a critical evaluation of the resolution properties of b spline and compact finite difference methods resolution properties of b spline and compact finite difference schemes are compared using fourier analysis in periodic domains and tests based on solution of the wave and heat equations in finite domains with uniform and nonuniform grids results show that compact finite difference schemes have a higher convergence rate and in some cases better resolution however b spline schemes have a more straightforward and robust formulation particularly near boundaries on nonuniform meshes

development of noncentered wavenumber based optimized interpolation schemes with amplification control for overlapping grids a noncentered interpolation technique has been constructed to perform simulations using overlapping grids for complex geometries high order centered lagrange polynomial interpolations and interpolations optimized in the fourier space are first generalized to the noncentered case these noncentered interpolations either generate significant dispersion errors or strongly amplify high wavenumber components accordingly a noncentered high order wavenumber based optimized interpolation method is developed with the addition of a nonlinear constraint for the control of the amplitude amplification induced by decentering high order piecewise polynomial regressions of the obtained interpolation coefficients are performed the time stability of the method is investigated in the 1 d case when the interpolation method is used in conjunction with explicit high order differencing filtering schemes as well as a 6 step runge kutta time integration algorithm a criterion is formulated to predict its stability as a function of the filtering strength and the courant friedrichs lewy constant finally 1 d convection simulations are presented to illustrate the stability and the accuracy of the developed noncentered interpolations

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

scattering of sound from axisymetric sources by multiple circular cylinders a general analytic method for calculating the scattering of sound by multiple rigid circular cylinders arranged in an arbitrary parallel configuration is presented the sound scattered by this collection of cylinders is generated by a time periodic spatially distributed axisymmetric source located within the domain of interest a hankel transform method is used to calculate the incident field while separation of variables is used to obtain the scattered fields from each cylinder in the collection the unknown scattering coefficients are determined through the use of general addition theorems that allows the various fields to be readily transformed between coordinate systems the method is validated using various two three and four cylinder configurations and the number of coefficients that must be retained in the truncated series is examined benchmark configurations consisting of two and three cylinder systems with cylinders of varying radii are also presented these solutions have been used to validate computational aeroacoustic solvers developed for complex geometries

a generalized finite difference gfd ale scheme for incompressible flows around moving solid bodies on hybrid meshfree cartesian grids a scheme using the mesh free generalized finite differencing gfd on flows past moving bodies is proposed the aim is to devise a method to simulate flow past an immersed moving body that avoids the intensive remeshing of the computational domain and minimizes data interpolation associated with the established computational fluid methodologies as such procedures are time consuming and are a significant source of error in flow simulation in the present scheme the moving body is embedded and enveloped by a cloud of mesh free nodes which convects with the motion of the body against a background of cartesian nodes the generalized finite difference gfd method with weighted least squares wls approximation is used to discretize the two dimensional viscous incompressible navier stokes equations at the mesh free nodes while standard finite difference approximations are applied elsewhere the convecting motion of the mesh free nodes is treated by the arbitrary lagrangian eulerian ale formulation of the flow equations which are solved by a second order crank nicolson based projection method the proposed numerical scheme was tested on a number of problems including the decaying vortex flow external flows past moving bodies and body driven flows in enclosures

direct numerical simulations of flow past an array of distributed roughness elements direct numerical simulation was used to describe the subsonic flow past an array of distributed cylindrical roughness elements mounted on a flat plate solutions were obtained for element heights corresponding to a roughness based reynolds number re k of both 202 and 334 the numerical method used a sixth order accurate centered compact finite difference scheme to represent spatial derivatives which was used in conjunction with a tenth order low pass pade type nondispersive filter operator to maintain stability an implicit approximately factored time marching algorithm was employed and newton like subiterations were applied to achieve second order temporal accuracy calculations were carried out on a massively parallel computing platform using domain decomposition to distribute subzones on individual processors a high order overset grid approach preserved spatial accuracy on the mesh system used to represent the roughness elements features of the flowfields are described and results of the computations are compared with experimentally measured velocity components of the time mean flowfield which are available only for re k 202 flow about the elements is characterized by a system of two weak corotating horseshoe vortices for re k 334 an unstable shear layer emanating from the top of the cylindrical element generated nonlinear unsteady disturbances of sufficient amplitude to produce explosive bypass transition downstream of the array the re k 202 case displayed exponential growth of turbulence energy in the streamwise direction which may eventually result in transition

reconstruction of turbulent fluctuations using a hybrid rans les approach a hybrid rans les approach is presented and assessed considering several flows which can be interpreted as the most general case of the nlde approach as defined by morris et al p j morris l n long a bangalore and a wang j comput phys 133 56 1997 a decomposition into three parts of the exact solution of the navier stokes equations is considered mean flow resolved fluctuations and unresolved subgrid fluctuations the mean flow is computed using a classical rans method while resolved fluctuations are derived from a les method this approach is succesfully assessed on the stationary and the pulsed plane channel flow configurations the case of the flow around a low pressure turbine blade is discussed in a second step the hybrid method is also demonstrated to be robust with respect to several sources of error and provides good results in the case of a coarsened grid simulation a reduced computational domain simulation and nonconsistent mean flow fields

two dimensional unsteady navier stokes solution method with moving overset grids a simple numerical algorithm to localize intergrid boundary points and to interpolate unsteady solution variables across two dimensional structured overset grids is presented overset grids are allowed to move in time relative to each other intergrid boundary points are localized in a triangular stencil on the donor grid by a directional search algorithm the final parameters of the search algorithm give the interpolation weights at the intergrid boundary point numerical results are presented for steady and unsteady viscous flow solutions over an airfoil undergoing a sinusoidal flapping motion computed flowfields demonstrate the accuracy of the method and excellent agreement is obtained against the single grid solutions the method is independent of numerical solution algorithms and it may easily be implemented on any two dimensional single block flow solver to make it a multiblock zonal solver with arbitrarily overset overlapping computational grids

quasi disjoint pentadiagonal matrix systems for the parallelization of compact finite difference schemes and filters this paper proposes a novel systematic approach for the parallelization of pentadiagonal compact finite difference schemes and filters based on domain decomposition the proposed approach allows a pentadiagonal banded matrix system to be split into quasi disjoint subsystems by using a linear algebraic transformation technique as a result the inversion of pentadiagonal matrices can be implemented within each subdomain in an independent manner subject to a conventional halo exchange process the proposed matrix transformation leads to new subdomain boundary sb compact schemes and filters that require three halo terms to exchange with neighboring subdomains the internode communication overhead in the present approach is equivalent to that of standard explicit schemes and filters based on seven point discretization stencils the new sb compact schemes and filters demand additional arithmetic operations compared to the original serial ones however it is shown that the additional cost becomes sufficiently low by choosing optimal sizes of their discretization stencils compared to earlier published results the proposed sb compact schemes and filters successfully reduce parallelization artifacts arising from subdomain boundaries to a level sufficiently negligible for sophisticated aeroacoustic simulations without degrading parallel efficiency the overall performance and parallel efficiency of the proposed approach are demonstrated by stringent benchmark tests

a new family of fourth order locally one dimensional schemes for the three dimensional wave equation in this paper we present a new family of locally one dimensional lod schemes with fourth order accuracy in both space and time for the three dimensional 3d acoustic wave equation it is well known that high order explicit schemes offer efficient time steppings but with restrictive cfl conditions implicit discretization gives unconditional stable schemes but with inefficient time steppings our lod schemes can be seen as a compromise between explicit and implicit schemes in the way that our schemes have more relax cfl conditions compared with traditional explicit schemes and only require solutions of tridiagonal systems of linear algebraic equations which is more efficient than implicit schemes moreover the new scheme is four layer in time and three layer in space so that boundary conditions can be imposed in the classical way the computations of the initial conditions for the three intermediate time layers are explicitly constructed furthermore the stability condition is derived and the restriction for the time step is given explicitly finally numerical examples are completed to show the performance of our new method

energy stable numerical methods for hyperbolic partial differential equations using overlapping domain decomposition overlapping domain decomposition methods otherwise known as overset grid or chimera methods are useful for simplifying the discretization of partial differential equations in or around complex geometries though in wide use such methods are prone to numerical instability unless numerical diffusion or some other form of regularization is used especially for higher order methods to address this shortcoming high order provably energy stable overlapping domain decomposition methods are derived for hyperbolic initial boundary value problems the overlap is treated by splitting the domain into pieces and using generalized summation by parts derivative operators and polynomial interpolation new implicit and explicit operators are derived that do not require regularization for stability in the linear limit applications to linear and nonlinear problems in one and two dimensions are presented where it is found the explicit operators are preferred to the implicit ones

acoustic source terms for the linearized euler equations in conservative form a rather novel approach to predict jet noise is the stochastic noise generation and radiation sngr method the sngr method uses the linear euler equations as an acoustic analogy together with source terms which are modeled in other studies the euler equations on primitive form are used in the present work the linear euler equations on conservative form are used due to this new source terms have to be derived for the conservative set of equations a formal derivation of the correct source terms for the linear euler equations on conservative form is presented simplified versions of the derived source terms are also developed to validate the derived source terms a direct simulation of a forced 2d mixing layer is carried out the solutions to the linearized euler equations with source terms are compared to the solution of the direct simulation and show a good agreement all simulations are performed using tam and webb s fourth order drp scheme and a four step fourth order runge kutta time marching technique artificial selective damping introduced through the numerical scheme is used to avoid spurious waves absorbing boundary conditions based on characteristic variables engquist and majda are used at the free boundaries and a buffer layer is added at the outflow

on the automorphism group of polyhedral graphs a 4 6 fullerene is a three connected cubic planar graph whose faces are squares and hexagons in this paper for a given 4 6 fullerene graph f we compute the order of automorphism group f we also study some spectral properties of fullerene graphs via their automorphism group

artificial dissipation models for the euler equations various artificial dissipation models that are used with central difference algorithms for the euler equations are analyzed for their effect on accuracy stability and convergence rates in particular linear and nonlinear models are investigated using an implicit approximate factorization code arc2d for transonic airfoils fully implicit application of the dissipation models is shown to improve robustness and convergence rates the treatment of dissipation models at boundaries will be examined it will be shown that accurate error free solutions with sharp shocks can be obtained using a central difference algorithm coupled with an appropriate nonlinear artificial dissipation model i introduction t he solution of the euler equations using numerical techniques requires the use of either a differencing method with inherent dissipation or the addition of dissipation terms to a nondissipative scheme this is because the euler equations do not provide any natural dissipation mechanism such as viscosity in the navier stokes equations that would eliminate high frequencies which are caused by nonlinearitie s and especially shocks a variety of numerical algorithms and computer codes for the euler equations have been developed methods such as maccormack s1 explicit

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

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

on the use of a high order overlapping grid method for coupling in cfd caa this paper presents a theoretical analysis and two applications of a high order overlapping grid method for coupling cartesian and curvilinear grids developed in order to simulate aerodynamic noise first the overlapping grid method based on lagrange interpolating polynomials is described and a theoretical analysis of the interpolation operator is then carried out it shows that the interpolation generates spurious modes that depend on the wavenumbers of the signal besides it also gives the optimal conditions in which interpolation can be applied then an application to the simulation of the aeroacoustic noise generated by the vortex shedding behind a cylinder is presented during this simulation it appears that interpolation can create some spurious acoustic modes in regions where hydrodynamic fluctuations are significant as predicted by the theoretical analysis it is shown that these spurious modes disappear when a refined cartesian grid is used 26 points per wavelength of the vortex shedding were found to be adequate in this study at last the simulation of the aerodynamic noise of a three element high lift wing profile has then been carried out for this application the main acoustic source at the slat trailing edge is represented analytically the propagation of the generated acoustic wave is simulated with a mean flow at rest and with a steady turbulent mean flow computed by rans the first application allows us to assess the method by comparing the results to a reference solution the second one shows that the influence of a non uniform mean flow on the directivity of an acoustic source can be observed in complex geometries this application therefore shows that the proposed coupling method is well adapted to complex geometries that are usually met in industrial applications

high order compact finite difference methods on general overset grids this work investigates the coupling of a very high order finite difference algorithm for the solution of conservation laws on general curvilinear meshes with overset grid techniques originally developed to address complex geometric configurations the solver portion of the algorithm is based on pade type compact finite differences of up to sixth order with up to 10th order filters employed to remove spurious waves generated by grid non uniformities boundary conditions and flow non linearities the overset grid approach is utilized as both a domain decomposition paradigm for implementation of the algorithm on massively parallel machines and as a means for handling geometric complexity in the computational domain two key features have been implemented in the current work the ability of the high order algorithm to accommodate holes cut in grids by the overset grid approach and the use of high order interpolation at non coincident grid overlaps several high order high accuracy interpolation methods were considered and a high order explicit non optimized lagrangian method was found to be the most accurate and robust for this application several two dimensional benchmark problems were examined to validate the interpolation methods and the overall algorithm these included grid to grid interpolation of analytic test functions the inviscid convection of a vortex laminar flow over single and double cylinder configurations and the scattering of acoustic waves from one and three cylinder configurations the employment of the overset grid techniques coupled with high order interpolation at overset boundaries was found to be an effective way of employing the high order algorithm for more complex geometries than was previously possible

grid adaptation using chimera composite overlapping meshes the objective of this paper is to perform grid adaptation using composite over lapping meshes in regions of large gradient to capture the salient features accurately during computation the chimera grid scheme a multiple overset mesh technique is used in combination with a navier stokes solver the numerical solution is first converged to a steady state based on an initial coarse mesh solution adaptive enhancement is then performed by using a secondary fine grid system which oversets on top of the base grid in the high gradient region but without requiring the mesh boundaries to join in any special way communications through boundary interfaces between those separated grids are carried out using tri linear interpolation applications to the euler equations for shock reflections and to a shock wave boundary layer interaction problem are tested with the present method the salient features are well resolved

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

constructing fullerene graphs from their eigenvalues and angles we discuss means of constructing fullerene graphs from their eigenvalues and angles an algorithm for such a construction is given

proper general decomposition pgd for the resolution of navier stokes equations in this work the pgd method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions in the first stage the equations of stokes and burgers will be solved then we will solve the navier stokes problem in the case of the lid driven cavity for different reynolds numbers re 100 1000 and 10 000 finally the pgd method will be compared to the standard resolution technique both in terms of cpu time and accuracy

solving navier stokes equation for flow past cylinders using single block structured and overset grids many complex fluid motions are driven by physical processes of instability transition and turbulence dependent upon nonlinear mechanisms here we solve the flow past cylinder s using single block structured and overset grids by computing navier stokes equation in two dimensions the suitability of a compact scheme in discretizing convection and diffusion terms are investigated first by looking at relevant numerical properties also for the overset grid method one of the methods is identified that shows the best results in minimizing interpolation error at sub domain boundaries for an analytical test function we provide extensive comparisons with experimental and other computational results for flow past a single cylinder utilizing both single block structured and chimera or overset grids apart from showing instability of this flow calculated by these methods we also compare the computed vorticity and velocity data using these two grids by employing the proper orthogonal decomposition pod we have analyzed and developed an overset grid method with compact scheme that does not need any filtering to control error this has been ascertained by performing pod analysis to show that the developed method is capable of handling complex geometries we have computed flow past two cylinders in side by side arrangement results obtained capture the known flow characteristics for this arrangement well using relatively fewer number of grid points

a wavenumber based extrapolation and interpolation method for use in conjunction with high order finite difference schemes the errors incurred in using extrapolation and interpolation in large scale computations are analyzed and quantified in the wavenumber space if a large extrapolation stencil is used the errors in the low wavenumbers can be significantly reduced however the errors in the high wavenumbers are at the same time greatly increased the opposite is true if the stencil size is reduced based on the wavenumber analysis an optimized extrapolation and interpolation method is proposed the optimization is carried out over a selected band of wavenumbers it is known that extrapolation often leads to numerical instability the instability is the result of large error amplification in the high wavenumber range to reduce the tendency to trigger numerical instability it is proposed that an extra constraint be imposed on the optimized extrapolation method the added constraint aims to reduce error amplification over the high wavenumbers numerical examples are provided to illustrate that accurate and stable numerical results can be obtained in large scale simulation using a high order finite difference scheme and the proposed optimized extrapolation method when the same problems are recomputed using the familiar high order polynomials extrapolation method in the lagrange form in one case the numerical results are plagued by large errors and ultimately instability in another problem it is found that the use of the lagrange polynomials extrapolation method would lead immediately to numerical instability

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

b spline method and zonal grids for simulations of complex turbulent flows a numerical technique for computations of turbulent flows is described the technique is based on b splines and allows grid embedding in physically significant flow regions numerical tests which include solutions of nonlinear advection diffusion equations and computations of flow over a circular cylinder at reynolds numbers up to 300 indicate that the method is accurate and efficient in computations of flow over a cylinder the lift drag and base suction coefficients agree well with existing experimental data and previous numerical simulations

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

polyhedral graphs via their automorphism groups abstract a polyhedral graph is a three connected simple planar graph an automorphism of a graph is a bijection on its vertices which preserves the edge set in this paper we compute the automorphism group of cubic polyhedral graphs whose faces are triangles quadrangles pentagons and hexagons in continuing we classify all cubic polyhedral graphs with cayley graph structure

time space domain dispersion relation based finite difference method with arbitrary even order accuracy for the 2d acoustic wave equation high order finite difference fd methods have been widely used for numerical solution of acoustic wave equations it has been reported that the modeling accuracy is of 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation under the same discretization the present version of time space domain dispersion relation based fd method can improve the accuracy from 2nd order to 2m th order along eight directions for the 2d acoustic wave equation to increase the accuracy further we propose a new fd stencil for 2d acoustic wave equation modeling this new time space domain dispersion relation based fd stencil can reach the same arbitrary even order accuracy along all directions and is more accurate and more stable than the conventional one for the same m dispersion analysis and modeling examples demonstrate its advantages

remarks on characteristic coefficients of fullerene graphs abstract in this paper we focus our attention on fullerene graphs and we determine some coefficients of their characteristic polynomial we also compute their characteristic coefficients respect to the width of fullerenes

on the use of higher order finite difference schemes on curvilinear and deforming meshes this study enables the use of very high order finite difference schemes for the solution of conservation laws on stretched curvilinear and deforming meshes to illustrate these procedures we focus on up to 6th order pade type spatial discretizations coupled with up to 10th order low pass filters these are combined with explicit and implicit time integration methods to examine wave propagation and wall bounded flows described by the navier stokes equations it is shown that without the incorporation of the filter application of the high order compact scheme to nonsmooth meshes results in spurious oscillations which inhibit their applicability inclusion of the discriminating low pass high order filter restores the advantages of high order approach even in the presence of large grid discontinuities when three dimensional curvilinear meshes are employed the use of standard metric evaluation procedures significantly degrades accuracy since freestream preservation is violated to overcome this problem a simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes for dynamically deforming grids an effective numerical treatment is described to evaluate expressions containing the time varying transformation metrics with these techniques metric cancellation is guaranteed regardless of the manner in which grid speeds are defined the efficacy of the new procedures is demonstrated by solving several model problems as well as by application to flow past a rapidly pitching airfoil and past a flexible panel

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

projective normality of the moduli space of rank 2 vector bundles on a generic curve we prove that the embedding to the projective space by the generalized theta divisors of the moduli space of rank 2 vector bundles on a generic curve is projective normal