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

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

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

repetition free words abstract we give a systematical treatment of repetition free words and repetition free homomorphisms we study the global structure of the set of repetition free infinite words and the behaviour of the density function of repetition free words generally we consider finite or infinite words which do not contain given words as subwords

repetiveness of languages generated by morphisms abstract we study the repetition of subwords in languages generated by morphisms fundamental to our approach is the notion of quasi repetitive elements using these elements we present a new characterization for repetitive morphisms from which we derive a simple proof for the fact that a d0l language is repetitive if and only if it is strongly repetitive ehrenfeucht and rozenberg inform and control 59 1983 13 35 from this proof we obtain a structurally simple polynomial time algorithm for deciding whether such a language is repetitive from further results on quasi repetitive elements we obtain as a consequence a complete characterization for all those morphisms on a two letter alphabet that are repetitive a result which is closely related to a result of seebold bull eatcs 36 1988 137 151 on the d0l periodicity problem finally we characterize those morphisms f on a two letter alphabet for which the language l f generated by f or the language sl f of subwords of l f are context free or even regular

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

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

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

repetition of subwords in dol languages a language k is repetitive if for each positive integer n there exists a word w such that w n is a subword of k language k is called strongly repetitive if there exists a word w such that for each positive integer n w n is a subword of k it is shown that it is decidable whether an arbitrary dol language is repetitive it is also shown that if a dol language is repetitive then it is strongly repetitive

invariance theorems concerning reflection at numerical boundaries abstract downwind computational boundaries in the numerical approximation of hyperbolic equations are in general not transparent and they create spurious reflection a useful measure of this is given by the energy or usual sum of squares of the reflected solution in response to an arbitrary solution which originates from within the computing domain we prove in that respect a somewhat unexpected property namely that for those full discretizations which are obtained by applying to a space discretization of the equations an energy conservative discrete time marching method the energy reflected at the boundary is independent of the value of t and is strictly equal to the energy reflected in the semidiscrete case this is verified in numerical experiments optimal boundary equations may be defined in the semidiscrete case of those which maximize the rate of convergence to zero of the reflected energy when h 0 a corollary of the preceding result is that those boundary equations remain optimal in the same sense when used in an energy conservative full discretization moreover this convergence result continues to hold when a nonconservative but stable i e dissipative time discretization is used this is verified numerically with the first order adams predictor corrector method the results of this paper are derived with the mathematics of the simple three point central difference discretization of spatial derivatives obvious generalizations to other cases are mentioned at the end of the paper

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

absorbing boundary conditions for free surface waves in this paper the use of absorbing boundary conditions is investigated for the numerical simulation of gravity waves on an incompressible inviscid fluid in three dimensions a review of existing methods is given for linear and nonlinear waves after which first and second order partial differential equations are introduced as absorbing boundary conditions for the linearized model well posedness is investigated and it is shown that the reflection properties of the second order equation are superior to those of the first order equation

lower bounds on generalization errors for drifting rules the problem of generalization by single layer perceptrons is studied in the case of time dependent rules lower bounds for the generalization errors within the single presentation of examples case are obtained for randomly drifting rules these bounds suggest a learning algorithm which uses knowledge of the error itself since this error is not readily available it has to be estimated through a mechanism of self evaluation the capacity of incorporating recency information into the error estimate is highly desirable the mechanism proposed has the advantage beyond good performance of being self adaptive in the sense that it adjusts to changes in the unknown drift rate of the rule the performance of the rule is also studied for sudden changes in an attempt to mimic the so called wisconsin test

synchronizing delay for binary uniform morphisms circular d0l systems are those with finite synchronizing delay we introduce a tool called graph of overhangs which can be used to find the minimal value of synchronizing delay of a given d0l system by studying the graphs of overhangs a general upper bound on the minimal value of a synchronizing delay of a circular d0l system with a binary uniform morphism is given

on measuring similarity between different two layered networks in this paper we present a method for calculating the generalization error of two layered networks is the fraction of the input space for which two networks yield different answers therefore it is a good index to measure the similarity between them the method presented here is an extension of work reported previously it is applied here to the case of a single layer perceptron which can be regarded as a particular two layered perceptron that tries to imitate a two layered network the particular realizations of such a two layered network that are analysed here are the parity machine the and machine and the committee machine we have also compared the input output mapping of a committee and a parity machine

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

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

on line learning in the committee machine the dynamics of learning from examples in the k 3 non overlapping committee machine with single presentation of examples is studied the optimal algorithm in the sense of mean generalization is obtained from a variational analysis of the differential equations which describe the dynamics the agreement of the theoretical predictions and the results of numerical simulations is excellent the optimized dynamics has the extra advantage with respect to the non optimized cases in that it uncouples the differential equations which describe the evolution of the relevant parameters i e the student teacher overlap and the norm of the student synaptic vector this in turn translates into the possibility of constructing useful practical optimized on line algorithms for the optimal algorithm the generalization error decays as approximately 0 88 alpha 1 the same nominal error as for the simple perceptron with optimized dynamics

compact implicit maccormack type schemes with high accuracy in this work the maccormack methodology is extended to implicit compact differencing schemes a prefactorization method is developed which splits the implicit matrix into two independent upper and lower matrices which are easier to invert using this method a new class of high order accurate compact maccormack type schemes is derived two fourth order schemes are described and results are shown for three linear and nonlinear caa benchmark problems

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

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

on line learning in parity machines the optimal algorithm for on line learning in the tree k parity machine is studied we introduce a set of recursion relations for the relevant probability distributions which permit study of the general k case the generalization error curve is determined and shown to decay to zero for large as even in the presence of noise there is no critical noise level the dynamics of on line learning is studied analytically near the origin in the absence of previous knowledge the learning dynamics has a fixed point at previous knowledge is needed in at least k 1 branches for the learning to take place

absorbing boundary conditions for the discretization schemes of the one dimensional wave equation when computing a partial differential equation it is often necessary to introduce artificial boundaries here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension fitting to the discretization scheme and stable moreover we give error estimates on the reflected part

the influence of the mass matrix on the dispersive nature of the semi discrete second order wave equation abstract the application of discrete solution methods to the second order wave equation can yield a dispersive representation of the non dispersive wave propagation problem resulting in a phase speed that depends not only upon the wavelength of the signal being propagated but also upon the direction of propagation in this work the dependence of the dispersive errors on the wave propagation direction mesh aspect ratio and wave number is investigated with the goal of understanding and hopefully reducing the phase and group errors associated with the two dimensional bilinear finite element an analysis of the dispersive effects associated with the consistent row sum lumped and higher order mass matrices has led to a reduced coupling penta diagonal mass matrix that yields improved phase and group errors with respect to wavelength and propagation direction the influence of row sum lumping the finite element mass matrix is demonstrated to always introduce lagging phase and group error while a linear combination of the lumped and consistent mass matrices i e a higher order mass matrix is shown to improve the dispersion characteristics of both the reduced and full integration element using a 2 5 phase error metric the higher order mass matrix can reduce the mesh resolution requirements relative to the lumped mass matrix by nearly a factor of two in one dimension a factor of 4 in two dimensions similarly the reduced coupling mass matrix can reduce the grid requirements by a factor of approximately four relative to the lumped mass matrix a factor of 16 in two dimensions although the reduced coupling mass matrix requires additional storage relative to the lumped mass the factor of 16 reduction in grid requirements promises tremendous overall savings for computing a fixed wavelength spectrum or alternatively for resolving shorter wavelengths with a fixed grid resolution

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

on the ability of the optimal perceptron to generalise a linearly separable boolean function is derived from a set of examples by a perceptron with optimal stability the probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method

on the subword complexity of square free dol languages abstract the subword complexity of a language k is the function which to every positive integer n assigns the number of different subwords of length n occuring in words of k a language k is square free if no word in it contains a subword of the form xx where x is a nonempty word the best upper and lower bounds on the subword complexity of infinite square free dol languages are established

optimal generalization in perceptions a new learning algorithm for the one layer perceptron is presented it aims to maximize the generalization gain per example analytical results are obtained for the case of single presentation of each example the weight attached to a hebbian term is a function of the expected stability of the example in the teacher perceptron this leads to the obtention of upper bounds for the generalization ability this scheme can be iterated and the results of numerical simulations show that it converges within errors to the theoretical optimal generalization ability of the bayes algorithm analytical and numerical results for an algorithm with maximized generalization in the learning strategy with selection of examples are obtained and it is proved that as expected orthogonal selection is optimal exponential decay of the generalization error is obtained for the single presentation of selected examples

some remarks about stabilizers we continue our study of stabilizers of infinite words over finite alphabets begun in d krieger on stabilizers of infinite words theoret comput sci 400 2008 169 181 let w be an aperiodic infinite word over a finite alphabet and let stab w be its stabilizer we show that stab w can be partitioned into the monoid of morphisms that stabilize w by finite fixed points and the ideal of morphisms that stabilize w by iteration we also settle a conjecture given in the paper mentioned above by showing that in some cases stab w is infinitely generated if the aforementioned ideal is nonempty then it contains either polynomially growing morphisms or exponentially growing morphisms but not both moreover in the polynomial case the degree of the polynomial is fixed we also show how to compute the polynomial degree from the dependency graph of a polynomially growing morphism

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

generalization performance of bayes optimal classification algorithm for learning a perceptron the generalization error of the bayes optimal classification algorithm when learning a perceptron from noise free random training examples is calculated exactly using methods of statistical mechanics it is shown that if an assumption of replica symmetry is made then in the thermodynamic limit the error of the bayes optimal algorithm is less than the error of a canonical stochastic learning algorithm by a factor approaching ensuremath surd 2 as the ratio of the number of training examples to perceptron weights grows in addition it is shown that approximations to the generalization error of the bayes optimal algorithm can be achieved by learning algorithms that use a two layer neutral net to learn a perceptron

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

bispecial factors in circular non pushy d0l languages we study bispecial factors in fixed points of morphisms in particular we propose a simple method of finding all bispecial words of non pushy circular d0l systems this method can be formulated as an algorithm moreover we prove that non pushy circular d0l systems are exactly those with finite critical exponents

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

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

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

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

languages d0l systems sets of curves and surface automorphisms we introduce extend and apply some relationships between formal language theory and surface theory first we show how boundaries of languages topologized with the cantor metric can be mapped to sets of curves on surfaces namely laminations second we present how and when endomorphisms of free monoids i e substitutions can be mapped to automorphisms of surfaces so that d0l systems correspond to iterations of these automorphisms third we apply these ideas to construct sets of non periodic irreducible automorphisms of surfaces following 14 35 43 so that the involved proofs do not use any differential or algebraic tools but accordingly substitution and d0l system theory of 11 12 31 32 37 mainly the decidability for a d0l language 12 to be strongly repetitive or not this construction also yields effective symbolic descriptions of the stable sets of curves associated with these automorphisms

on critical exponents in fixed points of non erasing morphisms let s be an alphabet of size t let f s s be a non erasing morphism let w be an infinite fixed point of f and let e w be the critical exponent of w we prove that if e w is finite then for a uniform f it is rational and for a non uniform f it lies in the field extension q r l 1 l where r l 1 l are the eigenvalues of the incidence matrix of f in particular e w is algebraic of degree at most t under certain conditions our proof implies an algorithm for computing e w

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