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

some results about the chaotic behavior of cellular automata we study the behavior of cellular automata ca for short in the cantor besicovitch and weyl topologies we solve an open problem about the existence of transitive ca in the besicovitch topology the proof of this result has some interest of its own since it is obtained by using kolmogorov complexity to our knowledge it is the first result about discrete dynamical systems obtained using kolmogorov complexity we also prove that in the besicovitch topology every ca has either a unique periodic point thus a fixed point or an uncountable set of periodic points this result underlines the fact that ca have a great degree of stability it may be considered a further step towards the understanding of ca periodic behavior moreover we prove that in the besicovitch topology there is a special set of configurations the set of toeplitz configurations that plays a role similar to that of spatially periodic configurations in the cantor topology that is it is dense and has a central role in the study of surjectivity and injectivity finally it is shown that the set of spatially quasi periodic configurations is not dense in the weyl topology

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

lanczos algorithm for the quadratic eigenvalue problem in engineering applications abstract a new approach of employing the lanczos two sided recursion to solve the quadratic eigenvalue problem is presented the methodology employed retains the n order quadratic problems as posed without the need to use the method of matrix augmentation traditionally used to cast the problem as a linear eigenvalue problem of order 2n pursuing the concept of finding a characteristic matrix to yield an eigenvalue square matrix the lanczos recursion has been devised to solve the quadratic eigenvalue problem appropriate proofs showing the lanczos biorthogonal transformation are included the example problems presented validate the algorithm and show the effectiveness of the method for large quadratic eigenproblems

on the effect of numerical errors in large eddy simulations of turbulent flows aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods analytical and numerical studies show that aliasing errors are more destructive for spectral and high order finite difference calculations than for low order finite difference simulations numerical errors have different effects for different forms of the nonlinear terms in the navier stokes equations for divergence and convective forms spectral methods are energy conserving only if dealiasing is performed for skew symmetric and rotational forms both spectral and finite difference methods are energy conserving even in the presence of aliasing errors it is shown that discrepancies between the results of dealiased spectral and standard nondialiased finite difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences the relative importance of aliasing and truncation errors as compared to subgrid scale model terms in large eddy simulations is analyzed and discussed for low order finite difference simulations truncation errors can exceed the magnitude of the subgrid scale term

regular article on a class of pade finite volume methods a class of pade finite volume methods providing an improved spectral resolution is presented and compared with well known methods the formulation is based on the sliding averages of the variables and allows the computation of derivatives of all orders using the fourier analysis these methods are examined with respect to i order of accuracy ii spectral resolution iii boundary conditions and iv stability

numerical algorithms for undamped gyroscopic systems abstract the solutions of a gyroscopic vibrating system oscillating about an equilibrium position with no external applied forces and no damping forces are completely determined by the quadratic eigenvalue problem 2 i m i g k x i 0 for i 1 2 n where m g and k are real n n matrices and m is symmetric positive definite denoted by m 0 g is skew symmetric and either k 0 or k 0 gyroscopic system in motion about a stable equilibrium position with k 0 are well understood two lanczos type algorithms the pseudo skew symmetric lanczos algorithm and the j lanczos algorithm are studied for computing some extreme eigenpairs for solving gyroscopic systems in motion about an unstable equilibrium position with k 0 shift and invert strategies error bounds implementation issues and numerical results for both algorithms are presented in details

implicit high order accurate in space algorithms for the navier stokes equations high order accurate alternating direction implicit schemes have been developed these schemes can be used to obtain efe ciently high resolution numerical solutions of the compressible euler and navier stokes equations implicit solvers compatible with both centered and upwind schemes for the discretization of the convective terms arediscussed forcenteredspacediscretizationoftheconvectivee uxes fourth orderspatialaccuracyoftheimplicit operators is obtained at no additional computing cost by performing compact space differentiation this implicit compact scheme which requires block tridiagonal matrix inversions is used to obtain viscous and inviscid e ow solutions in curvilinear coordinates in addition high order accurate in space implicit operators are obtained for diagonalized schemes which require scalar matrix inversions both high order algorithms improve convergence characteristics for subsonic and transonic e ow compared to second order accurate in space implicit schemes high order accurate in space implicit algorithms compatible with upwind high resolution shock capturing total variation diminishing schemes are also developed third order accurate upwind biased diagonalized algorithms were found to be as accurate and more efe cient than the standard block tridiagonal implicit solvers

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

a staggered compact finite difference formulation for the compressible navier stokes equations in this paper a compact high order up to 12th order numerical method to solve the compressible navier stokes equations will be presented a staggered arrangement of the variables has been used it is shown that the method is not only very accurate but numerically also very stable even in the case that not all the energy containing scales in the flow are resolved this in contrast to standard collocated compact finite difference methods some results for a turbulent non reacting and a reacting jet with a reynolds number of 10 000 and a mach number of 0 5 are reported

energy preserving turbulent simulations at a reduced computational cost energy conserving discretizations are widely regarded as a fundamental requirement for high fidelity simulations of turbulent flows the skew symmetric splitting of the nonlinear term is a well known approach to obtain semi discrete conservation of energy in the inviscid limit however its computation is roughly twice as expensive as that of the divergence or advective forms alone a novel time advancement strategy that retains the conservation properties of skew symmetric based schemes at a reduced computational cost has been developed this method is based on properly constructed runge kutta schemes in which a different form advective or divergence for the convective term is adopted at each stage a general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew symmetric based techniques

a generalized lanczos scheme it is shown in this paper how the lanczos algorithm can be generalized so that it applies to both symmetric and skew symmetric matrices and corresponding generalized eigenvalue problems

an analysis of numerical errors in large eddy simulations of turbulence the reliability of numerical simulations of turbulence depend on our ability to quantify and control discretization errors in the classical literature on error analysis typically simple linear equations are studied estimates of errors derived from such analyses depend on the assumption that each dependent variable can be characterized by a unique amplitude and scale of spatial variation that can be normalized to unity this assumption is not valid for strongly nonlinear problems such as turbulence where nonlinear interactions rapidly redistribute energy resulting in the appearance of a broad continuous spectrum of amplitudes in such situations the numerical error as well as the subgrid model can change with grid spacing in a complicated manner that cannot be inferred from the results of classical error analysis in this paper a formalism for analyzing errors in such nonlinear problems is developed in the context of finite difference approximations for the navier stokes equations when the flow is fully turbulent analytical expressions for the power spectra of these errors are derived by exploiting the joint normal approximation for turbulent velocity fields these results are applied to large eddy simulation of turbulence to obtain quantitative bounds on the magnitude of numerical errors an assessment of the significance of these errors in made by comparing their magnitudes with that of the nonlinear and subgrid terms one method of controlling the errors is suggested and its effectiveness evaluated through quantitative measures although explicit evaluations are presented only for large eddy simulation the expressions derived for the power spectra of errors are applicable to direct numerical simulation as well

multidimensional optimization of finite difference schemes for computational aeroacoustics because of the long propagation distances computational aeroacoustics schemes must propagate the waves at the correct wave speeds and lower the isotropy error as much as possible the spatial differencing schemes are most frequently analyzed and optimized for one dimensional test cases therefore in multidimensional problems such optimized schemes may not have isotropic behavior in this work optimized finite difference schemes for multidimensional computational aeroacoustics are derived which are designed to have improved isotropy compared to existing schemes the derivation is performed based on both taylor series expansion and fourier analysis various explicit centered finite difference schemes and the associated boundary stencils have been derived and analyzed the isotropy corrector factor a parameter of the schemes can be determined by minimizing the integrated error between the phase or group velocities on different spatial directions the order of accuracy of the optimized schemes is the same as that of the classical schemes the advantage being in reducing the isotropy error the present schemes are restricted to equally spaced cartesian grids so the generalized curvilinear transformation method and cartesian grid methods are good candidates the optimized schemes are tested by solving various multidimensional problems of aeroacoustics

on the spectrum of an operator pencil with applications to wave propagation in periodic and frequency dependent materials we study wave propagation in periodic and frequency dependent materials when the medium in a frequency interval is characterized by a real valued permittivity the spectral parameter relates to the quasi momentum which leads to spectral analysis of a quadratic operator pencil where frequency is a parameter we show that the underlying operator has a discrete spectrum where the eigenvalues are symmetrically placed with respect to the real and imaginary axis moreover we discretize the operator pencil with finite elements and use a krylov space method to compute eigenvalues of the resulting large sparse matrix pencil

analysis of the symmetric lanczos algorithm with reorthogonalization methods we present an error analysis of the symmetric lanczos algorithm in finite precision arithmetic the loss of orthogonality among the computed lanczos vectors is explained with the help of a recurrence formula a backward error analysis shows that semiorthogonality among the lanczos vectors is enough to guarantee the accuracy of the computed quantities up to machine precision the results of this analysis are then extended to the more general case of the lanczos algorithm with a semiorthogonalization strategy based on the recurrence formula a new reorthogonalization method called partial reorthogonalization is introduced we show that both partial reorthogonalization and selective orthogonalization as introduced by parlett and scott 15 are semiorthogonalization strategies finally we discuss the application of our results to the solution of linear systems of equations and to the eigenvalue problem

solving sparse symmetric generalized eigenvalue problems without factorization in this paper we discuss an iterative technique for finding the algebraically smallest or largest eigenvalue of the generalized eigenvalue problem a lambda m where a and m are real symmetric and m is positive definite we assume that a and m are such that it is undesirable to factor the matrix a sigma m for any value of sigma we prove that the algorithm is globally convergent and that convergence is asymptotically quadratic finally we discuss the modifications required in the algorithm to make it computationally feasible

complex shift and invert strategies for real matrices abstract when using an iterative method for solving a generalized nonsymmetric eigenvalue problem of the form fu mu where f and m are real banded matrices it is often desirable to work with the shifted and inverted operator b f m 1 m in order to enhance the eigenvalue separation and improve efficiency unfortunately the shift is generally complex and so is the matrix b the question then is whether it is possible to avoid complex arithmetic while preserving any advantages of bandedness of the pair f m for the classical problem where m i and f is banded complex arithmetic can be avoided by using double shifts i e by working with the real matrix bb whose bandwidth is double that of f this satisfactory solution extends to the case where m is diagonal as well in the generalized case the answer to the above question is negative in the sense that complex arithmetic can be avoided only at the expense of losing the advantage of bandedness one solution is to factor the shifted matrix f m in complex arithmetic but employ real arithmetic subsequently in the iterative procedure this paper examines several approaches and discusses their respective merits under various circumstances

combined compact difference scheme for the time fractional convection diffusion equation with variable coefficients combined compact finite difference scheme for solving the time fractional convection diffusion reaction equation is given the primary unknown and its derivative are solved simultaneously the coefficients of the fractional partial differential equation are variable both dirichlet and robin boundary conditions are discussed and the algorithm gives high order convergence results fourth order combined compact finite difference scheme is given for solving the time fractional convection diffusion reaction equation with variable coefficients we introduce the flux as a new variable and transform the original equation into a system of two equations compact difference is used as a high order approximation for spatial derivatives of integer order in the coupled partial differential equations the caputo fractional derivative is discretized by a high order approximation both dirichlet and robin boundary conditions are discussed convergence analysis is given for the problem of integer order with constant coefficients under some assumption numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

spectral compact difference hybrid computation of passive scalar in isotropic turbulence an accurate and efficient hybrid numerical method is developed for direct numerical simulation of passive scalar in homogeneous turbulence for the schmidt number 1 and 50 the hybrid method uses the standard fourier spectral method for the incompressible navier stokes equation and the combined compact difference scheme for the passive scalar transport equation accuracy of the method is carefully examined by comparing with the full spectral method regarding the spectra probability density function field structure of the passive scalar and is found to be very satisfactory the computational time for the hybrid method is decreased by 26 for the schmidt number 1 when compared to the full spectral method and by 77 for the schmidt number 50 when the number of grid points for the velocity field is reduced under the scale separation

cellular non deterministic automata and partial differential equations abstract we define cellular non deterministic automata cnda in the spirit of non deterministic automata theory they are different from the well known stochastic automata we propose the concept of deterministic superautomata to analyze the dynamical behavior of a cnda and show especially that a cnda can be embedded in a deterministic cellular automaton as an application we discuss a connection between certain partial differential equations and cnda

projective block lanczos algorithm for dense hermitian eigensystems projection operators are used to effect deflation by restriction and it is argued that this is an optimal lanczos algorithm for memory minimization algorithmic optimization is constrained to dense hermitian eigensystems where a significant number of the extreme eigenvectors must be obtained reliably and completely the defining constraints are operator algebra without a matrix representation and semi orthogonalization without storage of krylov vectors other semi orthogonalization strategies for lanczos algorithms and conjugate gradient techniques are evaluated within these constraints large scale sparse complex numerical experiments are performed on clusters of magnetic dipoles a quantum many body system that is not block diagonalizable plane wave density functional theory of beryllium clusters provides examples of dense complex eigensystems use of preconditioners and spectral transformations is evaluated in a preprocessor prior to a high accuracy self consistent field calculation

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 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

short note turbulent channel flow simulations in convecting reference frames we perform finite difference numerical simulations of turbulent channel flow in stationary and moving reference frames the flow statistics computed in the stationary frame exhibit significant discrepancies with respect to reference spectral data in terms of energy spectra and convection velocity of disturbances especially in the near wall region on the other hand simulations performed in a reference frame moving with the bulk flow velocity suffer from this shortcoming to a much lesser extent we propose an explanation of these observations by assuming the scalar burgers equation as working model the analysis highlights the lack of galilean invariance of finite difference schemes which exhibit a leading dispersive error in the presence of bulk advection the numerically computed energy spectra exhibit the same features as the channel spectra supporting the notion that dispersive effects are responsible for the blockage of the mechanism of energy cascade to high wavenumbers

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

a robust colocated implicit algorithm for direct numerical simulation of compressible turbulent flows a non dissipative robust implicit algorithm is proposed for direct numerical and large eddy simulation of compressible turbulent flows the algorithm addresses the problems caused by low mach numbers and under resolved high reynolds numbers it colocates variables in space to allow easy extension to unstructured grids and discretely conserves mass momentum and total energy the navier stokes equations are non dimensionalized using an incompressible scaling for pressure and the energy equation is used to obtain an expression for the velocity divergence a pressure correction approach is used to solve the resulting equations such that the discrete divergence is constrained by the energy equation as a result the discrete equations analytically reduce to the incompressible equations at very low mach number i e the algorithm overcomes the acoustic time scale limit without preconditioning or solution of an implicit system of equations the algorithm discretely conserves kinetic energy in the incompressible inviscid limit and is robust for inviscid compressible turbulence on the convective time scale these properties make it well suited for dns les of compressible turbulent flows results are shown for acoustic propagation the incompressible taylor problem periodic shock tube problem and isotropic turbulence

compact high order numerical schemes for scalar hyperbolic partial differential equations abstract this work is a comparative study on the numerical behavior of compact high order schemes for discretizing scalar hyperbolic problems to this aim we investigate two possibilities a high order hermitian scheme hups based upon the scalar variable and its first derivatives in space and a classical high order upwind scheme cups based upon the scalar variable and a wide numerical stencil as a reference scheme to study the theoretical properties of both schemes we use a spectral analysis based on a fourier transform of the numerical solution in a one dimensional context this analysis enables us to investigate the impact of the spurious components generated by the hermitian schemes as demonstrated by the spectral analysis in some circumstances this spurious component may deeply alter the accuracy of the scheme and even in some cases destroy its consistency thus by identifying the salient features of a hermitian scheme we try to design efficients hermitian schemes for multi dimensional purposes a two dimensional fourier analysis and some scalar numerical tests allow to highlight two high order versions of a hermitian scheme hups4 5 of which the numerical properties are compared with a classical fifth order scheme cups5 based upon a wider stencil

higher entropy conservation and numerical stability of compressible turbulence simulations we present a numerical formulation for the treatment of nonlinear instabilities in shock free compressible turbulence simulations the formulation is high order and contains no artificial dissipation numerical stability is enhanced through semi discrete satisfaction of global conservation properties stemming from the second law of thermodynamics and the entropy equation the numerical implementation is achieved using a conservative skew symmetric splitting of the nonlinear terms the robustness of the method is demonstrated by performing unresolved numerical simulations and large eddy simulations of compressible isotropic turbulence at a very high reynolds number results show the scheme is capable of capturing the statistical equilibrium of low mach number compressible turbulent fluctuations at infinite reynolds number comparisons with the entropy splitting technique j comput phys 162 2000 33 j comput phys 178 2002 307 staggered method j comput phys 191 2 2003 392 and skew symmetric like schemes j comput phys 161 2000 114 confirm the superiority of the current approach we also discuss a flaw in the skew symmetric splitting implemented in the literature very good results are obtained based on the proper splitting

a new method for diagonalising large matrices the structure and implementation of a new general iterative method for diagonalising large matrices the residual minimisation direct inversion in the iterative subspace method of bendt and zunger are described and contrasted with other more commonly used iterative techniques the method requires the direct diagonalisation of only a small submatrix does not require the storage of the large matrix and provides eigensolutions to within a prescribed precision in a rapidly convergent iterative procedure numerical results for two rather different matrices a real 50 50 non diagonally dominant matrix and a complex hermitian 181 181 matrix corresponding to the pseudopotential band structure of a semiconductor in a plane wave basis set are used to compare the new method with the competing methods the new method converges quickly and should be the most efficient for very large matrices in terms both of computation time and central storage requirements it is quite insensitive to the properties of the matrices used this technique makes possible efficient solution of a variety of quantum mechanical matrix problems where large basis set expansions are required

a solution of the eigenproblem for undamped gyroscopic systems with the lanczos algorithm this paper presents an efficient numerical solution of the quadratic eigenproblem arising in the analysis of gyroscopic systems such problems are known to reduce to a generalized linear eigenproblem defined by two real non singular matrices one symmetric and one skew symmetric for this class of problems the general lanczos algorithm for unsymmetric matrices is shown to simplify considerably and yields an efficient solution of the problem full advantage can be taken of the sparsity of the matrices and of the specific nature of gyroscopic systems numerical examples are presented which demonstrate the efficiency and accuracy of the solution procedure

on the effective accuracy of spectral like optimized finite difference schemes for computational aeroacoustics the present article aims at highlighting the strengths and weaknesses of the so called spectral like optimized explicit central finite difference schemes when the latter are used for numerically approximating spatial derivatives in aeroacoustics evolution problems with that view we first remind how differential operators can be approximated using explicit central finite difference schemes the possible spectral like optimization of the latter is then discussed the advantages and drawbacks of such an optimization being theoretically studied before they are numerically quantified for doing so two popular spectral like optimized schemes are assessed via a direct comparison against their standard counterparts such a comparative exercise being conducted for several academic test cases at the end general conclusions are drawn which allows us discussing the way spectral like optimized schemes shall be preferred or not to standard ones when it comes to simulate real life aeroacoustics problems

a curtis hedlund lyndon theorem for besicovitch and weyl spaces global functions of cellular automata on state spaces equipped with the cantor topology are well characterized by the curtis hedlund lyndon theorem in this paper we develop a characterization of global functions of cellular automata on z if the state space is equipped by weyl and besicovitch topology the necessary and sufficient condition for a function to be the global map of a cellular automaton is 1 a strong localization property a condition that strengthen lipschitz continuity 2 the set of cantor periodic states are positively invariant and 3 the function commutes in the weyl besicovitch sense with the shift operator

improved algorithms for the lowest few eigenvalues and associated eigenvectors of large matrices abstract some modifications of davidson s eigenvalue algorithm are discussed and their performances on a number of test cases are assessed they are found to offer improvements over the original algorithm a method for solving the equations stemming from the quasi degenerate variational perturbation theory is presented solutions can be obtained as simply as with the eigenvalue algorithm for both ground and excited states

the lanczos algorithm with partial reorthogonalization the lanczos algorithm is becoming accepted as a powerful tool for finding the cigenvalues and for solving linear systems of equations any practical implementation of the algorithm suffers however from roundoff errors which usually cause the lanczos vectors to lose their mutual orthogonality in order to maintain some level of orthogonality full reorthogonalization fro and selective orthogonalization so have been used in the past as a remedy here partial reorthogonalization pro is proposed as a new method for maintain ing semiorthogonality among the lanczos vectors pro is based on a simple recurrence which allows us to monitor the loss of orthogonality among the lanczos vectors directly wvithout computing the inner products based on the information from the recurrence reorthogonalizations occur only when necessary thus substantial savings are made as compared to fro in some numerical examples we apply the lanczos algorithm with pro to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the lanczos vectors the results obtained compare favorably with the conjugate gradient method

computing eigenvalues of very large symmetric matrices an implementation of a lanczos algorithm with no reorthogonalization many engineering and scientific applications require the computation of eigenvalues and eigenvectors of very large symmetric or hermitian matrices we describe a lanczos procedure which allows us to compute either few or many eigenvalues of such matrices in any intervals specified by the user this procedure can even be used to compute all of the eigenvalues the desired eigenvalues are computed as eigenvalues of an associated symmetric tridiagonal matrix tm whose order depends upon the distribution of the eigenvalues in the given matrix a and upon which portions of the spectrum of a are desired the storage requirements depend linearly upon the order of a if the the storage required to generate the products ax is also linear in the order the amount of computation required depends directly upon the distribution of the desired eigenvalues and upon the cost of computing ax numerical results for a very large matrix of order 4900 demonstrate that this procedure can be used on very large matrices

on the sensitivity of additive cellular automata in besicovitch topologies we prove that additive cellular automata in the besicovitch topology that have a willson limit set of hausdorff dimension strictly bigger than 1 are sensitive to initial conditions

a family of high order finite difference schemes with good spectral resolution this paper presents a family of finite difference schemes for the first and second derivatives of smooth functions the schemes are hermitian and symmetric and may be considered a more general version of the standard compact pade schemes discussed by lele they are different from the standard pade schemes in that the first and second derivatives are evaluated simultaneously for the same stencil width the proposed schemes are two orders higher in accuracy and have significantly better spectral representation eigenvalue analysis and numerical solutions of the one dimensional advection equation are used to demonstrate the numerical stability of the schemes the computational cost of computing both derivatives is assessed and shown to be essentially the same as the standard pade schemes the proposed schemes appear to be attractive alternatives to the standard pade schemes for computations of the navier stokes equations

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

new methods for calculations of the lowest eigenvalues of the real symmetric generalized eigenvalue problem a new iterative method based on a newton correction vector for extension of the krylov subspace its diagonal and band versions are proposed for calculation of selected lowest eigenvalues and corresponding eigenvectors of the generalized symmetric eigenvalue problem additionally diagonal and band jacobi davidson methods are introduced test calculations show that the new iterative method usually converges faster than quadratic near a solution the new iterative method along with its band version uses a smaller number of iterative steps to obtain a solution compared to the jacobi davidson band jacobi davidson method and generalized davidson method correspondingly the diagonal version of the new method preserves an advantage over the diagonal jacobi davidson and the davidson method

compact finite volume schemes on boundary fitted grids the paper focuses on the development of a framework for high order compact finite volume discretization of the three dimensional scalar advection diffusion equation in order to deal with irregular domains a coordinate transformation is applied between a curvilinear non orthogonal grid in the physical space and the computational space advective fluxes are computed by the fifth order upwind scheme introduced by pirozzoli s pirozzoli conservative hybrid compact weno schemes for shock turbulence interaction j comp phys 178 2002 81 while the coupled derivative scheme m h kobayashi on a class of pade finite volume methods j comp phys 156 1999 137 is used for the discretization of the diffusive fluxes numerical tests include unsteady diffusion over a distorted grid linear free surface gravity waves in a irregular domain and the advection of a scalar field the proposed methodology attains high order formal accuracy and shows very favorable resolution characteristics for the simulation of problems with a wide range of length scales

a new compact spectral scheme for turbulence simulations we propose a new kind of compact difference scheme for the computation of the first and second derivatives in the simulation of high reynolds number turbulent flows the scheme combines and truncates the pseudospectral representation of derivative for convergence acceleration comparison of the wave resolution property with available optimized compact schemes minimizing the prescribed wave resolution error reveals our scheme s superiority for the same size of stencils without introducing optimization parameters an accompanying boundary scheme is also proposed with the stability analysis the proposed scheme is tested for the evaluation of derivatives of a function that decays very slowly in the wavenumber space and for the simulation of three dimensional isotropic turbulence

a new combined stable and dispersion relation preserving compact scheme for non periodic problems a new compact scheme is presented for computing wave propagation problems and navier stokes equation a combined compact difference scheme is developed for non periodic problems called nccd henceforth that simultaneously evaluates first and second derivatives improving an existing combined compact difference ccd scheme following the methodologies in sengupta et al t k sengupta s k sircar a dipankar high accuracy schemes for dns and acoustics j sci comput 26 2 2006 151 193 stability and dispersion relation preservation drp property analysis is performed here for general ccd schemes for the first time emphasizing their utility in uni and bi directional wave propagation problems that is relevant to acoustic wave propagation problems we highlight a specific points in parameter space those give rise to least phase and dispersion errors for non periodic wave problems b the solution error of ccd nccd schemes in solving stommel ocean model an elliptic p d e and c the effectiveness of the nccd scheme in solving navier stokes equation for the benchmark lid driven cavity problem at high reynolds numbers showing that the present method is capable of providing very accurate solution using far fewer points as compared to existing solutions in the literature

further improvement and analysis of ccd scheme dissipation discretization and de aliasing properties in this paper we further analyze a combined compact difference ccd scheme proposed recently t k sengupta v lakshmanan v v s n vijay a new combined stable and dispersion relation preserving compact scheme for non periodic problems j comput phys 228 8 2009 3048 3071 for its dissipation discretization properties to show that its superiority also helps in controlling aliasing error for a benchmark internal flow however application of the same ccd method to study the receptivity of a boundary layer experiencing adverse pressure gradient is not successful this is traced to the nature of the equilibrium flow where the better dissipation property is not helpful in the inviscid part of the flow where the aliasing problems continue to persist a further modification is proposed to the ccd method here to solve complex physical problems requiring information on higher order disturbance quantities as in problems of flow receptivity and instability

an implicit compact scheme solver with application to chemically reacting flows a novel stable implicit compact scheme solver that is higher order in space suitable for modeling steady state and time dependent phenomena on nonuniform grids for one dimensional configurations is presented several properties of compact scheme discretizations are introduced to develop efficient algorithms for jacobian matrix generation and jacobian vector multiplication using a new component form for jacobian operations composite nonuniform grids are introduced that enable the implicit compact scheme solver to achieve sixth order accuracy a robust newton s method is employed with explicit generation of jacobian matrices superior resolution characteristics of the implicit compact scheme solver are demonstrated with several steady state and time dependent problems for the burgers equation the example of the solution of stiff flame problem is given an analysis of spectral properties of jacobian matrices is presented which shows that the condition number and the eigenvalue distributions behave similarly to those found in jacobians associated with low order discretizations two sparsification strategies are developed for the systematic approximation of a dense jacobian aimed at the practical implementation of linear system preconditioning through partial jacobians

high order non uniform grid schemes for numerical simulation of hypersonic boundary layer stability and transition the direct numerical simulation of receptivity instability and transition of hypersonic boundary layers requires high order accurate schemes because lower order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields the main limiting factor in the application of high order schemes to practical boundary layer flow problems is the numerical instability of high order boundary closure schemes on the wall this paper presents a family of high order non uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary layer transition by using an appropriate grid stretching and clustering grid points near the boundary high order schemes with stable boundary closures can be obtained the order of the schemes ranges from first order at the lowest to the global spectral collocation method at the highest the accuracy and stability of the new high order numerical schemes is tested by numerical simulations of the linear wave equation and two dimensional incompressible flat plate boundary layer flows the high order non uniform grid schemes up to the 11th order are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge the steady and unsteady results show that the new high order schemes are stable and are able to produce high accuracy for computations of the nonlinear two dimensional navier stokes equations for the wall bounded supersonic flow

modification of the liu davidson method for obtaining one or simultaneously several eigensolutions of a large real symmetric matrix a new and efficient iteration method for obtaining simultaneously several eigensolutions and even for obtaining only one solution of a large real symmetric matrix is presented by modifying the simultaneous expansion method by davidson and liu the method is basically the ritz iteration method to correct trial vector s simultaneously using correction vectors however the number of the correction vectors determined in each iteration need not be the same as the number of the desired solutions it is advantageous for the former number to be smaller than the latter when many eigensolutions are sought in addition trial vectors which need not be obtained exactly nor be corrected are included so as to bound the desired solutions which are to be obtained exactly from above and or below when interior solutions are sought the correction vector space may be kept constant throughout the iterations when the correction vector space has room enough to include the trial vectors of the previous iterations the old trial vectors can serve as expansion vectors the performance of the algorithm is tested for five matrices and is compared with the original liu davidson algorithm the results demonstrate that the present algorithm attains efficient use of memory space and reduction of iteration cycles arithmetic operations and i o processings

generalized besicovitch and weyl spaces topology patterns and sliding block codes the besicovitch and weyl topologies on the space of configurations take a point of view completely different from the usual product topology as such the properties of the former are much different from that of the latter the one dimensional case has already been the subject of thorough studies we extend it to greater dimensions and also to more general groups

the compact and crank nicolson adi schemes for two dimensional semilinear multidelay parabolic equations this paper deals with numerical solutions of initial boundary value problems of the two dimensional semilinear multidelay parabolic equations two types of alternating direction implicit adi schemes are suggested the unique solvability convergence and unconditional stability of the schemes are analyzed and hence the corresponding criteria are established especially by using the discrete energy method it is proven that the compact adi scheme can attain second order accuracy in time and fourth order accuracy in space and the crank nicolson adi scheme has second order accuracy in both time and space numerical experiments are performed to verify the efficiency and accuracy of the both schemes

a fourth order accurate finite volume compact method for the incompressible navier stokes solutions abstract this paper presents a finite volume fourth order accurate compact scheme for discretization of the incompressible navier stokes equations in primitive variable formulation the numerical method of integrating the navier stokes equations comprises a compact finite volume formulation of the average convective and diffusive fluxes the pressure velocity coupling is achieved via the coupled solution of the resulting system of equations the solution of the coupled set of equations is performed with an implicit newton krylov matrix free method for stationary problems for simulation of unsteady flows a standard fourth order runge kutta method was used for temporal discretization and the velocity pressure coupling was ensured at each stage also using the matrix free method several incompressible viscous steady and unsteady flow problems have been computed to assess the robustness and accuracy of the proposed method

robust preconditioning of large sparse symmetric eigenvalue problems abstract iterative methods for solving large sparse symmetric eigenvalue problems often encounter convergence difficulties because of ill conditioning the generalized davidson method is a well known technique which uses eigenvalue preconditioning to surmount these difficulties preconditioning the eigenvalue problem entails more subtleties than for linear systems in addition the use of an accurate conventional preconditioner i e as used in linear systems may cause deterioration of convergence or convergence to the wrong eigenvalue the purpose of this paper is to assess the quality of eigenvalue preconditioning and to propose strategies to improve robustness numerical experiments for some ill conditioned cases confirm the robustness of the approach

a fast solver of the shallow water equations on a sphere using a combined compact difference scheme the spectral like three point combined compact difference sp ccd scheme is developed for the shallow water models on a sphere which is originally proposed by chu and fan j comput phys 140 1998 370 a standard test set proposed by williamson et al j comput phys 102 1992 211 is used to evaluate the accuracy of the new method it is shown that the accuracy of the method is comparable to that of the spherical harmonics model as a whole the relative efficiency of the method with respect to the spherical harmonics model appears from the resolution of 512 256 and it becomes significant for the higher resolutions the test results suggest that the method could be extended to numerical atmospheric models

an efficient time advancing strategy for energy preserving simulations energy conserving numerical methods are widely employed within the broad area of convection dominated systems semi discrete conservation of energy is usually obtained by adopting the so called skew symmetric splitting of the non linear convective term defined as a suitable average of the divergence and advective forms although generally allowing global conservation of kinetic energy it has the drawback of being roughly twice as expensive as standard divergence or advective forms alone in this paper a general theoretical framework has been developed to derive an efficient time advancement strategy in the context of explicit runge kutta schemes the novel technique retains the conservation properties of skew symmetric based discretizations at a reduced computational cost it is found that optimal energy conservation can be achieved by properly constructed runge kutta methods in which only divergence and advective forms for the convective term are used as a consequence a considerable improvement in computational efficiency over existing practices is achieved the overall procedure has proved to be able to produce new schemes with a specified order of accuracy on both solution and energy the effectiveness of the method as well as the asymptotic behavior of the schemes is demonstrated by numerical simulation of burgers equation

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 spectral numerical method for the navier stokes equations with applications to taylor couette flow a new spectral method for solving the incompressible navier stokes equations in a plane channel and between concentric cylinders is presented the method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step in addition the number of dependent variables is reduced resulting in a reduction in computer memory requirements several test problems have been computed for the channel flow and for flow between concentric cylinders including taylor couette flow with axisymmetric taylor vortices and wavy vortices in all cases agreement with available experimental and theoretical results is very good

large eddy simulation of a spatially evolving supersonic turbulent boundary layer flow several issues involving the large eddy simulation of wall bounded compressible turbulent flows are investigated a spatially evolving supersonic boundary layer is simulated using a high order accurate finite difference scheme and the dynamic subgrid scale model a parametric study suggests that the finite difference scheme has a detrimental effect on the resolution of the smaller scales due to excessive numerical dissipation from the spatial differencing also because the dynamic model uses the smaller resolved scale eddies to determine the model coefficients the predicted coefficients do not have the appropriate values the use of higher order schemes is found to better capture the smaller resolved scales and to improve substantially the quality of the results the effect of descretization errors on large eddy simulation needs to be addressed further before proceeding with large eddy simulation of flows of engineering interest

numerically stable formulations of convective terms for turbulent compressible flows abstract a systematic analysis of the discrete conservation properties of non dissipative central difference approximations of the compressible navier stokes equations is reported a generalized splitting of the nonlinear convective terms is considered and energy preserving formulations are fully characterized by deriving a two parameter family of split forms previously developed formulations reported in literature are shown to be particular members of this family novel splittings are introduced and discussed as well furthermore the conservation properties yielded by different choices for the energy equation i e total and internal energy entropy are analyzed thoroughly it is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family local conservation of primary invariants which is a fundamental property to build high fidelity shock capturing methods is also discussed in the paper numerical tests performed for the taylor green vortex at zero viscosity fully confirm the theoretical findings and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results

a low cost parallel implementation of direct numerical simulation of wall turbulence a numerical method for the direct numerical simulation of incompressible wall turbulence in rectangular and cylindrical geometries is presented the distinctive feature resides in its design being targeted towards an efficient distributed memory parallel computing on commodity hardware the adopted discretization is spectral in the two homogeneous directions fourth order accurate compact finite difference schemes over a variable spacing mesh in the wall normal direction are key to our parallel implementation the parallel algorithm is designed in such a way as to minimize data exchange among the computing machines and in particular to avoid taking a global transpose of the data during the pseudo spectral evaluation of the non linear terms the computing machines can then be connected to each other through low cost network devices the code is optimized for memory requirements which can moreover be subdivided among the computing nodes the layout of a simple dedicated and optimized computing system based on commodity hardware is described the performance of the numerical method on this computing system is evaluated and compared with that of other codes described in the literature as well as with that of the same code implementing a commonly employed strategy for the pseudo spectral calculation

stabilized non dissipative approximations of euler equations in generalized curvilinear coordinates we discuss stabilization strategies for finite difference approximations of the compressible euler equations in generalized curvilinear coordinates that do not rely on explicit upwinding or filtering of the physical variables our approach rather relies on a skew symmetric like splitting of the convective derivatives that guarantees preservation of kinetic energy in the semi discrete low mach number limit a locally conservative formulation allows efficient implementation and easy incorporation into existing compressible flow solvers the validity of the approach is tested for benchmark flow cases including the propagation of a cylindrical vortex and the head on collision of two vortex dipoles the tests support high accuracy and superior stability over conventional central discretization of the convective derivatives the potential use for dns les of turbulent compressible flows in complex geometries is discussed

construction of invariant compact finite difference schemes in this paper we propose a method which is based on equivariant moving frames for development of high order accurate invariant compact finite difference schemes that preserve lie symmetries of underlying partial differential equations in this method variable transformations that are obtained from the extended symmetry groups of partial differential equations pdes are used to transform independent and dependent variables and derivative terms of compact finite difference schemes constructed for these pdes such that the resulting schemes are invariant under the chosen symmetry groups the unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames in some cases owing to selection of convenient moving frames numerical representation of invariant or symmetry preserving compact numerical schemes is found to be notably simpler than that of standard noninvariant compact numerical schemes further the accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite difference algorithms application of the proposed method is demonstrated through construction of invariant compact finite difference schemes for some common linear and nonlinear pdes including the linear advection diffusion equation in one or two dimensions the inviscid burgers equation in one dimension viscous burgers equation in one or two dimensions spherical burgers equation in one dimension and shallow water equations in two dimensions results obtained from our numerical simulations indicate that invariant compact finite difference schemes not only inherit selected symmetry properties of underlying pdes but are also comparably more accurate than the standard noninvariant base numerical schemes considered here

a parallel davidson type algorithm for several eigenvalues in this paper we propose a new parallelization of the davidson algorithm adapted for many eigenvalues in our parallelization we use a relationship between two consecutive subspaces which allows us to calculate eigenvalues in the subspace through an arrowhead matrix theoretical timing estimates for the parallel algorithm are developed and compared against our numerical results on the paragon finally our algorithm is compared against another recent parallel algorithm for multiple eigenvalues but based on arnoldi parpack

davidson s method and preconditioning for generalized eigenvalue problems the generalized eigenvalue problem az sz where a is symmetric and s is symmetric positive definite arises in a number of different applications we look at methods for solution of large problems of this type and give an approach that can yield rapid convergence

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

combined compact difference scheme for linear second order partial differential equations with mixed derivative a combined compact difference scheme is proposed for linear second order partial differential equations with mixed derivative the scheme is based on a nine point stencil at the interior with sixth order accurate local truncation error fourier analysis is used to analyze the spectral resolution of the proposed scheme numerical tests demonstrate at least sixth order convergence rate with dirichlet boundary condition and fifth order with robin boundary condition a bonus is that high reynolds numbers do not interfere with the order of accuracy

a staggered grid high order accurate method for the incompressible navier stokes equations a high order accurate finite difference method for the numerical solution of the incompressible navier stokes equations is presented fourth order accurate discretizations of the convective and viscous fluxes are obtained on staggered meshes using explicit or compact finite difference formulas high order accuracy in time is obtained by marching the solution with runge kutta methods the incompressibility constraint is enforced for each runge kutta stage iteratively either by local pressure correction or by a poisson equation based global pressure correction method local pressure correction is carried out on cell by cell basis using a local fourth order accurate discrete analog of the continuity equation the global pressure correction is based on the numerical solution of a poisson type equation which is discretized to fourth order accuracy and solved using gmres in both cases the updated pressure is used to recompute the velocities in order to satisfy the incompressibility constraint to fourth order accuracy the accuracy and efficiency of the proposed method is demonstrated in test problems

high order accurate finite difference schemes based on symmetry preservation abstract in this paper we present a mathematical approach that is based on modified equations and the method of equivariant moving frames for construction of high order accurate invariant finite difference schemes that preserve lie symmetry groups of underlying partial differential equations pdes in the proposed approach invariant or symmetry preserving numerical schemes with a desired or fixed order of accuracy are constructed from some non invariant base numerical schemes modified forms of pdes are used to improve the order of accuracy of existing schemes and these modified forms are obtained through addition of defect correction terms to the original forms of pdes these defect correction terms of modified pdes that are noted from truncation error analysis are either completely removed from schemes or their representation is significantly simplified by considering convenient moving frames this feature of the proposed method can especially be useful to avoid cumbersome numerical representations when high order schemes are developed from low order ones via the method of modified equations the proposed method is demonstrated via construction of invariant numerical schemes with fixed and higher order of accuracy for some common linear and nonlinear problems including the linear advection diffusion equation in 1d and 2d inviscid burgers equation and viscous burgers equation and the performance of these invariant numerical schemes is further evaluated our results indicate that such invariant numerical schemes obtained from existing base numerical schemes have the potential to significantly improve the quality of results not only in terms of desired higher order accuracy but also in the context of preservation of appropriate symmetry properties of underlying pdes

comparison of conservative and rotational forms in large eddy simulation of turbulent channel flow abstract fully developed turbulent channel flow is studied numerically using large eddy simulation les with a fourier finite difference method a prominent feature of this paper lies in the use of a conservative form of the arakawa type for the convective terms in the navier stokes equations the results are compared with earlier ones of moin and kim 1 based on the rotational form of the convective terms noticeable differences are found in quantities such as turbulence intensities two point correlations etc the difference is most prominent in the grid scale portion of the turbulent shear stress the present results agree well with the recent direct simulation of a mildly curved channel flow by moser and moin 2 using the fourier chebyshev polynomial expansions an estimate is made of the error inherent in the rotational form combined with the second order central finite difference method the relationship between the energy production mechanism and helicity is pointed out

the shift inverted j lanczos algorithm for the numerical solutions of large sparse algebraic riccati equations abstract the goal of solving an algebraic riccati equation is to find the stable invariant subspace corresponding to all the eigenvalues lying in the open left half plane the purpose of this paper is to propose a structure preserving lanczos type algorithm incorporated with shift and invert techniques named shift inverted j lanczos algorithm for computing the stable invariant subspace for large sparse hamiltonian matrices the algorithm is based on the j tridiagonalization procedure of a hamiltonian matrix using symplectic similarity transformations we give a detailed analysis on the convergence behavior of the j lanczos algorithm and present error bound analysis and paige type theorem numerical results for the proposed algorithm applied to a practical example arising from the position and velocity control for a string of high speed vehicles are reported

use of indefinite pencils for computing damped natural modes abstract a quadratic eigenvalue problem with symmetric positive definite coefficient matrices may be reduced to linear form while retaining symmetry in the new coefficients but neither of them will be positive definite formally the symmetric lanczos algorithm and subspace iteration may be used to compute some eigenpairs of the linear problem the trouble is that the basis vectors are orthogonal with respect to an indefinite inner product so there is no assurance that they will be linearly independent nevertheless this is an attractive way to solve the original problem and we discuss how to implement it and how it relates to the unsymmetric lanczos procedures we discuss complex origin shifts reorthogonalization and error bounds several methods for solving the reduced problem are mentioned but we have no fully satisfactory technique some dangers are described and examples are given comparing our lanczos program with a modified subspace iteration