in two dimensional phase space or 2 x 2 v has been summarized in appendix

a critical comparison of eulerian grid based vlasov solvers a common problem with direct vlasov solvers is ensuring that the distribution function remains positive a related problem is to guarantee that the numerical scheme does not introduce false oscillations in velocity space in this paper we use a variety of schemes to assess the importance of these issues and to determine an optimal strategy for eulerian split approaches to vlasov solvers from these tests we conclude that maintaining positivity is less important than correctly dissipating the fine scale structure which arises naturally in the solution to many vlasov problems furthermore we show that there are distinct advantages to using high order schemes i e third order rather than second a natural choice which satisfies all of these requirements is the piecewise parabolic method ppm which is applied here to vlasov s equation for the first time

convergence analysis of a discontinuous galerkin strang splitting approximation for the vlasov poisson equations a rigorous convergence analysis of the strang splitting algorithm with a discontinuous galerkin approximation in space for the vlasov poisson equations is provided it is shown that under suitable assumptions the error is of order mathcal o tau 2 h q h q tau where tau is the size of a time step h is the cell size and q the order of the discontinuous galerkin approximation in order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed

outflow boundary conditions for the fourier transformed three dimensional vlasov maxwell system a problem with the solution of the vlasov equation is its tendency to become filamented oscillatory in velocity space which in numerical simulations can give rise to unphysical oscillations and recurrence effects we present a three dimensional vlasov maxwell solver three spatial and velocity dimensions plus time in which the vlasov equation is fourier transformed in velocity space and the resulting equations solved numerically by designing absorbing outflow boundary conditions in the fourier transformed velocity space the highest fourier modes in velocity space are removed from the numerical solution this introduces a dissipative effect in velocity space and the numerical recurrence effect is strongly reduced the well posedness of the boundary conditions is proved analytically while the stability of the numerical implementation is assessed by long time numerical simulations well known wave modes in magnetized plasmas are shown to be reproduced by the numerical scheme

heisenberg algebra umbral calculus and orthogonal polynomials umbral calculus can be viewed as an abstract theory of the heisenberg commutation relation p m 1 in ordinary quantum mechanics p is the derivative and m the coordinate operator here we shall realize p as a second order differential operator and m as a first order integral one we show that this makes it possible to solve large classes of differential and integrodifferential equations and to introduce new classes of orthogonal polynomials related to laguerre polynomials these polynomials are particularly well suited for describing the so called flatenned beams in laser theory

smoluchowski dynamics and the ergodic nonergodic transition we use the recently introduced theory for the kinetics of systems of classical particles to investigate systems driven by smoluchowski dynamics we investigate the existence of ergodic nonergodic ene transitions near the liquid glass transition we develop a self consistent perturbation theory in terms of an effective two body potential and work to second order in this potential at second order we have an explicit relationship between the static structure factor and the effective potential and choose the static structure factor in the case of hard spheres to be given by the solution of the percus yevick approximation for hard spheres then using the analytically determined ene equation for the ergodicity function we find an ene transition for packing fraction greater than a critical value 0 76 which is physically unaccessible the existence of a linear fluctuation dissipation theorem in the problem is shown and used to great advantage

degenerate mittag leffler polynomials we present several explicit formulas and recurrence relations for the degenerate mittag leffler polynomials we also give several connections between mittag leffler polynomials and other known families of polynomials