practical aspects of higher order numerical schemes for wave propagation phenomena this paper examines practical issues related to the use of compact difference based fourth and sixth order schemes for wave propagation phenomena with focus on maxwell s equations of electromagnetics an outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches transformed plane method and physical space differencing in the first technique the truncation error expansion for the sixth order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available physical space differencing formulas are derived for the five point stencil by expressing the coefficients in terms of local spacing ratios the order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes to ensure stability for a broad range of problems fourier analysis is employed to develop a single parameter family of up to tenth order tridiagonal based spatial filters the implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1 d cavity where they are employed to suppress a late time instability the paper concludes after demonstrating the application of the scheme to several 3 d canonical problems utilizing cartesian as well as curvilinear meshes published in 1999 by john wiley sons ltd this article is a u s government work and is in the public domain in the united states

error dynamics the propagation of a signal in a continuous medium and the associated evolution of error is of prime importance in many applications of applied physics there have been many efforts in analyzing error dynamics using method attributed to von neumann 1 2 that is readily applied for linear equations and in quasilinearized form for non linear equations the main assumption for linear problems is that the error and the signal follow the same dynamics while this appears intuitively correct the main aim behind this work is to show that this is not correct for discrete computing due to dispersion or phase error or when the numerical method is not strictly neutrally stable we demonstrate the above with the help of the linear advection equation for the analysis of space time discretization schemes the linear advection equation as a model that represents many flows and wave phenomena is used

a new alternating bi diagonal compact scheme for non uniform grids a new compact scheme has been developed for any non uniform grid the compact scheme has been developed for spatial discretization and is analyzed here in conjunction with four stage fourth order runge kutta rk4 scheme for time integration while solving the one dimensional convection equation the space time discretization combination is calibrated by subjecting the system to global spectral analysis gsa which was developed by the authors group here the compact scheme has been obtained by using a combination of two bi diagonal schemes the novel aspect of this scheme is its application in the physical plane directly without the necessity of mapping or transformations some typical cases for problems in acoustics as well as fluid mechanics have been studied here and potential use in large eddy simulations les has been demonstrated by solving navier stokes equation for lid driven cavity

on support varieties and the humphreys conjecture in type a abstract let g be a reductive algebraic group scheme defined over f p and let g 1 denote the frobenius kernel of g to each finite dimensional g module m one can define the support variety v g 1 m which can be regarded as a g stable closed subvariety of the nilpotent cone a g module is called a tilting module if it has both good and weyl filtrations in 1997 it was conjectured by j e humphreys that when p h the support varieties of the indecomposable tilting modules align with the nilpotent orbits given by the lusztig bijection in this paper we shall verify this conjecture when g s l n 1 and p n 1

multiscale autoregressive models and wavelets the multiscale autoregressive mar framework was introduced to support the development of optimal multiscale statistical signal processing its power resides in the fast and flexible algorithms to which it leads while the mar framework was originally motivated by wavelets the link between these two worlds has been previously established only in the simple case of the haar wavelet the first contribution of this paper is to provide a unification of the mar framework and all compactly supported wavelets as well as a new view of the multiscale stochastic realization problem the second contribution of this paper is to develop wavelet based approximate internal mar models for stochastic processes this will be done by incorporating a powerful synthesis algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients taking advantage of the statistical machinery provided by the mar framework we will illustrate the application of our models to sample path generation and estimation from noisy irregular and sparse measurements

spurious waves in discrete computation of wave phenomena and flow problems in the present work we focus on spurious propagating disturbances q waves to establish the existence of q waves in computations we compare properties of different numerical methods drawn from finite difference finite volume and finite element methods existence and properties of q waves are demonstrated with propagation of wave packets following one dimensional 1d convection equation skewed wave propagation and by solution of linearized rotating shallow water wave equation lrswe specific numerical experiments are performed with parameters that convert a wave packet into a q wave we also show the case where q waves are created additionally to physical disturbances those propagate downstream formation of q waves are shown in the case of a discrete shielded vortex in the uniform flow and incompressible transitional flow past an aerofoil by solving the navier stokes equation in performing this exercise we establish critical wavenumber range beyond which q waves are created relevance of this information for dns and les is discussed we have further discussed the case of spurious caustics in discrete computing

representation theoretic support spaces for finite group schemes we introduce the space p g of abelian p points of a finite group scheme over an algebraically closed field of characteristic p 0 we construct a homeomorphism g p g proj g from p g to the projectivization of the cohomology variety for any finite group g for an elementary abelian p group respectively an infinitesimal group scheme p g can be identified with the projectivization of the variety of cyclic shifted subgroups resp variety of 1 parameter subgroups for a finite dimensional g module m g restricts to a homeomorphism p g m proj g m thereby giving a representation theoretic interpretation of the cohomological support variety

a dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations in this work we have derived an optimized upwind compact difference scheme for achieving excellent spatial resolution the derived numerical scheme adds numerical diffusion which is strictly restricted to a high wavenumber region in order to control numerical instabilities as well as to achieve de aliasing ability more importantly the derived numerical scheme has excellent dispersion relation preserving drp property the applicability of the proposed scheme in simulating real flow problems has been demonstrated by solving flow inside a lid driven cavity a transitional flow past ag24 aerofoil and a two dimensional decaying turbulence flow

a positivity preserving variational method for multi dimensional convectiondiffusionreaction equation a new positivity preserving variational ppv procedure is proposed to solve the convectiondiffusionreaction cdr equation through the generalization of stabilized finite element methods the present variational procedure offers minimal phase and amplitude errors for different regimes associated with convection diffusion and reaction effects by means of fourier analysis we first review the shortcomings of the galerkin least squares gls and the subgrid scale sgs methods during the change in sign of the reaction coefficient that motivates us for the present linear stabilization as a combined gls sgs methodology discrete upwind operator with a solution dependent nonlinear term is then introduced in high gradient regions which enables the positivity preserving property in the variational formulation direct extension to multi dimensions is carried out by considering the principle streamline and crosswind directions the efficacy of the method is demonstrated by the systematic accuracy and stability analyses in one and two dimensions results show the reduction of oscillations in the solution in one and two dimensional cases and a remarkable reduction in the phase error is observed for the cases with negative reaction coefficient the proposed formulation provides a superior solution in the reaction dominated as well as the convection dominated regimes due to the minimization of spurious oscillations and accurate capturing of the high gradient regions

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

finite difference schemes for long time integration a general method for constructing finite difference schemes for longtime integration problems is presented it is demonstrated for discretizations of first and second space derivatives however the approach is not limited to these cases the schemes are constructed so as to minimize the global truncation error taking into account the initial data the resulting second order compact schemes can be used for integration times fourfold or more longer than previously studied schemes with similar computational complexity a similar approach was used to obtain improved integration schemes

analysis of central and upwind compact schemes central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space numerical stability and dispersion relation preservation a von neumann matrix spectral analysis is developed here to analyze spatial discretization schemes for any explicit and implicit schemes to investigate the full domain simultaneously this allows one to evaluate various boundary closures and their effects on the domain interior the same method can be used for stability analysis performed for the semi discrete initial boundary value problems ibvp this analysis tells one about the stability for every resolved length scale some well known compact schemes that were found to be g k s and time stable are shown here to be unstable for selective length scales by this analysis this is attributed to boundary closure and we suggest special boundary treatment to remove this shortcoming to demonstrate the asymptotic stability of the resultant schemes numerical solution of the wave equation is compared with analytical solution furthermore some of these schemes are used to solve two dimensional navier stokes equation and a computational acoustic problem to check their ability to solve problems for long time it is found that those schemes that were found unstable for the wave equation are unsuitable for solving incompressible navier stokes equation in contrast the proposed compact schemes with improved boundary closure and an explicit higher order upwind scheme produced correct results the numerical solution for the acoustic problem is compared with the exact solution and the quality of the match shows that the used compact scheme has the requisite drp property

fractal estimation using models on multiscale trees we estimate the hurst parameter h of fractional brownian motion or by extension the fractal exponent spl phi of stochastic processes having 1 f sup spl phi like spectra by applying a multiresolution framework this framework admits an efficient likelihood function evaluation allowing us to compute the maximum likelihood estimate of this fractal parameter with relative ease in addition to yielding results that compare well with other proposed methods and in contrast with other approaches our method is directly applicable with at most very simple modification in a variety of other contexts including fractal estimation given irregularly sampled data or nonstationary measurement noise and the estimation of fractal parameters for 2 d random fields

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

analysis of anisotropy of numerical wave solutions by high accuracy finite difference methods fourier analysis is used to quantitatively assess the resolution and in particular the isotropy of wave solution using finite difference spatial discretization schemes along with fourth order runge kutta temporal scheme aspect ratio of the grid in two dimension along with the angle of wave propagation are the parameters varied to qualitatively and quantitatively assess the anisotropy of the solutions for a a skewed one dimensional wave convecting in two dimensions following the standard convection equation and b a wave propagating following the two dimensional linearized rotating shallow water equations results show the effect of changing the aspect ratio and the propagation angle on the directional nature of the solution as obtained by different methods for the above non dispersive and dispersive wave system

high order numerical sommerfeld boundary conditions theory and experiments abstract we develop high order non reflecting boundary equations for a semidiscrete approximation of the simple hyperbolic advection equation u 1 cu x 0 these boundary equation are based on a discrete interpretation of sommerfeld s radiation condition for a second order wave equation which is associated with the semi discrete equation the performance of these schemes is expressed by an exact measure of the energy reflected at the boundary for low order cases the discrete sommerfeld boundary equations are identical with the standard finite difference equations but for high orders of approximation starting with 4 points the discrete sommerfeld schemes differ from standard finite differences it is shown and verified experimentally that the discrete sommerfeld schemes are optimal in the sense that they produce the least amount of reflected energy moreover it is known theoretically and we verify experimentally that the reflected energy remains invariat when the semi discrete equation are time discretized with the trapezoidal crank nicolson method the corresponding fully discrete boundary equations are thus also optimal in the sense that they minimize the reflected energy

global spectral analysis of multi level time integration schemes an analysis is reported here for three time level integration methods following the global spectral analysis gsa described in high accuracy computing methods t k sengupta cambridge univ press usa the focus is on the second order adamsbashforth ab2 and the extrapolation in time ext2 methods careful distinction is made for the first time step at t 0 by either euler forward or four stage fourth order rungekutta rk4 time schemes the latter is used to solve a benchmark aeroacoustic problem several one dimensional wave propagation models are analyzed pure advection and advection diffusion equations various spatial discretizations are discussed including fourier spectral method attention is paid to the presence of physical and numerical modes as noted in the quadratic equation obtained from the difference equation for the model 1d convection equation it is shown that ab2 method is less stable and accurate than ext2 method with respect to numerical dissipation and dispersion this is true for the methods in which the physical mode dominates over the numerical mode presented analysis provides useful guide to analyze any three time level methods

the cohomology of line bundles on the three dimensional flag variety we give a recursive description of the characters of the cohomology of the line bundles of the three dimensional flag variety over an algebraically closed field of characteristic p the recursion involves also certain rank two bundles and we calculate their cohomology at the same time the method of proof is to adapt an expansion formula valid generally for homogeneous vector bundles on flag varieties g b to the case in which g is the special linear group of degree 3 in order to carry this out it is necessary to calculate explicitly the module of invariants for the action of the first infinitesimal subgroup of the unipotent radical of a borel subgroup of g on certain tilting modules

differentiating the weyl generic dimension formula with applications to support varieties abstract the authors compute the support varieties of all irreducible modules for the small quantum group u g where g is a finite dimensional simple complex lie algebra and is a primitive l th root of unity with l larger than the coxeter number of g the calculation employs the prior calculations and techniques of ostrik and of nakano parshall and vella as well as deep results involving the validity of the lusztig character formula for quantum groups and the positivity of parabolic kazhdan lusztig polynomials for the affine weyl group analogous support variety calculations are provided for the first frobenius kernel g 1 of a reductive algebraic group scheme g defined over the prime field f p

support varieties of line bundle cohomology groups for sl3 k abstract let g sl 3 k where k is a field of characteristic p 0 and let x t be any weight with corresponding line bundle l on g b in this paper we compute the support varieties for all modules of the form h i h i g b l over the first frobenius kernel g 1 the calculation involves certain recursive character formulas given by donkin cf 5 which can be used to compute the characters of the line bundle cohomology groups in the case where is a p regular weight and m h i 0 for some i these formulas are used to show that any pth root of unity is not a root of the generic dimension of m to handle the case where is not p regular we employ techniques similar to those used by drupieski nakano and parshall cf 7 to show that the module h i is not projective over g 1 whenever it is nonzero and lies outside of the steinberg block

multiscale systems kalman filters and riccati equations an algorithm analogous to the rauch tung striebel algorithm spl minus consisting of a fine to coarse kalman filter like sweep followed by a coarse to fine smoothing step spl minus was developed previously by the authors ibid vol 39 no 3 p 464 78 1994 in this paper they present a detailed system theoretic analysis of this filter and of the new scale recursive riccati equation associated with it while this analysis is similar in spirit to that for standard kalman filters the structure of the dyadic tree leads to several significant differences in particular the structure of the kalman filter error dynamics leads to the formulation of an ml version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the hamiltonian for the smoothing problem in addition the notion of stability for dynamics requires some care as do the concepts of reachability and observability using these system theoretic constructs the stability and steady state behavior of the fine to coarse kalman filter and its riccati equation are analysed

error growth and phase lag analysis for high courant numbers abstract the type of numerical scheme used in numerical computation of fluid flow largely affects the accuracy of the solution in order to obtain accurate numerical solution in a transient analysis implicit schemes are known to be unconditionally stable for linear systems while non linearity in advection systems often limits the stability nevertheless the implicit schemes for non linear systems allow larger courant numbers than explicit schemes however these assertions have been addressed based on a simple numerical error growth analysis which does not account for dispersion and phase lag errors even in linear systems thus motivated by the correct error dynamics given in sengupta et al 2007 it has been shown here that stable and accurate numerical solutions are possible for higher courant numbers while their low values do not always lead to stable and accurate solutions here higher courant numbers can be achieved by finer grid spacing keeping the time step same time integration is performed by implicit scheme for which numerical stability is guaranteed and thus a higher courant number implies higher resolution of the solution this has been proven through a test case of the propagation of acoustic wave in water by considering the latter to be a compressible medium by solving unsteady navier stokes equations this observation is elucidated by evaluating the error growth with first order in time and second order in space schemes with the latter represented by fully implicit form at t n 1 t the results show that for high courant numbers the value of error growth is very close to unity and confirms to the needed neutral stability of the obtained numerical solution furthermore the value of phase error nearly vanishes and reveals that numerical waves travel with nearly the same speed as physical waves

global spectral analysis for convection diffusion reaction equation in one and two dimensions effects of numerical anti diffusion and dispersion abstract convection diffusion reaction cdr equation plays a central role in many disciplines of engineering science and finances as a consequence importance of analysis of numerical methods for the accurate solution of cdr equation has motivated the present research we have used the global spectral analysis to characterize all the three important physical processes in terms of the non dimensional numerical parameters namely the non dimensional wavenumber kh courant friedrich lewy cfl number n c the peclet number pe and the damkohler number da for the purpose of illustration we have focused on two space time discretization schemes known for accuracy and robustness the basic properties relate to numerical issues arising for the numerical amplification factor numerical diffusion coefficient numerical phase speed and numerical group velocity with the help of model one dimensional 1d and two dimensional 2d cdr equations we have reported the numerical property charts for the cases i when all the processes of convection diffusion and reaction are of same order with critical numerical behaviour enforcing low values of da for the 1d cdr equation studied here ii the 2d cdr equation considered is diffusion reaction dominated and as a consequence this enforces da to be larger we have thoroughly analyzed these cases to identify the essential roles of anti diffusion on the critical n c and pe values which in turn decides admissible space and time steps to be used with the discretization schemes the property charts have been used to calibrate the analysis with two model equations one of which has an exact solution for a 1d cdr equation and the second case for the 2d cdr equation has numerical solution available in the literature these cases help to identify the importance of such analysis in explaining the utility of the choice one can exercise in fixing the numerical parameters this also identifies and explains some hitherto unknown numerical problems for cdr equation and their alleviation techniques

a new flux vector splitting compact finite volume scheme a new high resolution finite volume method is proposed here that uses flux vector splitting as the building block to represent the physical processes the high accuracy projection in space is obtained here inter alia by using some compact schemes of sengupta et al 1 and lele 2 that have extremely high accuracy in spectral plane this produces a new class of finite volume fv scheme for spatial discretization that has been analyzed for spectral accuracy numerical stability and dispersion relation preservation drp property following the full domain analysis method of 1 using two different time integration strategies these combination of spatial and temporal methods have been tested for two linear wave problems reported in 1 3 and the solution of burgers equation is compared with the analytical results presented in adams and shariff 4 the shock capturing method is based on the proposed technique in 5 that does not require the usage of any nonlinear differencing techniques or limiters to show that the proposed method can also handle irregular grids the burgers equation is solved using non uniform grids also to demonstrate the utility of the proposed scheme to solve practical problems the benchmark problem of nonlinear wave propagation in a one dimensional shock tube has been solved with the initial data as given in 23 by solving the euler equation the analysis and comparison of this flux vector splitting scheme with other well known methods clearly demonstrate the superior scale resolution of the proposed method at the same time providing a neutrally stable scheme when used with four stage runge kutta scheme also the same scheme displays extended ranges of wave numbers and circular frequencies over which drp property is valid the computed results of three test cases and their comparison with analytical results clearly reveal that the presented method can be used for practical problems

grid optimized dispersion relation preserving schemes on general geometries for computational aeroacoustics abstract in this paper we investigate the dispersion relation preserving property of a finite difference scheme on general geometries for computational aeroacoustics where nondispersive and nondissipative properties are of critical importance the analysis pertains to the application of the optimization algorithm of the dispersion relation preserving drp scheme in wave number space to the general geometries nonuniform cartesian and curvilinear grids in many computational aeroacoustics applications the drp schemes have often been favored for their accuracy and efficiency drp schemes however are implemented only on uniform cartesian grids practical problems in aeroacoustics however are seldom confined to uniform cartesian grids with the associated computing grids often being nonuniform cartesian or curvilinear grid optimized dispersion relation preserving godrp finite difference schemes are proposed based on optimization that gives finite difference equations locally the same dispersion relation as the original partial differential equations on the grid points in the nonuniform cartesian or curvilinear mesh this local dispersion relation preserving property guarantees global accuracy of numerical schemes in the wave number space over the full domain the basic idea behind mathematical formulations of godrp schemes is that the optimization in the wave number space is carried out not on the computational domain but on the physical domain because the properties of cartesian and curvilinear grids differ whether the coupling of the coordinate variables between the physical and computational domains exists or not different mathematical formulations are developed for each grid type to investigate the effectiveness of godrp schemes a sequence of benchmark problems is executed through many numerical test problems it is shown that the use of godrp schemes can broaden the application area of conventional drp schemes to aeroacoustic phenomena enhancing both the speed and accuracy of the computation using nonuniform cartesian or curvilinear grids

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

on the spurious solutions in the high order finite difference methods for eigenvalue problems in this paper the origins of spurious solutions occurring in the high order finite difference methods are studied based on a uniform mesh spurious modes are found in the high order one sided finite difference discretizations of many eigenvalue problems spurious modes are classified as spectral pollution and non spectral pollution the latter can be partially avoided by mesh refinement while the former persists when the mesh is refined through numerical studies of some prototype eigenvalue problems such as those of the helmholtz and beam equations we show that perfect central differentiation schemes do not produce any spurious modes nevertheless high order central difference schemes encounter difficulties in implementing complex boundary conditions we further show that the central difference schemes will produce spurious modes as well if asymmetric approximation is involved in boundary treatments in general central difference schemes are less likely to produce spurious modes than one sided difference schemes 2007 elsevier b v all rights reserved

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100