a hermite upwind weno scheme for solving hyperbolic conservation laws this paper proposes a new weno procedure to compute problems containing both discontinuities and a large disparity of characteristic scales in a one dimensional context the weno procedure is defined on a three points stencil and designed to be sixth order in regions of smoothness we define a finite volume discretization in which we consider the cell averages of the variable and its first derivative as discrete unknowns the reconstruction of their point values is then ensured by a unique sixth order hermite polynomial this polynomial is considered as a symmetric and convex combination by ideal weights of three fourth order polynomials a central polynomial defined on the three points stencil is combined with two polynomials based on the left and the right two points stencils the symmetric nature of such an interpolation has an important consequence the choice of ideal weights has no influence on the properties of the discretization this advantage enables to formulate the hermite interpolation for non uniform meshes following the methodology of the classic weno procedure nonlinear weights are then defined to deal with the peculiarities of the hermite interpolation near discontinuities we define a new procedure in order for the nonlinear weights to smoothly evolve between the ideal weights in regions of smoothness and one sided weights otherwise the resulting scheme is a sixth order weno method based on central hermite interpolation and tvd runge kutta time integration we call this scheme the hcweno6 scheme numerical experiments in the scalar and the 1d euler cases make it possible to check and to validate the options selected in these experiments we emphasize the resolution power of the method by computing test cases that model realistic aero acoustic problems

rank 2 symmetric hyperbolic kac moody algebras and hilbert modular forms in this paper we study rank 2 symmetric hyperbolic kac moody algebras h a with the cartan matrices 2 a a2 a 3 and their automorphic correction in terms of hilbert modular forms we associate a family of h a s to the quadratic field q p for each odd prime p and show that there exists a chain of embeddings in each family when p 5 13 17 we show that the first h a in each family i e h 3 h 11 h 66 is contained in a generalized kac moody superalgebra whose denominator function is a hilbert modular form given by a borcherds product hence our results provide automorphic correction for those h a s we also compute asymptotic formulas for the root multiplicities of the generalized kac moody superalgebras using the fact that the exponents in the borcherds products are fourier coefficients of weakly holomorphic modular forms of weight 0

partial synchronization from symmetry towards stability abstract in this paper we study the existence and stability of linear invariant manifolds in a network of coupled identical dynamical systems symmetry under permutation of different units of the network is helpful to construct explicit formulae for linear invariant manifolds of the network in order to classify them and to examine their stability through lyapunov s direct method

the streamline subgrid integration method i quasi monotonic second order transport schemes a new finite volume method has been developed for conservative and monotonic transport in multiple dimensions without any sort of dimension splitting to emphasize the space time integrity of the fluid system the streamline subgrid integration ssi method is a combination of the semi lagrangian and the finite volume methods with generalized multidimensional subgrid distributions second order transport schemes are constructed in two dimensions and their extensions to three dimensions are also discussed spurious divergence is rigorously controlled with lagrangian control volumes monotonicity is well preserved for both compressible and incompressible flows and positive definiteness is always guaranteed the accuracy and the numerical properties of these schemes are evaluated with both continuous and discontinuous solutions in flows of rigid motion and incompressible deformation where analytic solutions are available icosahedral geodesic grids are selected to demonstrate the general nature of the ssi method in spherical geometry that the numerical solutions are virtually not affected by the irregularity of grid structure we have found that the numerical solutions converge quickly to the analytic solutions and the ssi second order schemes are relatively more beneficial for high resolution applications

necessary and sufficient graphical conditions for formation control of unicycles the feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control the directed graph defined by the information flow plays a key role it is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node a similar result is given for formation stabilization to a line and to more general geometric arrangements

conservative rezoning remapping for general quadrilateral meshes a classic problem in lagrangian numerical hydrodynamics is the conservative transfer of quantities from an old distorted mesh to a new mesh the same problem arises whenever the mesh is changed as for example in adaptive mesh techniques this transfer of information is an interpolation process which is frequently called rezoning or remapping the general problem of conservative rezoning from one arbitrary mesh to another may be formulated as follows m sub k integral integral integral sub v sub k rho r dv that is we compute the mass m sub k of each cell of the new mesh by integrating the known density distribution in the old mesh over the cell volume v sub k a direct intregation is generally prohibitive we show however that is is possible to convert this integral to a surface integral by the appropriate use of the divergence theorem thus greatly reducing the complexity of the problem for two dimensional generall quadrilateral meshes the resulting method is exact and particularly simple

a semi lagrangian discontinuous galerkin dg local dg method for solving convection diffusion equations abstract in this paper we propose an efficient high order semi lagrangian sl discontinuous galerkin dg method for solving linear convection diffusion equations the method generalizes our previous work on developing the sldg method for transport equations 5 making it capable of handling additional diffusion and source terms within the dg framework the solution is evolved along the characteristics while the diffusion term is discretized by the local dg ldg method and integrated along characteristics by implicit runge kutta methods together with source terms the proposed method is named the sldg ldg method and enjoys many attractive features of the dg and sl methods these include the uniformly high order accuracy e g third order in space and in time compact mass conservative and stability under large time stepping size an l 2 stability analysis is provided when the method is coupled with the first order backward euler discretization effectiveness of the method are demonstrated by a group of numerical tests in one and two dimensions

optimization based remap and transport a divide and conquer strategy for feature preserving discretizations this paper examines the application of optimization and control ideas to the formulation of feature preserving numerical methods with particular emphasis on the conservative and bound preserving remap constrained interpolation and transport advection of a single scalar quantity we present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization based remap obr of mass density the latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity subject to a system of linear inequality constraints the approximation of an exact mass update operator defines the target quantity which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds the latter are enforced by the system of linear inequalities in so doing the generic obr formulation separates accuracy considerations from the enforcement of physical properties we proceed to show how the generic obr formulation yields the recently introduced flux variable flux target fvft 1 and mass variable mass target mvmt 2 formulations of remap and then follow with a formal examination of their relationship using an intermediate flux variable mass target fvmt formulation we show the equivalence of fvft and mvmt optimal solutions to underscore the scope and the versatility of the generic obr formulation we introduce the notion of adaptable targets i e target quantities that reflect local solution properties extend fvft and mvmt to remap on the sphere and use obr to formulate adaptable conservative and bound preserving optimization based transport algorithms for cartesian and latitude longitude coordinate systems a selection of representative numerical examples on two dimensional grids demonstrates the computational properties of our approach

finite volume transport on various cubed sphere grids the performance of a multidimensional finite volume transport scheme is evaluated on the cubed sphere geometry advection tests with prescribed winds are used to evaluate a variety of cubed sphere projections and grid modifications including the gnomonic and conformal mappings as well as two numerically generated grids by an elliptic solver and spring dynamics we explore the impact of grid non orthogonality on advection tests over the corner singularities of the cubed sphere grids using some variations of the transport scheme including the piecewise parabolic method with alternative monotonicity constraints the advection tests revealed comparable or better accuracy to those of the original latitudinal longitudinal grid implementation it is found that slight deviations from orthogonality on the modified cubed sphere quasi orthogonal grids do not negatively impact the accuracy in fact the more uniform version of the quasi orthogonal cubed sphere grids provided better overall accuracy than the most orthogonal and therefore much less uniform conformal grid it is also shown that a simple non orthogonal extension to the transport equation enables the use of the highly non orthogonal and computationally more efficient gnomonic grid with acceptable accuracy

application of differential transformation to eigenvalue problems using differential transformation to solve eigenvalue problems is introduced in this study first differential transformation is introduced briefly second taking differential transformation of the strum liouville problem a set of difference equations is derived doing some simple mathematics operations on these operations we can get any ith eigenvalue and eigenfunction at a time finally two eigenvalue problems are solved by the present method the calculated results are compared closely with the results obtained by another analytical method

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

a conservative semi lagrangian multi tracer transport scheme cslam on the cubed sphere grid a conservative multi tracer transport algorithm on the cubed sphere based on the semi lagrangian approach cslam has been developed the scheme relies on backward trajectories and the resulting upstream cells polygons are approximated with great circle arcs biquadratic polynomial functions are used for approximating the density distribution in the cubed sphere grid cells the upstream surface integrals associated with the conservative semi lagrangian scheme are computed as line integrals by employing the gauss green theorem the line integrals are evaluated using a combination of exact integrals and high order gaussian quadrature the upstream cell trajectories information and computation of weights of integrals can be reused for each additional tracer the cslam scheme is extensively tested with various standard benchmark test cases of solid body rotation and deformational flow in both cartesian and spherical geometry and the results are compared with those of other published schemes the cslam scheme is accurate robust and moreover the edges and vertices of the cubed sphere discontinuities do not affect the overall accuracy of the scheme the cslam scheme exhibits excellent convergence properties and has an option for enforcing monotonicity the advantages of introducing cross terms in the fully two dimensional biquadratic density distribution functions are also examined in the context of cartesian as well as the cubed sphere grid which has six local sub domains with discontinuous edges and corners

vorticity divergence mass conserving semi lagrangian shallow water model using the reduced grid on the sphere the semi lagrangian semi implicit shallow water model on the sphere using the reduced latitude longitude grid is presented the key feature of the model is the vorticity divergence formulation on the unstaggered grid the new algorithm for the reconstruction of wind components from vorticity and divergence is described the mass conservative version of the model is developed the conservative cascade scheme ccs by nair et al is modified to provide a locally conservative semi lagrangian advection algorithm for the reduced grid some numerical advection tests are carried out to demonstrate the accuracy of the ccs with the reduced grid the ccs based discretization for the continuity equation and finite volume helmholtz problem solver are introduced to guarantee the mass conservation the results for shallow water tests on the sphere are presented the results for different versions of the model are compared they are also compared with the results for the same tests available in literature the impact of the reduced grid is analyzed the mass conservative version of the model using the reduced grid with up to 20 reduction of grid points number has approximately the same accuracy as its non conservative counterpart implemented on the regular latitude longitude grid

the cubed sphere a new gridding technique for the solution of partial differential equations in spherical geometry is presented the method is based on a decomposition of the sphere into six identical regions obtained by projecting the sides of a circumscribed cube onto a spherical surface by choosing the coordinate lines on each region to be arcs of great circles one obtains six coordinate systems which are free of any singularity and define the same metric taking full advantage of the symmetry properties of the decomposition a variation of the composite mesh finite difference method can be applied to couple the six grids and obtain with a high degree of efficiency very accurate numerical solutions of partial differential equations on the sphere the advantages of this new technique over both spectral and uniform longitude latitude grid point methods are discussed in the context of applications on serial and parallel architectures we present results of two test cases for numerical approximations to the shallow water equations in spherical geometry the linear advection of a cosine bell and the nonlinear evolution of a rossby haurwitz wave performance analysis for this latter case indicates that the new method can provide with substantial savings in execution times numerical solutions which are as accurate as those obtainable with the spectral transform method

optimization based limiters for the spectral element method abstract we introduce a new family of optimization based limiters for the h p spectral element method the native spectral element advection operator is oscillatory but due to its mimetic properties it is locally conservative and has a monotone property with respect to element averages we exploit this property to construct locally conservative quasimonotone and sign preserving limiters the quasimonotone limiter prevents all overshoots and undershoots at the element level but is not strictly non oscillatory it also maintains quasimonotonicity even with the addition of a dissipation term such as viscosity or hyperviscosity the limiters are based on a least squares formulation with equality and inequality constraints and are local to each element we evaluate the new limiters using a deformational flow test case for advection on the surface of the sphere we focus on mesh refinement for moderate p 3 and high order p 6 elements as expected the spectral element method obtains its formal order of accuracy for smooth problems without limiters for advection of fields with cusps and discontinuities the high order convergence is lost but in all cases p 6 outperforms p 3 for the same degrees of freedom

application of the parabolic spline method psm to a multi dimensional conservative semi lagrangian transport scheme slice the recently devised one dimensional parabolic spline method psm for efficient conservative and monotonic remapping is introduced into the semi lagrangian inherently conserving and efficient slice scheme for transport problems in multi dimensions to ensure mass conservation an integral form of the transport equation is used rather than the differential form of classical semi lagrangian schemes integrals within the slice scheme are computed using multiple sweeps of psm along flow dependent cascade directions to avoid the large timestep dependent splitting errors associated with traditional fixed direction splitting accuracy of the overall scheme including at large timestep is demonstrated using two dimensional test problems in both cartesian and spherical geometries and compared with that of the piecewise parabolic method ppm applied within the same slice framework

an analysis of 1d finite volume methods for geophysical problems on refined grids this paper examines high order unstaggered symmetric and upwind finite volume discretizations of the advection equation in the presence of an abrupt discontinuity in grid resolution an approach for characterizing the initial amplitude of a parasitic mode as well as its decay rate away from a grid resolution discontinuity is presented using a combination of numerical analysis and empirical studies it is shown that spurious parasitic modes which are artificially generated by the resolution discontinuity are mostly undamped by symmetric finite volume schemes but are quickly removed by upwind and semi lagrangian integrated mass slim schemes slope curvature limiting is insufficient to completely remove these modes especially at low forcing frequencies where the incident wave can act as a carrier of the parasitic mode increasing the order of accuracy of the reconstruction at the grid interface is effective at removing noise from the lowest frequency incident modes but insufficient at high frequencies it is shown that this analysis can be extended to the 1d linear shallow water equations via riemann invariants

3d conservative cascade semi lagrangian transport scheme using reduced latitude longitude grid ccs rg the article presents conservative cascade scheme using reduced grid ccs rg a finite volume semi lagrangian fvsl algorithm for numerical solution of 3d advection equation on the sphere the ccs rg is based on the cascade approach that allows to split 3d advected quantity mass remapping between the lagrangian and eulerian grids onto 3 1d remappings another feature is using a quasi uniform spaced reduced latitude longitude grid on the sphere the scheme is inherently mass conservative monotonic and stable with large time steps the ccs rg performance is tested with the advection cases from the dynamical core model intercomparison project test suite the impact of the reduced grids and two different monotonic filters on the solution is investigated the comparison with the results obtained by other schemes proves competitive accuracy of ccs rg in most cases the numerical solution obtained with various reduced grids is as accurate as the regular grid solution

semi discrete discontinuous galerkin methods and stage exceeding order strong stability preserving runge kutta time discretizations this paper investigates the use of a special class of strong stability preserving ssp runge kutta time discretization methods in conjunction with discontinuous galerkin dg finite element spatial discretizatons the class of ssp methods investigated here is defined by the property that the number of stages s is greater than the order k of the method from analysis cfl conditions for the linear l 2 stability of the methods defined using the s k ssp schemes are obtained that are less restrictive than those of the standard so called rkdg methods that use s k ssp runge kutta schemes the improvement in the cfl conditions for linear stability of the methods more than offsets the additional work introduced by the increased number of stages given that the cfl conditions for linear stability are what must be respected in practice in order to maintain high order accuracy the use of the s k ssp schemes results in rkdg methods that are more efficient than those previously defined furthermore with the application of a slope limiter the nonlinear stability properties of the forward euler method and the dg spatial discretization which have been previously proven are preserved with these methods under less restrictive cfl conditions than those required for linear stability thus more efficient rkdg methods that possess the same favorable accuracy and stability properties of the standard rkdg methods are obtained numerical results verify the cfl conditions for stability obtained from analysis and demonstrate the efficiency advantages of these new rkdg methods

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

incremental remapping as a transport advection algorithm there are two fundamentally different strategies for solving the standard transport or continuity equation corresponding to whether it is expressed as a partial differential equation or as an integral statement of conservation the more common approach is to discretize the partial differential equation and to march the solution forward in time the alternative method is to project cell volumes along lagrangian trajectories as far forward or backward in time as desired and then to remap the resulting density distribution onto some target mesh this latter approach is known as remapping remapping has many advantages not the least of which is that the time step is limited only by accuracy considerations but it tends to be expensive and complex in this paper we show that if the time step is made sufficiently short such that trajectories are confined to the nearest neighbor cells then the remapping may be written as a flux form transport algorithm and it becomes nearly as simple and efficient as standard transport schemes the resulting method called incremental remapping retains most of the advantages of general remapping these include a geometric basis for transport b compatibility of associated tracer transport with simple tracer advection i e retention of tracer monotonicity properties and c efficient handling of multiple tracers since each additional tracer adds only a relatively small incremental cost

a multistep flux corrected transport scheme a multistep flux corrected transport mfct scheme is developed to achieve conservative and monotonic tracer transports for multistep dynamical cores mfct extends zalesak two time level scheme to any multistep time differencing schemes by including multiple high order fluxes in the antidiffusive flux while computing the two time level low order monotone solution the multistep time differencing scheme used in this study is the third order adams bashforth ab3 scheme implemented in a finite volume icosahedral shallow water model the accuracy of ab3 mfct is quantified by the shape preserving advection experiments in non divergent flow as well as a cosine bell whose shape changes during advection in shear flow ab3 mfct has been shown to be insensitive to time step size this make ab3 mfct an attractive transport scheme for explicit high resolution model applications with small time step mfct is tested in shallow water model simulations to demonstrate that the use of mfct maintains positive definite tracer transport while at the same time conserving both fluid mass and tracer mass within round off errors in the ab3 dynamic core

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 conservative multi tracer transport scheme for spectral element spherical grids atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers for computational efficiency conservative multi tracer semi lagrangian type transport schemes are appropriate global models based on high order galerkin approach employ highly non uniform spectral element grids and semi lagrangian transport is a challenge on those grids a conservative semi lagrangian scheme spelt spectral element lagrangian transport employing a multi moment compact reconstruction procedure is developed for non uniform quadrilateral grids the scheme is based on a characteristic semi lagrangian method that avoids complex and expensive upstream area computations the spelt scheme has been implemented in the high order method modeling environment homme which is based on a cubed sphere grid with spectral element spatial discretization additionally we show the strong scalability and multi tracer efficiency using several benchmark tests the spelt solution can be made monotonic positivity preserving by combining the flux corrected transport algorithm which is demonstrated on a uniform resolution grid in particular spelt can be efficiently used for non uniform grids and provides accurate and stable results for high resolution meshes

reprint of atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers for computational efficiency conservative multi tracer semi lagrangian type transport schemes are appropriate global models based on high order galerkin approach employ highly non uniform spectral element grids and semi lagrangian transport is a challenge on those grids a conservative semi lagrangian scheme spelt spectral element lagrangian transport employing a multi moment compact reconstruction procedure is developed for non uniform quadrilateral grids the scheme is based on a characteristic semi lagrangian method that avoids complex and expensive upstream area computations the spelt scheme has been implemented in the high order method modeling environment homme which is based on a cubed sphere grid with spectral element spatial discretization additionally we show the strong scalability and multi tracer efficiency using several benchmark tests the spelt solution can be made monotonic positivity preserving by combining the flux corrected transport algorithm which is demonstrated on a uniform resolution grid in particular spelt can be efficiently used for non uniform grids and provides accurate and stable results for high resolution meshes

multi leader coordination algorithm for networks with switching topology and quantized information this paper analyzes the stability and convergence properties of a proportional integral protocol for coordination of a network of agents with dynamic information flow and quantized information exchange in the setup adopted each agent is only required to exchange its coordination state with its neighboring agents and the desired reference rate is only available to a group of leaders we show that the integral term of the protocol allows the agents to learn the reference rate rather than have it available a priori and also provides disturbance rejection capabilities the paper addresses the case where the graph that captures the underlying network topology is not connected during some interval of time or even fails to be connected at all times

h consensus of second order multi agent systems with asymmetric delays abstract this paper is concerned with the problem of consensus in the h sense for second order continuous time multi agent systems with multiple asymmetric time varying delays by using a model transformation approach and matrix theory we establish several conditions in terms of linear matrix inequalities such that consensus of multi agent systems can be achieved in the h sense the feasibility of the consensus conditions is also analyzed as an application we consider the case of intermittent measurement between agents numerical examples are presented to illustrate the theoretical results which can be applied to the case of negative information weights

optimization based mesh correction with volume and convexity constraints we consider the problem of finding a mesh such that 1 it is the closest with respect to a suitable metric to a given source mesh having the same connectivity and 2 the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh this volume correction problem arises in important simulation contexts such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi lagrangian transport schemes in this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective while the prescribed cell volumes mesh validity and or cell convexity specify the constraints we solve this problem numerically using a sequential quadratic programming sqp method whose performance scales with the mesh size to achieve scalable performance we develop a specialized multigrid based preconditioner for optimality systems that arise in the application of the sqp method to the volume correction problem numerical examples illustrate the importance of volume correction and showcase the accuracy robustness and scalability of our approach

tracer transport within an unstructured grid ocean model using characteristic discontinuous galerkin advection abstract in a previous article a characteristic discontinuous galerkin cdg advection scheme was presented for tracer transport lee et al 2016 the scheme is conservative unconditionally stable with respect to time step and scales sub linearly with the number of tracers being advected here we present the implementation of the cdg advection scheme for tracer transport within mpas ocean a boussinesque unstructured grid ocean model with an arbitrary lagrangian eulerian vertical coordinate the scheme is implemented in both the vertical and horizontal dimensions and special care is taken to ensure that the scheme remains conservative in the context of moving vertical layers consistency is ensured with respect to the dynamics by a renormalization of the fluxes with respect to the volume fluxes derived from the continuity equation for spherical implementations the intersection of the flux swept regions and the eulerian grid are determined for great circle arcs and the fluxes and element assembly are performed on the plane via a length preserving projection solutions are presented for a suite of test cases and comparisons made to the existing flux corrected transport scheme in mpas ocean

a forward trajectory global semi lagrangian transport scheme a forward trajectory semi lagrangian scheme for advection on the surface of the sphere is proposed the advection scheme utilizes the forward downstream trajectory originating at eulerian grid points and cascade interpolation a sequence of 1d interpolations to transfer data from the downstream lagrangian points to the eulerian points a new and more accurate algorithm determines pole values the resulting forward trajectory semi lagrangian scheme can easily incorporate high order trajectory integration methods this avoids the standard iterative process in a typical backward trajectory scheme two third order accurate schemes and a second order accurate scheme are presented a mass conservative version of the forward trajectory semi lagrangian scheme is also derived within the cascade interpolation framework mass from a lagrangian cell is transferred to the corresponding eulerian cell with two 1d remappings through an intermediate cell system mass in the polar region is redistributed by way of an efficient local approximation the resulting scheme is globally conservative but restricted to meridional courant number c 1

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

discontinuities and hysteresis in quantized average consensus we consider continuous time average consensus dynamics in which the agents states are communicated through uniform quantizers solutions to the resulting system are defined in the krasowskii sense and are proven to converge to conditions of practical consensus to cope with undesired chattering phenomena we introduce a hysteretic quantizer and we study the convergence properties of the resulting dynamics by a hybrid system approach

passivity as a design tool for group coordination we pursue a group coordination problem where the objective is to steer the differences between output variables of the group members to a prescribed compact set to stabilize this set we study a class of feedback rules that are implementable with local information available to each member when the information flow between neighboring members is bidirectional we show that the closed loop system exhibits a special interconnection structure which inherits the passivity properties of its components by exploiting this structure we develop a passivity based design framework which results in a broad class of feedback rules that encompass as special cases some of the existing formation stabilization and group agreement designs in the literature the passivity approach offers additional design flexibility compared to these special cases and systematically constructs a lurie type lyapunov function for the closed loop system we further study the robustness of these feedback laws in the presence of a time varying communication topology and present a persistency of excitation condition which allows the interconnection graph to lose connectivity pointwise in time as long as it is established in an integral sense

derivative riemann solvers for systems of conservation laws and ader methods in this paper we first briefly review the semi analytical method e f toro v a titarev solution of the generalized riemann problem for advection reaction equations proc roy soc london 458 2018 2002 271 281 for solving the derivative riemann problem for systems of hyperbolic conservation laws with source terms next we generalize it to hyperbolic systems for which the riemann problem solution is not available as an application example we implement the new derivative riemann solver in the high order finite volume ader advection schemes we provide numerical examples for the compressible euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes

short note a simple mass conserving semi lagrangian scheme for transport problems a mass conserving semi lagrangian sl scheme is achieved with a combination of a simple explicit smoothness based mass correction and a standard non conservative interpolating sl scheme the resulting mass correction can be incorporated into any existing sl scheme with negligible extra cost a more selective and less damping monotonicity filter by comparison to traditional filters is also presented results from various tests from the literature show that in addition to mass conservation the proposed scheme has negligible impact on the overall accuracy of the standard non conservative sl scheme

on simplifying incremental remap based transport schemes the flux form incremental remapping transport scheme introduced by dukowicz and baumgardner 1 converts the transport problem into a remapping problem this involves identifying overlap areas between quadrilateral flux areas and regular square grid cells which is non trivial and leads to some algorithm complexity in the simpler swept area approach originally introduced by hirt et al 2 the search for overlap areas is eliminated even if the flux areas overlap several regular grid cells the resulting simplified scheme leads to a much simpler and robust algorithm we show that for sufficiently small courant numbers approximately cfl 1 2 the simplified or swept area scheme can be more accurate than the original incremental remapping scheme this is demonstrated through a von neumann stability analysis an error analysis and in idealized transport test cases on the sphere using the incremental remapping based scheme called ff cslam flux form version of the conservative semi lagrangian multi tracer scheme on the cubed sphere

the multi agent rendezvous problem part 2 the asynchronous case this paper is concerned with the collective behavior of a group of n 1 mobile autonomous agents labelled 1 through n which can all move in the plane each agent is able to continuously track the positions of all other agents currently within its sensing region where by an agent s sensing region we mean a closed disk of positive radius r centered at the agent s current position the multi agent rendezvous problem is to devise local control strategies one for each agent which without any active communication between agents cause all members of the group to eventually rendezvous at a single unspecified location this paper describes a family of unsynchronized strategies for solving the problem correctness is established appealing to the concept of analytic synchronization

automorphic correction of the hyperbolic kac moody algebra e10 in this paper we study automorphic correction of the hyperbolic kac moody algebra e10 using the borcherds product for o 10 2 attached to a weakly holomorphic modular form of weight 4 for sl2 z we also clarify some aspects of automorphic correction for lorentzian kac moody algebras and give heuristic reasons for the expectation that every lorentzian kac moody algebra has an automorphic correction

conservative rezoning algorithm for generalized two dimensional meshes a method is presented for transferring a conserved quantity q from one generalized mesh to another when the volumetric density of q is uniform within each cell of the original mesh

a weno limited ader dt finite volume scheme for efficient robust and communication avoiding multi dimensional transport the novel ader dt time discretization is applied to two dimensional transport in a quadrature free weno and fct limited finite volume context emphasis is placed on 1 the serial and parallel computational properties of ader dt and this framework and 2 the flexibility of ader dt and this framework in efficiently balancing accuracy with other constraints important to transport applications this study demonstrates a range of choices for the user when approaching their specific application while maintaining good parallel properties in this method genuine multi dimensionality single step and single stage time stepping strict positivity and a flexible range of limiting are all achieved with only one parallel synchronization and data exchange per time step in terms of parallel data transfers per simulated time interval this improves upon multi stage time stepping and post hoc filtering techniques such as hyperdiffusion this method is evaluated with standard transport test cases over a range of limiting options to demonstrate quantitatively and qualitatively what a user should expect when employing this method in their application

mesh free semi lagrangian methods for transport on a sphere using radial basis functions abstract we present three new semi lagrangian methods based on radial basis function rbf interpolation for numerically simulating transport on a sphere the methods are mesh free and are formulated entirely in cartesian coordinates thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface based coordinate systems for problems involving tracer transport in a given velocity field the semi lagrangian framework allows these new methods to avoid the use of any stabilization terms such as hyperviscosity during time integration thus reducing the number of parameters that have to be tuned the three new methods are based on interpolation using 1 global rbfs 2 local rbf stencils and 3 rbf partition of unity for the latter two of these methods we find that it is crucial to include some low degree spherical harmonics in the interpolants standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy efficiency conservation properties and dissipation dispersion errors for global rbfs spectral spatial convergence is observed for smooth solutions on quasi uniform nodes while high order accuracy is observed for the local rbf stencil and partition of unity approaches

an efficient quadrature rule on the cubed sphere abstract a new quadrature rule for functions defined on the sphere is introduced the nodes are defined as the points of the cubed sphere the associated weights are defined in analogy to the trapezoidal rule on each panel of the cubed sphere the formula enjoys a symmetry property ensuring that a proportion of 7 8 of all spherical harmonics is integrated exactly based on the remaining spherical harmonics it is possible to define modified weights giving an enhanced quadrature rule numerical results show that the new quadrature is competitive with classical rules of the literature this second quadrature rule is believed to be of interest for applied mathematicians physicists and engineers dealing with data located at the points of the cubed sphere

stability analysis of swarms in this note we specify an individual based continuous time model for swarm aggregation in n dimensional space and study its stability properties we show that the individuals autonomous agents or biological creatures will form a cohesive swarm in a finite time moreover we obtain an explicit bound on the swarm size which depends only on the parameters of the swarm model

persistent excitation in adaptive systems the importance of the concept of persistent excitation pe in adaptive identification and control has been recognized for some time recently it has become evident that it also plays a central role in many questions related to the robustness of adaptive systems there is every reason to believe that arguments involving this concept will continue to feature prominently in the analysis of most of the important problems of adaptive control hence there is a real need for a deeper understanding of the concept the paper is written with three objectives the first which is tutorial in nature is to provide a general framework for the discussion of persistent excitation and to collect results in the area which are scattered throughout the adaptive literature the second objective is to present some new results related to the uniform asymptotic stability u a s and robustness of adaptive systems and the relation of pe to the stability properties of a class of non linear systems the final objective is to di

multi moment ader taylor methods for systems of conservation laws with source terms in one dimension a new integration method combining the ader time discretization with a multi moment finite volume framework is introduced ader runtime is reduced by performing only one cauchy kowalewski c k procedure per cell per time step and by using the differential transform method for high order derivatives three methods are implemented 1 single moment weno weno 2 two moment hermite weno hweno and 3 entirely local multi moment mm loc mm loc evolves all moments sharing the locality of galerkin methods yet with a constant time step during p refinement five 1 d experiments validate the methods 1 linear advection 2 burger s equation shock 3 transient shallow water sw 4 steady state sw simulation and 5 sw shock weno and hweno methods showed expected polynomial h refinement convergence and successfully limited oscillations for shock experiments mm loc showed expected polynomial h refinement and exponential p refinement convergence for linear advection and showed sub exponential yet super polynomial convergence with p refinement in the sw case hweno accuracy was generally equal to or better than a five moment mm loc scheme mm loc was less accurate than rkdg at lower refinements but with greater h and p convergence rkdg accuracy is eventually surpassed the ader time integrator of mm loc also proved more accurate with p refinement at a cfl of unity than a semi discrete rk analog of mm loc being faster in serial and requiring less frequent inter node communication than galerkin methods the ader based mm loc and hweno schemes can be spatially refined and have the same runtime making them a competitive option for further investigation

algorithmic improvements for schemes using the ader time discretization abstract we detail several algorithmic changes to the ader multi moment finite volume methods ader mmfv of norman and finkel 2012 2 the dt recurrence relations are improved flux and source term differential transforms are saved each are expanded as polynomials quadrature is removed from the algorithm integration is performed analytically and a different riemann solver admitting direct use of time averaged fluxes is used these algorithmic changes were implemented and tested and smooth 1 d shallow water experiments confirm the same or slightly better accuracy as well as 2 3 lower runtimes compared to norman and finkel 2012 2

multidimensional method of lines transport for atmospheric flows over steep terrain using arbitrary meshes abstract including terrain in atmospheric models gives rise to mesh distortions near the lower boundary that can degrade accuracy and challenge the stability of transport schemes multidimensional transport schemes avoid splitting errors on distorted arbitrary meshes and method of lines schemes have a low computational cost because they perform reconstructions at fixed points this paper presents a multidimensional method of lines finite volume transport scheme cubicfit which is designed to be numerically stable on arbitrary meshes stability conditions derived from a von neumann stability analysis are imposed during model initialisation to obtain stability and improve accuracy in distorted regions of the mesh and near steeply sloping lower boundaries reconstruction calculations depend upon the mesh only needing just one vector multiply per face per time stage irrespective of the velocity field the cubicfit scheme is evaluated using three idealised numerical tests the first is a variant of a standard horizontal transport test on severely distorted terrain following meshes the second is a new test case that assesses accuracy near the ground by transporting a tracer at the lower boundary over steep terrain on terrain following meshes cut cell meshes and new slanted cell meshes that do not suffer from severe time step constraints associated with cut cells the third standard test deforms a tracer in a vortical flow on hexagonal icosahedral meshes and cubed sphere meshes in all tests cubicfit is stable and largely insensitive to mesh distortions and cubicfit results are more accurate than those obtained using a multidimensional linear upwind transport scheme the cubicfit scheme is second order convergent regardless of mesh distortions

on vector field reconstructions for semi lagrangian transport methods on geodesic staggered grids we analyse several vector reconstruction methods based on the knowledge of only specific pointwise vector components and extend their use to non structured polygonal c grids on the sphere the emphasis is on the reconstruction of the vector field at arbitrary locations on the sphere as required by semi lagrangian transport schemes this is done by first reconstructing the vector field to fixed locations followed by interpolations with generalized barycentric coordinates we derive a hybrid scheme combining the efficiency of perot s method with the accuracy of a least square scheme this method is second order accurate and has shown to be competitive and computationally efficient we analysed the vector reconstruction methods within a semi lagrangian transport method and demonstrated that second order accurate reconstructions are enough to fulfil the requirements for second order accurate semi lagrangian methods on icosahedral c grids

state agreement for continuous time coupled nonlinear systems two related problems are treated in continuous time first the state agreement problem is studied for coupled nonlinear differential equations the vector fields can switch within a finite family associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems generalizing the work of moreau under the assumption that the vector fields satisfy a certain subtangentiality condition it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time the proof uses nonsmooth analysis second the rendezvous problem for kinematic point mass mobile robots is studied when the robots fields of view have a fixed radius the circumcenter control law of ando e ieee trans robotics automation 15 1999 pp 818 828 is shown to solve the problem the rendezvous problem is a kind of state agreement problem but the interaction structure is state dependent

hermitian approximation of the spherical divergence on the cubed sphere previous work croisille 2013 showed that the cubed sphere grid offers a suitable discrete framework for extending hermitian compact operators collatz 1960 to the spherical setup in this paper we further investigate the design of high order accurate approximations of spherical differential operators on the cubed sphere with an emphasis on the spherical divergence of a tangent vector field the basic principle of this approximation relies on evaluating pointwise hermitian derivatives along a series of great circles covering the sphere several test cases demonstrate the very good accuracy of the approximate spherical divergence calculated with the new scheme

shallow water model on cubed sphere by multi moment finite volume method a global numerical model for shallow water flows on the cubed sphere grid is proposed in this paper the model is constructed by using the constrained interpolation profile multi moment finite volume method cip mm fvm two kinds of moments i e the point value pv and the volume integrated average via are defined and independently updated in the present model by different numerical formulations the lax friedrichs upwind splitting is used to update the pv moment in terms of a derivative riemann problem and a finite volume formulation derived by integrating the governing equations over each mesh element is used to predict the via moment the cubed sphere grid is applied to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry highly localized reconstruction in cip mm fvm is well suited for the cubed sphere grid especially in dealing with the discontinuity in the coordinates between different patches the mass conservation is completely achieved over the whole globe the numerical model has been verified by williamson s standard test set for shallow water equation model on sphere the results reveal that the present model is competitive to most existing ones

a class of deformational flow test cases for linear transport problems on the sphere a class of new benchmark deformational flow test cases for the two dimensional horizontal linear transport problems on the sphere is proposed the scalar field follows complex trajectories and undergoes severe deformation during the simulation however the flow reverses its course at half time and the scalar field returns to its initial position and shape this process makes the exact solution available at the end of the simulation and facilitates assessment of the accuracy of the underlying transport scheme a procedure to eliminate possible cancellations of errors when the flow reverses is proposed the test suite consists of four cases three are based on non divergent flow fields and one on a divergent flow the initial conditions are prescribed in terms of regular latitude longitude spherical coordinates and are easy to implement the divergent flow is specifically aimed for conservative global transport schemes to test for conservation consistency and monotonicity or positivity of limiters filters in a challenging flow environment in the context of semi lagrangian schemes the time varying flow fields can be used to test trajectory algorithms where the exact trajectories do not follow great circle arcs the characteristics of the test cases are demonstrated with two different transport schemes

connection graph stability method for synchronized coupled chaotic systems this paper elucidates the relation between network dynamics and graph theory a new general method to determine global stability of total synchronization in networks with different topologies is proposed this method combines the lyapunov function approach with graph theoretical reasoning in this context the main step is to establish a bound on the total length of all paths passing through an edge on the network connection graph in particular the method is applied to the study of synchronization in rings of 2k nearest neighbor coupled oscillators a rigorous bound is given for the minimum coupling strength sufficient for global synchronization of all oscillators this bound is explicitly linked with the average path length of the coupling graph contrary to the master stability function approach developed by pecora and carroll the connection graph stability method leads to global stability of synchronization and it permits not only constant but also time dependent interaction coefficients in a companion paper blinking model and synchronization in small world networks with a time varying coupling see this issue this method is extended to the blinking model of small world networks where in addition to the fixed 2 k nearest neighbor interactions all the remaining links are rapidly switched on and off independently of each other

high performance high resolution semi lagrangian tracer transport on a sphere current climate models have a limited ability to increase spatial resolution because numerical stability requires the time step to decrease we describe a semi lagrangian method for tracer transport that is stable for arbitrary courant numbers and we test a parallel implementation discretized on the cubed sphere the method includes a fixer that conserves mass and constrains tracers to a physical range of values the method shows third order convergence and maintains nonlinear tracer correlations to second order it shows optimal accuracy at courant numbers of 10 20 more than an order of magnitude higher than explicit methods we present parallel performance in terms of strong scaling weak scaling and spatial scaling where the time step stays constant while the resolution increases for a 0 2 o test with 100 tracers the implementation scales efficiently to 10 000 mpi tasks

weakly holomorphic modular forms and rank two hyperbolic kac moody algebras in this paper we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of fourier coefficients of the modular forms we use the results to construct automorphic correction of the rank 2 hyperbolic kac moody algebras h a a 4 5 6 through hilbert modular forms explicitly given by borcherds lifts of the weakly holomorphic modular forms we also compute asymptotics of the fourier coefficients as they are related to root multiplicities of the rank 2 hyperbolic kac moody algebras this work is a continuation of an earlier work of the first and second authors where automorphic correction was constructed for h a a 3 11 66

a layered analysis of consensus this paper introduces a simple notion of layering as a tool for analyzing well behaved runs of a given model of distributed computation using layering a model independent analysis of the consensus problem is performed and then applied to proving lower bounds and impossibility results for consensus in a number of familiar and less familiar models the proofs are simpler and more direct than existing ones and they expose a unified structure to the difficulty of reaching consensus in particular the proofs for the classical synchronous and asynchronous models now follow the same outline a new notion of connectivity among states in runs of a consensus protocol called potence connectivity is introduced this notion is more general than previous notions of connectivity used for this purpose and plays a key role in the uniform analysis of consensus

blinking model and synchronization in small world networks with a time varying coupling the paper proposes a new type of small world networks of cells with chaotic behavior this network consists of a regular lattice of cells with constant 2k nearest neighbor couplings and time dependent on off couplings between any other pair of cells in each time interval of duration such a coupling is switched on with probability p and the corresponding switching random variables are independent for different links and for different times at each moment the coupling structure corresponds to a small world graph but the shortcuts change from time interval to time interval which is a good model for many real world dynamical networks it is to be distinguished from networks that have randomly chosen shortcuts fixed in time here we apply the connection graph stability method developed in the companion paper connection graph stability method for synchronized coupled chaotic systems see this issue to the study of global synchronization in this network with the time varying coupling structure in the case where the on off switching is fast with respect to the characteristic synchronization time of the network the synchronization thresholds are explicitly linked with the average path length of the coupling graph and with the probability p of shortcut switchings in this blinking model we prove that for the blinking model a few random shortcut additions significantly lower the synchronization threshold together with the effective characteristic path length short interactions between cells as in the blinking model are important in practice to cite prominent examples computers networked over the internet interact by sending packets of information and neurons in our brain interact by sending short pulses called spikes the rare interaction between arbitrary nodes in the network greatly facilitates synchronization without loading the network much in this respect we believe that it is more efficient than a structure of fixed random connections

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper