enhanced convergence estimates for semi lagrangian schemes application to the vlasov poisson equation we prove enhanced error estimates for high order semi lagrangian discretizations of the vlasov poisson equation it provides new insights into optimal numerical strategies for the numerical solution of this problem the new error estimate o min frac delta x delta t 1 delta x p delta t 2 is based on advanced error estimates for semi lagrangian schemes also equal to shifted strang s schemes for the discretization of the advection equation

the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

high order hamiltonian splitting for the vlasov poisson equations we consider the vlasov poisson equation in a hamiltonian framework and derive new time splitting methods based on the decomposition of the hamiltonian functional between the kinetic and electric energy assuming smoothness of the solutions we study the order conditions of such methods it appears that these conditions are of runge kutta nystrom type in the one dimensional case the order conditions can be further simplified and efficient methods of order 6 with a reduced number of stages can be constructed in the general case high order methods can also be constructed using explicit computations of commutators numerical results are performed and show the benefit of using high order splitting schemes in that context complete and self contained proofs of convergence results and rigorous error estimates are also given

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

a semi lagrangian transport method for kinetic problems with application to dense to dilute polydisperse reacting spray flows abstract for sprays as described by a kinetic disperse phase model strongly coupled to the navier stokes equations the resolution strategy is constrained by accuracy objectives robustness needs and the computing architecture in order to leverage the good properties of the eulerian formalism we introduce a deterministic particle based numerical method to solve transport in physical space which is simple to adapt to the many types of closures and moment systems the method is inspired by the semi lagrangian schemes developed for gas dynamics we show how semi lagrangian formulations are relevant for a disperse phase far from equilibrium and where the particle particle coupling barely influences the transport i e when particle pressure is negligible the particle behavior is indeed close to free streaming the new method uses the assumption of parcel transport and avoids to compute fluxes and their limiters which makes it robust it is a deterministic resolution method so that it does not require efforts on statistical convergence noise control or post processing all couplings are done among data under the form of eulerian fields which allows one to use efficient algorithms and to anticipate the computational load this makes the method both accurate and efficient in the context of parallel computing after a complete verification of the new transport method on various academic test cases we demonstrate the overall strategy s ability to solve a strongly coupled liquid jet with fine spatial resolution and we apply it to the case of high fidelity large eddy simulation of a dense spray flow a fuel spray is simulated after atomization at diesel engine combustion chamber conditions the large parallel strongly coupled computation proves the efficiency of the method for dense polydisperse reacting spray flows

distinguished kahler metrics on hirzebruch surfaces let 5 be a hirzebruch surface n 1 using the family of ex tremal metrics on these surfaces constructed by calabi 1 we study a closely related scale invariant variational problem and show that only si admits an extremal kahler metric which is critical for this new functional applying a result of derdzinski 3 we prove that this metric cannot be conformally equiv alent to an einstein metric on siwhen n 2 we show there is a critical orbifold metric on the space obtained from 2 by blowing down the negative section

on twistor spaces of anti self dual hermitian surfaces we consider a complex surface m with anti self dual hermitian metric h and study the holomorphic properties of its twistor space z we show that the naturally defined divisor line bundle x is isomorphic to the 1 2 power of the canonical bundle of z if and only if there is a kahler metric of zero scalar curvature in the conformal class of h this has strong consequences on the geometry of m which were also found by c boyer 3 using completely different methods we also prove the existence of a very close relation between holomorphic vector fields on m and z in the case that m is compact and kahler

nonselfadjoint schrodinger operators with singular first order coefficients schrodinger operators of the form t i grad b x 2 a x grad q x in r m are considered where a b ate real vector valued functions and q is a scalar complex valued function it is shown that t is essentially quasi m accretive in l 2 r m if 1 x2223 1 a l 4 l div a l and re q 0 the proof is elementary

convergence analysis of strang splitting for vlasov type equations a rigorous convergence analysis of the strang splitting algorithm for vlasov type equations in the setting of abstract evolution equations is provided it is shown that under suitable assumptions the convergence is of second order in the time step tau as an example it is verified that the vlasov poisson equations in 1 1 dimensions fit into the framework of this analysis further numerical experiments for the latter case are presented

a momentum construction for circle invariant kahler metrics examples of kahler metrics of constant scalar curvature are relatively scarce over the past two decades several workers in geometry and physics have used symmetry reduction to construct complete kahler metrics of constant scalar curvature by ode methods one fruitful idea the calabi ansatz is to begin with an hermitian line bundle p l h m g m over a kahler manifold and to search for kahler forms p m dd c f t in some disk subbundle where t is the logarithm of the norm function and f is a function of one variable our main technical result theorem a is the calculation of the scalar curvature for an arbitrary kahler metric g arising from the calabi ansatz this suggests geometric hypotheses which we call a constancy to impose upon the base metric g m and hermitian structure h in order that the scalar curvature of g be specified by solving an ode we show that constancy is necessary and sufficient for the calabi ansatz to work in the following sense under the assumption of constancy the disk bundle admits a one parameter family of complete kahler metrics of constant scalar curvature that restrict to g m on the zero section theorems b and d an analogous result holds for the punctured disk bundle theorem c a simple criterion determines when such a metric is einstein conversely in the absence of constancy the calabi ansatz yields at most one metric of constant scalar curvature in either the disk bundle or the punctured disk bundle theorem e many of the metrics constructed here seem to be new including a complete negative einstein kahler metric on the disk subbundle of a stable vector bundle over a riemann surface of genus at least two and a complete scalar flat kahler metric on c 2

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

analysis of a new class of forward semi lagrangian schemes for the 1d vlasov poisson equations the vlasov equation is a kinetic model describing the evolution of a plasma which is a globally neutral gas of charged particles it is self consistently coupled with poisson s equation which rules the evolution of the electric field in this paper we introduce a new class of forward semi lagrangian schemes for the vlasov poisson system based on a cauchy kovalevsky ck procedure for the numerical solution of the characteristic curves exact conservation properties of the first moments of the distribution function for the schemes are derived and a convergence study is performed that applies as well for the ck scheme as for a more classical verlet scheme a l 1 convergence of the schemes will be proved error estimates in o left delta t 2 h 2 frac h 2 delta t right for verlet are obtained where t and h max x v are the discretization parameters

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

convergence of a semi lagrangian scheme for the one dimensional vlasov poisson system a semi lagrangian scheme is proposed for solving the periodic one dimensional vlasov poisson system in phase space on unstructured meshes the distribution function f t x v and the electric field e t x are shown to converge to the exact solution values in the l infty norm the rate of convergence is in o h4 3

a high order time splitting method based on integral deferred correction for semi lagrangian vlasov simulations semi lagrangian schemes with various splitting methods and with different reconstruction interpolation strategies have been applied to kinetic simulations for example the order of spatial accuracy of the algorithms proposed in qiu and christlieb 2010 29 is very high as high as ninth order however the temporal error is dominated by the operator splitting error which is second order for strang splitting it is therefore important to overcome such low order splitting error in order to have numerical algorithms that achieve higher orders of accuracy in both space and time in this paper we propose to use the integral deferred correction idc method to reduce the splitting error specifically the temporal order accuracy is increased by r with each correction loop in the idc framework where r 1 2r 1 2 for coupling the first order splitting and the strang splitting respectively the proposed algorithm is applied to the vlasov poisson system the guiding center model and two dimensional incompressible flow simulations in the vorticity stream function formulation we show numerically that the idc procedure can automatically increase the order of accuracy in time we also investigate numerical stability of the proposed algorithm via performing fourier analysis to a linear model problem

higher order time splitting for the linear vlasov equation many semi lagrangian schemes for solving the vlasov equation use a second order time splitting method proposed by cheng and knorr in this paper a fourth order time splitting method is proposed this is combined with cubic spline interpolation in x and v and a rigorous ell 2 error bound is obtained for the linear vlasov equation in arbitrary space dimension also this method and that of cheng and knorr are implemented and compared for a test case in one space dimension

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

ale scalar flat kahler metrics on non compact weighted projective spaces we construct new explicit toric scalar flat kahler ale metrics with orbifold singularities which we use to obtain smooth extremal kahler metrics on appropriate resolutions of orbifolds

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

energy conserving discontinuous galerkin spectral element method for the vlasov poisson system abstract we propose a new energy conserving spectral element discontinuous galerkin method for the approximation of the vlasov poisson system in arbitrary dimension using cartesian grids the method is derived from the one proposed in 4 with two modifications energy conservation is obtained by a suitable projection operator acting on the solution of the poisson problem rather than by solving multiple poisson problems and all the integrals appearing in the finite element formulation are approximated with gauss lobatto quadrature thereby yielding a spectral element formulation the resulting method has the following properties exact energy conservation up to errors introduced by the time discretization stability thanks to the use of upwind numerical fluxes high order accuracy and high locality for the time discretization we consider both runge kutta methods and exponential integrators and show results for 1d and 2d cases 2d and 4d in phase space respectively

singular first order perturbations of the heat equation we exhibit dimension independent conditions under which the formal operator a a v can be defined on such that its closure in l 2 r s dx is quasi m accretive here a is real so that is nonselfadjoint the method of proof is a generalized version of the argument employed in the portion of the author s thesis where term a was originally considered specifically we construct exp t as a limit of approximating semigroups since the thesis appeared kato has also dealt with the term a his conditions on a and v are similar to but more general than the conditions that appear here in addition he considers magnetic vector potentials of interest here is the semigroup method itself the conciseness of the arguments thereby produced and a relaxed condition on div a

conformal killing forms on riemannian manifolds conformal killing forms are a natural generalization of conformal vector fields on riemannian manifolds they are defined as sections in the kernel of a conformally invariant first order differential operator we show the existence of conformal killing forms on nearly kahler and weak g2 manifolds moreover we give a complete description of special conformal killing forms a further result is a sharp upper bound on the dimension of the space of conformal killing forms

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

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

conservative semi lagrangian schemes for vlasov equations conservative methods for the numerical solution of the vlasov equation are developed in the context of the one dimensional splitting in the case of constant advection these methods and the traditional semi lagrangian ones are proven to be equivalent but the conservative methods offer the possibility to add adequate filters in order to ensure the positivity in the non constant advection case they present an alternative to the traditional semi lagrangian schemes which can suffer from bad mass conservation in this time splitting setting

convergence of a difference scheme for the vlasov poisson fokker planck system in one dimension a finite difference scheme is proposed for the periodic one dimensional vlasov poisson fokker planck system which is related to the scheme in c cheng and g knorr em j comp phys 22 1976 pp 330 351 the density f left t x v right and the electric field e left t x right are shown to converge to the exact solution values in l 2 and l infty respectively the rate of convergence is essentially first order this scheme has been implemented in j schaeffer center for nonlinear analysis research report 97 na 004 carnegie mellon university 1997 and compares favorably with a random particle method

convergence of a high order semi lagrangian scheme with propagation of gradients for the one dimensional vlasov poisson system in this paper we give a proof of convergence of a new numerical method introduced in n besse and e sonnendrucker j comput phys 191 2003 pp 341 376 for the vlasov equation the numerical method is based on the semi lagrangian principle and the transport of the gradient of the statistical distribution function in order to get a high order and stable reconstruction these kinds of new schemes have been successfully implemented on unstructured meshes of four dimensional phase space cf n besse etude mathematique et numerique de l equation de vlasov sur des maillages non structures de l espace des phases these de l universite louis pasteur strasbourg france 2003 n besse j segre and e sonnendrucker transport theory statist phys 34 2005 pp 311 332 in order to make a rigorous proof of convergence of this method and simplify the convergence analysis we have considered the periodic one dimensional vlasov poisson system in phase space on a grid the distribution f t x v and the electric field are shown to converge to the exact solution values in h 1 norm the rate of convergence is of mathcal o delta t 2 frac delta x 4 alpha delta t frac delta v 4 alpha delta t alpha in n 2 alpha leq 1

finite volume transport schemes we analyze arbitrary order linear finite volume transport schemes and show asymptotic stability in l 1 and l for odd order schemes in dimension one it gives sharp fractional order estimates of convergence for bv solutions it shows odd order finite volume advection schemes are better than even order finite volume schemes therefore the gibbs phenomena is controlled for odd order finite volume schemes numerical experiments sustain the theoretical analysis in particular the oscillations of the lax wendroff scheme for small courant numbers are correlated with its non stability in l 1 a scheme of order three is proved to be stable in l 1 and l

the numerical analysis of random particle methods applied to vlasov poisson fokker planck kinetic equations we devise and study a random particle method for approximating vlasov poisson fokker planck systems such a proposed scheme takes into account the fact that the trajectories of a particle undergoing brownian motion due to collisions with the medium or background particles can be obtained as the solutions of stochastic differential equations i e the langevin equations these equations are the precise analogs of the deterministic hamiltonian system in the collisionless model the particle approximation in particular simulates the action of viscosity by the use of independent wiener processes brownian motions the error analysis generalizes that of ganguly and victory siam j numer anal 26 1989 pp 249 288 by incorporating into the consistency estimates the sampling errors generated by the random motion of the particles these particular errors are the dominant component of the consistency errors and are estimated by applying bennett s inequality for estimating tail probabilities for standard

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

convergence of an adaptive semi lagrangian scheme for the vlasov poisson system an adaptive semi lagrangian scheme for solving the cauchy problem associated to the periodic 1 1 dimensional vlasov poisson system in the two dimensional phase space is proposed and analyzed a key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow the accuracy of the scheme is monitored by a prescribed tolerance parameter e which represents the local interpolation error at each time step in the l metric the numerical solutions are proved to converge in l towards the exact ones as e and t tend to zero provided the initial data is lipschitz and has a finite total curvature or in other words that it belongs to w 1 infty cap w 2 1 the rate of convergence is mathcal o delta t 2 varepsilon delta t which should be compared to the results of besse who recently established in siam j numer anal 42 1 350 382 2004 similar rates for a uniform semi lagrangian scheme but requiring that the initial data are in mathcal c 2 several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations

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

uniform asymptotic stability of strang s explicit compact schemes for linear advection we consider a family of explicit compact schemes for advection in one dimension the order is arbitrarily high these stencils may be called strang s stencils after the seminal work of strang j math phys 41 1962 pp 147 154 we prove that odd order schemes are stable in all l q under cfl one the strategy of the proof is similar to the one of thomee j differential equations 1 1965 pp 273 292 with a careful verification that all sharp estimates on the amplification factor are independent of the cfl number this is possible based on a general representation formula for the amplification factor numerical results in one dimension confirm the analysis

the convergence analysis of fully discretized particle methods for solving vlasov poisson systems error estimates for fully discretized particle or simulation methods for multidimensional vlasov poisson systems are provided the treatment requires stability results which generalize to the vlasov poisson context those of j t beale and a majda math comp 39 1982 pp 29 52 and those of c anderson and c greengard siam j numer anal 22 1985 pp 413 440 for studying the convergence of vortex methods with these stability results the total error at each timestep can be studied this approach produces estimates better than those which would be obtained by considering the errors in the semidiscrete approximation siam j numer anal 26 1989 pp 249 288 separate from the errors incurred in the time discretization of the mollified hamiltonian trajectory equations

symplectic integration with processing a general study the number of conditions to be satisfied by the operators k and p in symplectic integrators with processing given by epe hke p is determined for a hamiltonian of the form h cal a cal b the conditions for k are explicitly written up to order six and used to obtain more efficient methods with fewer evaluations per step than other symplectic integrators special cases in which the number of conditions for the kernel is drastically reduced are also studied it is shown that the kernel completely determines the optimal method one can obtain by processing

convergence of a semi lagrangian scheme for the reduced vlasov maxwell system for laser plasma interaction the subject matter of this paper concerns the numerical approximation of reduced vlasov maxwell models by semi lagrangian schemes such reduced systems have been introduced recently in the literature for studying the laser plasma interaction we recall the main existence and uniqueness results on these topics we present the semi lagrangian scheme and finally we establish the convergence of this scheme

conformally einstein maxwell kahler metrics and structure of the automorphism group let m g be a compact kahler manifold and f a positive smooth function such that its hamiltonian vector field k j mathrm grad g f for the kahler form omega g is a holomorphic killing vector field we say that the pair g f is conformally einstein maxwell kahler metric if the conformal metric tilde g f 2 g has constant scalar curvature in this paper we prove a reductiveness result of the reduced lie algebra of holomorphic vector fields for conformally einstein maxwell kahler manifolds extending the lichnerowicz matsushima theorem for constant scalar curvature kahler manifolds more generally we consider extensions of calabi functional and extremal kahler metrics and prove an extension of calabi s theorem on the structure of the lie algebra of holomorphic vector fields for extremal kahler manifolds the proof uses a hessian formula for the calabi functional under the set up of donaldson fujiki picture

bernstein gelfand gelfand sequences the bernstein gelfand gelfand sequences extend the complexes of homogeneous vector bundles to curved cartan geometries

convergence of classes of high order semi lagrangian schemes for the vlasov poisson system abstract in this paper we present some classes of high order semi lagran gian schemes for solving the periodic one dimensional vlasov poisson system in phase space on uniform grids we prove that the distribution function f t x v and the electric field e t x converge in the l 2 norm with a rate of displaystyle mathcal o left delta t 2 h m 1 frac h m 1 delta t right where m is the degree of the polynomial reconstruction and delta t and h are respectively the time and the phase space discretization parameters

composition methods for differential equations with processing we construct numerical integrators for differential equations up to order 12 obtained by composition of basic integrators the following cases are considered i composition for a system separable in two solvable parts ii composition using as basic methods a first order integrator and its adjoint iii composition using second order symmetric methods and iv composition using fourth order symmetric methods each scheme is implemented with a processor or corrector to improve their efficiency and this can be done virtually cost free

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

convergence of a finite volume scheme for the vlasov poisson system we propose a finite volume scheme to discretize the one dimensional vlasov poisson system we prove that if the initial data are positive bounded continuous and have their first moment bounded then the numerical approximation converges to the weak solution of the system for the weak topology of l infty moreover if the initial data belong to bv the convergence is strong in c 0 0 t l 1 loc to prove the convergence of the discrete electric field we obtain an estimation in w 1 infty omega t then we have begin eqnarray f h t x v rightharpoonup f t x v textrm in l infty q t mbox weak star quad textrm as h rightarrow 0 e h t x rightarrow e t x textrm in c 0 overline omega t quad textrm as h rightarrow 0 end eqnarray where e f is the unique weak solution of the vlasov poisson system

nonoscillatory interpolation methods applied to vlasov based models we demonstrate the ability of nonoscillatory interpolation strategies for solving efficiently the transport phase in kinetic systems with applications to charged particle transport in plasmas and semiconductors pointwise weighted essentially nonoscillatory pweno interpolation is applied to obtain semi lagrangian and flux balance methods that together with splitting techniques form the building blocks of our numerical approach these methods do not present the restrictive cfl condition typical of finite difference methods with explicit time solvers and moreover they provide reliable results controlling parasite oscillations from classical polynomial interpolation while giving highly accurate approximations of smooth parts of the solutions we perform and compare these methods in different benchmark problems for vlasov or collisional models for charged particle transport