the chain collocation method a spectrally accurate calculus of forms preserving in the discrete realm the underlying geometric topological and algebraic structures at stake in partial differential equations has proven to be a fruitful guiding principle for numerical methods in a variety of fields such as elasticity electromagnetism or fluid mechanics however structure preserving methods have traditionally used spaces of piecewise polynomial basis functions for differential forms yet in many problems where solutions are smoothly varying in space a spectral numerical treatment is called for in an effort to provide structure preserving numerical tools with spectral accuracy on logically rectangular grids over periodic or bounded domains we present a spectral extension of the discrete exterior calculus dec with resulting computational tools extending well known collocation based spectral methods its efficient implementation using fast fourier transforms is provided as well

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

a spectral chebyshev method for linear stability analysis of one dimensional exact solutions of gas dynamics we present a spectral numerical method for solving one dimensional systems of partial differential equations pdes which arise from linearization of the euler equations about an exact solution depending on space and time a two domain chebyshev collocation method is used matching of quantities is performed in the space of characteristic variables as suggested by kopriva appl numer math 2 1986 221 j comput phys 125 1996 244 time dependent boundary conditions are handled following an approach proposed by thompson j comput phys 68 1987 1 89 1990 439 an exact numerical stability analysis valid for any explicit three step third order nondegenerate runge kutta scheme is provided the numerical method is tested against exact solutions for the three fundamental modes of a compressible flow entropy vorticity and acoustic modes

a divergence free multidomain spectral solver of the navier stokes equations in geometries of high aspect ratio in order to solve with high accuracy the incompressible navier stokes equations in geometries of high aspect ratio one has developed a spectral multidomain algorithm well adapted to the parallel computing in cases of 2d problems a chebyshev collocation method and an extended influence matrix technique are used in each subdomain to solve the generalized stokes problem which results from the discretization in time the continuity conditions needed at the interfaces of each subdomain are computed by using again an influence matrix the setup of which ensures all the necessary compatibility conditions especially the incompressibility this work is also described for 3d problems with one homogeneous direction a study of accuracy versus the number of subdomains is presented as well as an example of an application concerned with rayleigh benard convection in a cavity of large aspect ratio

adaptive spectral methods with application to mixing layer computations this paper reports some experiments on the use of adaptive chebyshev pseudospectral methods for compressible mixing layer computations different functionals measuring the optimality of the polynomial approximation are discussed and compared in particular we address the problem of the practical computation of the various functionals the utility of the self adaptive method is then demonstrated by some examples from compressible mixing layer calculations

application of the spectral multidomain method to the navier stokes equations an application of the spectral multidomain method to the two dimensional time dependent incompressible navier stokes equations is presented the governing equations are discretized on a non staggered stretched mesh with a mixed finite difference chebyshev method and are integrated by a time splitting procedure the accuracy of the multidomain method is demonstrated for two different channel flow configurations one with a backward facing step and the other with a rectangular step the method is then applied to the investigation of the effects of an isolated two dimensional roughness element on the spatial development of instability waves in boundary layers

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations

high order discontinuous element based schemes for the inviscid shallow water equations spectral multidomain penalty and discontinuous galerkin methods two commonly used types of high order accuracyelement based schemes collocationbased spectral multidomain penalty methods smpm and nodal discontinuous galerkin methods dgm are compared in the framework of the inviscid shallow water equations differences and similarities in formulation are identified with the primary difference being the dissipative term in the rusanov form of the numerical flux for the dgm that provides additional numerical stability however it should be emphasized that to arrive at this equivalence between smpm and dgm requires making specific choices in the construction of both methods these choices are addressed in general both methods offer a multitude of choices in the penalty terms used to introduce boundary conditions and stabilize the numerical solution the resulting specialized class of smpm and dgm are then applied to a suite of six commonly considered geophysical flow test cases three linear and three non linear we also include results for a classical continuous galerkin i e spectral element method for comparison both the analysis and numerical experiments show that the smpm and dgm are essentially identical both methods can be shown to be equivalent for very special choices of quadrature rules and riemann solvers in the dgm along with special choices in the type of penalty term in the smpm although we only focus our studies on the inviscid shallow water equations the results presented should be applicable to other systems of nonlinear hyperbolic equations such as the compressible euler equations and extendable to the compressible and incompressible navier stokes equations where viscous terms are included

compatibility conditions for time dependent partial differential equations and the rate of convergence of chebyshev and fourier spectral methods abstract compatibility conditions for partial differential equations pdes are an infinite set of relations between the initial conditions the pde and the boundary conditions which are necessary and sufficient for the solution to be c that is infinitely differentiable everywhere on the computational domain including the boundaries since the performance of chebyshev spectral and spectral element methods is dramatically reduced when the solution is not c one would expect that the compatibility conditions would be a major theme in the spectral literature instead it has been completely ignored therefore we pursue three goals here first we present a proof of the compatibility conditions in a simplified form that does not require functional analysis second we analyze the connection between the compatibility conditions and the rate of convergence of chebyshev methods lastly we describe strategies for slightly adjusting initial conditions so that the compatibility conditions are satisfied

multidomain spectral solution of the euler gas dynamics equations abstract we present interface treatments for computing the euler gas dynamics equations by a multidomain chebyshev spectral collocation method these treatments are suitable for use at the corners of subdomains and for patched or overlapping subdomains they can be applied to interfaces placed in subsonic supersonic or transonic regions of a flow computed results indicate that the solutions are spectrally accurate can be more accurate and more efficient than a single domain calculation and that reflections of waves at interfaces are not significant

permutation matrices related to sudoku we consider a class of permutation matrices associated with the popular game of sudoku and prove some connectivity and existence results for this matrix class

spectral solution of the viscous blunt body problem 2 multidomain approximation we present steady solutions of high speed viscous flows over blunt bodies using a multidomain chebyshev spectral collocation method the region within the shock layer is divided into subdomains so that internal layers can be well resolved in the interiors of the subdomains the solution is approximated by chebyshev collocation at interfaces between subdomains the advective terms are upwinded and the viscous terms are treated by a penalty method the method is applied to five flows in the mach number range 5 25 and reynolds number range from 2 x 10 3 to 83 x 10 4 based on nose radius results are compared with experimental data and to a finite difference solution

an efficient correction procedure via reconstruction for simulation of viscous flow on moving and deforming domains in this paper we report the development of a new parallel solver using the correction procedure via reconstruction cpr for viscous flows on moving and deforming grids by employing an accurate treatment of flux derivatives for moving and deforming unstructured grids consisting of all quadrilateral cells it is found that the geometric conservation law is not explicitly required the free stream preservation is automatically satisfied the cpr code is verified using a benchmark case for a moving inviscid vortex on moving and deforming grids the optimal orders of accuracy are obtained it is subsequently employed to study viscous flows on moving and deforming grids the cpr method is faster than and nearly as accurate as the sd method for solving viscous flow problems with moving boundaries

computational framework for coupling compressible and low mach number codes a hybrid multicode computational method is developed that allows combining compressible and low mach number codes into a single computational solver the proposed methodology can be used for integrated simulations of multicomponent engineering problems the unsteady coupling between the two codes is performed via exchanging time dependent state information through the interfaces using overlapping meshes the proposed coupling procedure is validated on laminar steady and unsteady test problems both constant density and variable density regimes of the low mach number code are investigated results of the test cases performed in frameworks of one way coupling and two way coupling are documented several methods of implementation of interface conditions for the compressible code are compared by looking at the levels of dilatation as well as numerical errors the injection method when all the variables are simply transformed from one code to another through an interpolation procedure is chosen for its superior stability accuracy and simplicity of implementation numerical results for the coupled calculations obtained with the injection method show good agreement with both standalone calculations and analytical solutions

multiple positive solutions for a semilinear equation with critical exponent and prescribed singularity abstract this paper is devoted to the study of a semilinear equation of the form u v x u u 2 2 u h x u h 0 1 where r n n 3 is an open bounded domain with smooth boundary 0 n 2 2 2 2 2 n n 2 h x 0 and v x has prescribed finitely many singular points using variational methods we establish some existence and multiplicity of positive solutions for the problem

towards efficient tracking of inertial particles with high order multidomain methods this paper develops an efficient particle tracking algorithm to be used in fluid simulations approximated by a high order multidomain discretization of the navier stokes equations we discuss how to locate a particle s host subdomain how to interpolate the flow field to its location and how to integrate its motion in time a search algorithm for the nearest subdomain and quadrature point tuned to a typical quadrilateral isoparametric spectral subdomain takes advantage of the inverse of the linear blending equation we show that to compute particle laden flows a sixth order lagrangian polynomial that uses points solely within a subdomain is sufficiently accurate to interpolate the carrier phase variables to the particle position time integration of particles with a lower order adams bashforth scheme rather than the fourth order runge kutta scheme often used for the integration of the carrier phase increases computational efficiency while maintaining engineering accuracy we verify the tracking algorithm with numerical tests on a steady channel flow and an unsteady backward facing step flow

shock fitted multidomain solution of supersonic flows abstract shock fitting is presented as a way to compute shocks with spectral methods with fitting spectral approximation is performed on smooth portions of the flow and a precise description of a discontinuous shock is possible coupled with spectral domain decomposition one has the ability to resolve efficiently and without stretching transformations boundary layers and other regions in the flow where increased local resolution is needed a supersonic blunt body problem is solved to provide an example of the use of the approach comparisons between computations and experimental measurements are made for body surface quantities shock shape and body normal direction profiles

a spectral multidomain method for the numerical simulation of turbulent flows the primitive variable formulation of the unsteady incompressible navier stokes equations in three space dimensions is discretized with a combined fourier legendre spectral method a semi implicit pressure correction scheme is applied to decouple the velocity from the pressure the arising elliptic scalar problems are first diagonalized in the periodic fourier direction and then solved by a multidomain legendre collocation method in the two remaining space coordinates in particular both an iterative and a direct version of the so called projection decomposition method pdm are introduced to separate the equations for the internal nodes from the ones governing the interface unknowns the pdm method first introduced by v agoshkov and e ovtchinnikov and later applied to spectral methods by p gervasio e ovtchinnikov and a quarteroni is a domain decomposition technique for elliptic boundary value problems which is based on a galerkin approximation of the steklov poincare equation for the unknown variables associated to the grid points lying on the interface between subdomains after having shown the exponential convergence of the proposed discretization technique some issues on the efficient implementation of the method are given finally as an illustration of the potentialities of the algorithm for the numerical simulation of turbulent flows the results of a direct numerical simulation dns of a fully turbulent plane channel flow are presented

stable boundary conditions and difference schemas for navier stokes equations the navier stokes equations at high reynolds numbers can be viewed as an incompletely elliptic perturbation of the euler equations by using the entropy function for the euler equations as a measure of energy for the navier stokes equations we are able to obtain nonlinear energy estimates for the mixed initial boundary value problem these estimates are used to derive boundary conditions that guarantee l 2 boundedness even when the reynolds number tends to infinity finally we propose a new difference scheme for modeling the navier stokes equations in multidimensions for which we are able to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation

reduced navier stokes equations near a flow boundary we derive a hierarchy of pdes for the leading order evolution of wall based quantities such as the skin friction and the wall pressure gradient in two dimensional fluid flows the resulting reduced navier stokes rns equations are defined on the boundary of the flow and hence have reduced spatial dimensionality compared to the navier stokes equations this spatial reduction speeds up numerical computations and makes the equations attractive candidates for flow control design we prove that members of the rns hierarchy are well posed if appended with boundary conditions obtained from wall based sensors we also derive the lowest order rns equations for three dimensional flows for several benchmark problems our numerical simulations show close finite time agreement between the solutions of rns and those of the full navier stokes equations c 2006 elsevier b v all rights reserved

a simple efficient and high order accurate curved sliding mesh interface approach to spectral difference method on coupled rotating and stationary domains this paper presents a simple efficient and high order accurate sliding mesh interface approach to the spectral difference sd method we demonstrate the approach by solving the two dimensional compressible navier stokes equations on quadrilateral grids this approach is an extension of the straight mortar method originally designed for stationary domains 7 8 our sliding method creates curved dynamic mortars on sliding mesh interfaces to couple rotating and stationary domains on the nonconforming sliding mesh interfaces the related variables are first projected from cell faces to mortars to compute common fluxes and then the common fluxes are projected back from the mortars to the cell faces to ensure conservation to verify the spatial order of accuracy of the sliding mesh spectral difference ssd method both inviscid and viscous flow cases are tested it is shown that the ssd method preserves the high order accuracy of the sd method meanwhile the ssd method is found to be very efficient in terms of computational cost this novel sliding mesh interface method is very suitable for parallel processing with domain decomposition it can be applied to a wide range of problems such as the hydrodynamics of marine propellers the aerodynamics of rotorcraft wind turbines and oscillating wing power generators etc

a high order solver for unsteady incompressible navier stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping we report development of a high order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping the method falls under the class of methods now referred to as flux reconstruction correction procedure via reconstruction the governing equations employ chorin s classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems an implicit non linear lower upper symmetric gauss seidel scheme with backward euler discretization is used to efficiently march the solution in pseudo time while a second order backward euler discretization is used to march in physical time we verify and validate implementation of the high order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems the current implicit time stepping scheme is proven effective in satisfying the divergence free constraint on the velocity field in the artificial compressibility formulation within the context of the high order flux reconstruction method this compact high order method is very suitable for parallel computing and can easily be extended to moving and deforming grids

a staggered grid multidomain spectral method for the compressible navier stokes equations we present a flexible non conforming staggered grid chebyshev spectral multidomain method for the solution of the compressible navier stokes equations in this method subdomain corners are not included in the approximation thereby simplifying the subdomain connectivity to allow for local refinement of the polynomial order or subdomain subdivision non conforming subdomains are treated by a mortar method spectral accuracy is shown in one and two dimensional linear and non linear problems application is made to four compressible flow problems the couette flow a steady boundary layer over a flat plate steady transonic flow in a nozzle and unsteady flow over a cylinder at a reynolds number of 75

a triangular spectral element method applications to the incompressible navier stokes equations abstract encouraged by the success of spectral elements methods in computational fluid dynamics and p type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers we introduce here a complete formulation using a modal basis which has been implemented in a new code ne t r the new basis has the following properties jacobi polynomials of mixed weights semi orthogonality hierarchical structure generalized tensor warped product variable order and a new apex co ordinate system allowing automated integration with gaussian quadrature we have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations we use this notation to formulate the linear advection and helmholtz equations in an efficient manner and show that we recover the following properties well conditioned matrices asymptotic operation count of o n 3 scaling o n 2 of the spectral radius of the weak convective operator and exponential convergence using polynomials up to 40 having constructed and numerically analysed these equations we are then able to solve the incompressible navier stokes equations using a high order splitting scheme we demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the stokes and navier stokes problems

unstructured spectral element methods for simulation of turbulent flows in this paper we present a spectral element fourier algorithm for simulating incompressible turbulent flows in complex geometries using unstructured quadrilateral meshes to this end we compare two different interface formulations for extending the conforming spectral element method in order to allow for surgical mesh refinement and still retain spectral accuracy the zanolli iterative procedure and variational patching based on auxiliary mortar functions we present an interpretation of the original mortar element method as a patching scheme and develop direct and iterative solution techniques that make the method efficient for simulations of turbulent flows the properties of the new method are analyzed in detail by studying the eigenspectra of the advection and diffusion operators we then present numerical results that illustrate the flexibility as well as the exponential convergence of the new algorithm for nonconforming discretizations we conclude with simulation studies of the turbulent cylinder wake at re 1000 external flow and turbulent flow over riblets at re 3280 internal flow

a weak legendre collocation spectral method for the solution of the incompressible navier stokes equations in unstructured quadrilateral subdomains a weak legendre spectral method is developed for the solution of the primitive variable formulation of the unsteady incompressible navier stokes equations in general two dimensional and axisymmetric geometries a semi implicit projection method is utilized for the temporal approximation and for decoupling the velocity field from the pressure field a series of elliptic boundary value problems arises from the above procedure each of which is spatially discretized by a weak collocation method in multiple nonoverlapping subdomains in particular a modified variational formulation of the partial differential equations is presented which leads after discretization to a weak multidomain approximation of the corresponding problems a weak formalism for the influence matrix technique is also developed which is consistent with the spatial discretization scheme and successfully separate the equations for the internal nodes from the ones governing the interface unknowns a method of dealing with the singularity problem faced by the weak formulation at axisymmetric problems is proposed while a combination of direct methods is studied for tackling effectively the linear algebraic systems resulting from the full discretization exponential convergence is demonstrated for a plethora of stokes and navier stokes simulations

on the outflow conditions for spectral solution of the viscous blunt body problem the purpose of this paper is to study and identify suitable outflow boundary conditions for the numerical simulation of viscous supersonic hypersonic flow over blunt bodies governed by the compressible navier stokes equations with an emphasis motivated primarily by the use of spectral methods without any filtering the subsonic supersonic composition of the outflow boundary requires a dual boundary treatment for well posedness all compatibility relations modified to undertake the hyperbolic parabolic behaviour of the governing equations are used for the supersonic part of the outflow regarding the unknown downstream information in the subsonic region different subsonic outflow conditions in the sense of the viscous blunt body problem are examined a verification procedure is conducted to make out the distinctive effect of each outflow condition on the solution detailed comparisons are performed to examine the accuracy and performance of the outflow conditions considered for two model geometries of different surface curvature variations numerical simulations indicate a noticeable influence of pressure from subsonic portion to supersonic portion of the boundary layer it is demonstrated that two approaches for imposing subsonic outflow conditions namely 1 extrapolating all flow variables and 2 extrapolation of pressure along with using proper compatibility relations are more suitable than the others for accurate numerical simulation of viscous high speed flows over blunt bodies using spectral collocation methods

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

global coefficient adjustment method for neumann condition in explicit chebyshev collocation method and its application to compressible navier stokes equations the present paper consists of two parts in part 1 a new technique of treating neumann boundary conditions with an explicit chebyshev collocation method is developed any neumann boundary condition can be satisfied by adjusting all the chebyshev coefficients of a solution which results in a small influence on the solution and its derivatives except at the boundary comparisons between the new technique and several traditional ones are made for a one dimensional advection diffusion problem which confirms the superiority of the new technique the spectral accuracy of the new technique is also demonstrated in part 2 a chebyshev collocation code for the compressible navier stokes equations is developed to solve the high speed flows around a sphere good resolutions are obtained in the boundary layer by using the new technique and comparison between the calculation and the experiment shows good agreement

a variational formulation for the chebyshev pseudospectral approximation of neumann problems a collocation scheme is proposed on the basis of the chebyshev nodes for the approximation of elliptic problems with dirichlet neumann boundary conditions deriving from a suitable variational formulation in weighted sobolev spaces

spectral solution of the viscous blunt body problem the viscous blunt body problem is solved with a shock fitted chebyshev spectral method no explicit artificial viscosity or filtering is needed to obtain smooth converged solutions the method is applied to two problems first results for the flow over a right circular cylinder in the mach number range of 5 5 6 0 are compared with experimental data second a solution for a mach 25 flow over a hyperbolic cone is compared with a viscous shock layer calculation

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

existence of positive solutions of some semilinear elliptic equations with singular coefficients in this paper we consider the semilinear elliptic problem in a bounded domain omega subseteq r n delta u frac mu x alpha u 2 ast alpha 1 f x g u in omega u 0 in omega u 0 on partial omega where mu geq 0 0 leq alpha leq 2 2 asterik alpha 2 n alpha n 2 f omega rightarrow r is measurable f 0 a e having a lower order singularity than x 2 at the origin and g r rightarrow r is either linear or superlinear for 1 p n we characterize a class of singular functions im p for which the embedding w 0 1 p omega hookrightarrow l p omega f is compact when p 2 alpha 2 f in im 2 and 0 leq mu frac 1 2 n 2 2 we prove that the linear problem has h 0 1 discrete spectrum by improving the hardy inequality we show that for f belonging to a certain subclass of im 2 the first eigenvalue goes to a positive number as mu approaches frac 1 2 n 2 2 furthermore when g is superlinear we show that for the same subclass of im 2 the functional corresponding to the differential equation satisfies the palais smale condition if alpha 2 and a brezis nirenberg type of phenomenon occurs for the case 0 leq alpha 2

a godunov type method in lagrangian coordinates for computing linearly perturbed planar symmetric flows of gas dynamics linear stability studies of complex flows require that efficient numerical methods be devised for predicting growth rates of multi dimensional perturbations for one dimensional 1d basic flow i e of planar cylindrical or spherical symmetry a general numerical approach is viable which consists in solving simultaneously the one dimensional equations of gas dynamics and their linearized forms for three dimensional perturbations extensions of artificial viscosity methods have thus been used in the past more recently equations aux derivees partielles et applications articles dedies a j l lions 1998 godunov type schemes for single fluid flows of gas dynamics and magnetohydrodynamics have been proposed pursuing this effort we introduce within the lagrangian perturbation approach a class of godunov type schemes which is well suited for solving multi material problems of gas dynamics these schemes tire developed here for the planar symmetric case and comprise two second order extensions the numerical capabilities of these methods are illustrated by computations of richtmyer meshkov instabilities occurring at a single material interface a systematic comparison of numerically computed growth rates with results of the linear theory for the richtmyer meshkov instability is provided

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

a spectral anelastic navier stokes solver for a stratified two miscible layer system in infinite horizontal channel we present a consistent chebyshev fourier fourier pseudo spectral algorithm for the numerical solution of anelastic models these models satisfy the constraint i u i 0 where is a mean density profile and u the velocity field the choice of the decoupling method is discussed and the uzawa approach is generalized to such a constraint the pressure operator properties are detailed and the solution of the zeroth fourier mode case is discussed the solution of linear systems is obtained and refined using an iterative method the uzawa algorithm is embedded in an auto adaptive multidomain approach in order to handle steep evolving gradients the algorithm is reduced to the boussinesq approximation the characteristics and performance of the resulting numerical code are analyzed the validation includes comparisons with linear stability results and with the waddell et al 55 single mode experiment we also compute the nonlinear growth rate of a boussinesq turbulent mixing layer a comparison between boussinesq and anelastic models is then sketched out finally four 3d simulations are carried out with the anelastic and boussinesq models these results illustrate the capability of the entire method to handle stiff problems such as the rayleigh taylor configuration

the shannon entropy of sudoku matrices we study properties of an ensemble of sudoku matrices a special type of doubly stochastic matrix when normalized using their statistically averaged singular values the determinants are very near

stability analysis of spectral methods for hyperbolic initial boundary value systems we treat a constant coefficient hyperbolic system in one space variable with zero initial data dissipative boundary conditions are imposed at the two points x pm 1 this problem is discretized either by a spectral or pseudospectral approximation in space we demonstrate sufficient conditions under which the spectral numerical solution is stable moreover these conditions have to be checked only for scalar equations the stability theorems take the form of explicit bounds for the norm of the solution in terms of the boundary data the dependence of these bounds on n the number of points in the domain or equivalently the degree of the polynomials involved is investigated for a class of standard spectral methods including chebyshev and legendre collocations

the cfl condition for spectral approximations to hyperbolic initial boundary value problems abstract study the stability of spectral approximations to scalar hyperbolic initial boundary value problems with variable coefficients time is discretized by explicit multi level or runge kutta methods of order or 3 forward euler time differencing is included and we study spatial discretizations by spectral and pseudospectral approximations associated with the general family of jacobi polynomials prove that these fully explicit spectral approximations are stable provided their time step delta is restricted by the cfl like condition delta const 1 n sq where n equals teh spatial number of degrees of freedom give two independent proofs of this results depending on two different choices of appropriate l sq weighted norms in both approaches the proofs hinge on a certain inverse inequality interesting for its own sake the result confirms the commonly held belief that the above cfl stability restriction which is extensively used in practical implementations guarantees the stability and hence the convergence of fully explicit spectral approximations in the non periodic case keywords hyperbolic equations spectral approximations stability condition

calculation of the number of all pairs of disjoint s permutation matrices the concept of s permutation matrix is considered a general formula for counting all disjoint pairs of n2 n2 s permutation matrices as a function of the positive integer n is formulated and proven in this paper to do that the graph theory techniques have been used it has been shown that to count the number of disjoint pairs of n2 n2 s permutation matrices it is sufficient to obtain some numerical characteristics of all n n bipartite graphs

a dynamical pseudo spectral domain decomposition technique a dynamical spectral domain decomposition method is presented in each subdomain a transformation of coordinate is used both the locations of the interfaces and the parameters of the mappings are dynamically adapted by minimizing theh 2 norm of the calculated solution we show on some functions that the total norm of the chebyshev series depends on the location of the interfaces moreover there exists a minimum that defines the best location of the interface this defines a dynamical generation of chebyshev collocation points this numerical method is applied on partial differential equations and it is shown that both the overall accuracy and the matching at the interfaces are improved with respect to a fixed interface calculation this algorithm is then used for the numerical solution of the time dependent full navier stokes equations the solution technique consists in a fourier chebyshev collocation method combined with a matching method the computational domain is decomposed into subdomains in the vertical direction in each subdomain a coordinate transform is used and the locations of the interfaces are dynamically determined the elliptic problems coming from the viscous terms are solved by means of the chebyshev acceleration method density is matched with an upwind procedure whereas the velocities the temperature and the concentration are handled with the influence matrix method numerical examples are carried out on the compressible kelvin helmholtz and rayleigh taylor flows

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

chebyshev collocation method and multi domain decomposition for navier stokes equations in complex curved geometries a general multidomain decomposition is proposed for the numerical solution of the 2d incompressible stationary navier stokes equations the solution technique consists in a chebyshev orthogonal collocation method preconditioned by a standard galerkin finite element technique the preconditioned system is then solved through a richardson procedure the domain of interest is decomposed into quadrilaterals curved when needed a gordon transfinite interpolation performs the curvilinear grid generation of the obtained simply connected planar subdomains the interface conditions naturally incorporated into the finite element approach relate neighbour subdomains through the normal jump of appropriate fluxes across internal boundaries where an integral form ofc 1 continuity is consequently achieved at convergence of the iterative processes the study of model stokes problems demonstrates that the current method still behaves spectrally in distorted geometries for curvilinear distortion a loss of several orders of magnitude is observed in the solution accuracy even when the distortion is very limited finally some results of flow simulation in a constricted channel are proposed to illustrate the abilities of the present method to treat navier stokes problems

four solutions of an inhomogeneous elliptic equation with critical exponent and singular term abstract in this paper we prove the existence of four nontrivial solutions of u x 2 u u 2 2 u x 2 u f x x 0 and show that at least one of them is sign changing our results extend some previous works such as g tarantello multiplicity results for an inhomogeneous neumann problem with critical exponent manuscripta math 81 1993 51 78 d kang and y deng multiple solutions for inhomogeneous elliptic problems involving critical sobolev hardy exponents j math anal appl 60 2005 729 753 n hirano and n shioji a multiplicty result including a sign changing solution for an inhomogeneous neumann problem with critical exponent proc roy soc edinburgh 137a 2007 333 347

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

multidomain spectral solution of compressible viscous flows abstract we develop a nonoverlapping multidomain spectral collocation method to solve compressible viscous flows at the interfaces the advection terms are treated with a characteristic correction method the diffusion terms are treated with a penalty method spectral accuracy is demonstrated on linear model problems in one and two space dimensions the method is applied to a subsonic and a supersonic flow over a flat plate the results are compared to solutions of the boundary layer equations which show that two digit accuracy in the adiabatic plate temperature is obtained with 16 points in the boundary layer for a freestream mach number of two a second application is to a transonic flow in a two dimensional converging diverging nozzle where the computed results are compared to experimental data

a strategy for the efficient simulation of viscous compressible flows using a multi domain pseudospectral method abstract a new multi domain pseudospectral method is developed to simulate viscous compressible flow in a quasi one dimensional nozzle the flow variables at the interface points are advanced in time using the same second order time marching scheme as the interior points the spatial derivatives of the inviscid flux vector are evaluated at the interface points using a spectrally accurate modification of van leer s flux vector splitting method the derivatives of the viscous flux vector are evaluated alternately from neighboring sub domains the scheme is found to be spectrally accurate shock waves are resolved without oscillations for moderate reynolds numbers

multiple solutions for inhomogeneous elliptic problems involving critical sobolev hardy exponents abstract let r n be a smooth bounded domain such that 0 n 3 0 s 2 2 s 2 n s n 2 is the critical sobolev hardy exponent f x is a given function by using the ekeland s variational principle and the mountain pass lemma we prove the existence of multiple solutions for the singular critical inhomogeneous problem u u x 2 u 2 s 2 x s u u f x with dirichlet boundary condition on under some assumptions on f x and

multiplicity result for a class of elliptic equations with singular term abstract we consider the existence of nontrivial solutions of the equation u x 2 u u 2 2 u x 2 u f x u x 0 u h 0 1 where 0 is a smooth bounded domain in r n n 3 by variational methods and nehari set techniques we show that this problem under some additional hypotheses on 0 0 0 0 1 and f l has four nontrivial solutions in h 0 1 and that least one of them is sign changing

an improvement of fractional step methods for the incompressible navier stokes equations abstract a numerical method for computing three dimensional unsteady incompressible flows is presented the method is a predictor corrector technique combined with a fractional step method each time step is advanced in three sub steps the novel feature of the present scheme is that the poisson equation for the pressure is solved only at the final sub step resulting in substantial savings in computing time it is shown that the method allows a larger cfl number and reduces the computing cost without loss of accuracy by satisfying the continuity equation only at the last sub step numerical solutions for the decaying vortices and flow over a backward facing step are obtained and compared with analytical and other numerical results

a direct solver with o n complexity for variable coefficient elliptic pdes discretized via a high order composite spectral collocation method a numerical method for solving elliptic pdes with variable coefficients on two dimensional domains is presented the method is based on high order composite spectral approximations and is designed for problems with smooth solutions the resulting system of linear equations is solved using a direct as opposed to iterative solver that has optimal o n complexity for all stages of the computation when applied to problems with nonoscillatory solutions such as the laplace and the stokes equations numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of 10 10 or better for challenging problems such as highly oscillatory helmholtz problems and convection dominated convection diffusion equations in terms of speed it is demonstrated that a problem with a nonoscillatory solution that was discretized using 10 8 nodes can be solved in 115 minutes on a personal workstation with two quad core 3 3 ghz cpus since the solver is direct and the solution

computation of electromagnetic scattering with a non conforming discontinuous spectral element method in this paper we solve electromagnetic scattering problems by approximating maxwell s equations in the time domain with a high order quadrilateral discontinuous spectral element method dsem the method is a collocation form of the discontinuous galerkin method for hyperbolic systems where the solution is approximated by a tensor product legendre expansion and inner products are replaced with gauss legendre quadratures to increase flexibility of the method we use a mortar element method to couple element faces mortars provide a means for coupling element faces along which the polynomial orders differ which allows the flexibility to choose the approximation order within an element by considering only local resolution requirements mortars also permit local subdivision of a mesh by connecting element faces that do not share a full side we present evidence showing that the convergence of the non conforming approximations is spectral along with examples of their use copyright 2001 john wiley sons ltd

a conservative staggered grid chebyshev multidomain method for compressible flows we present a new multidomain spectral collocation method that uses a staggered grid for the solution of compressible flow problems the solution unknowns are defined at the nodes of a gauss quadrature rule the fluxes are evaluated at the nodes of a gauss lobatto rule the method is conservative free stream preserving and exponentially accurate a significant advantage of the method is that subdomain corners are not included in the approximation making solutions in complex geometries easier to compute

short note a comparison of computational efficiencies of spectral difference method and correction procedure via reconstruction we report computational efficiencies of two types of numerical solvers the first type uses the spectral difference sd method and the second one uses the correction procedure via reconstruction cpr in this paper we employ the lumped g 2 scheme proposed by huynh as an example of the cpr approach the solvers deal with both inviscid euler equations and navier stokes equations on 2d unstructured grids which are comprised of all quadrilateral cells both types of solvers are programmed using fortran 90 with similar management of data structures we employ identical time marching schemes for both sd and cpr methods spatial 3rd and 4th order of accuracy for both methods is demonstrated by a study of a moving inviscid vortex the comparisons were directed to measure the computational efficiency of both sd and cpr methods in spatial discretization with respect to the fourth order methods cpr is 27 faster than sd for inviscid flow and more promisingly cpr is over 40 faster than sd for viscous flow

a conservative staggered grid chebyshev multidomain method for compressible flows ii a semi structured method we present a chebyshev multidomain method that can solve systems of hyperbolic equations in conservation form on an unrestricted quadrilateral subdivision of a domain within each subdomain the solutions and fluxes are approximated by a staggered grid chebyshev method thus the method is unstructured in terms of the subdomain decomposition but strongly structured within the subdomains communication between subdomains is done by a mortar method in such a way that the method is globally conservative the method is applied to both linear and nonlinear test problems and spectral accuracy is demonstrated

on the cardinality of a factor set of binary matrices abstract the work considers an equivalence relation in the set of all n m binary matrices in each element of the factor set generated by this relation we define the concept of canonical binary matrix namely the minimal element with respect to the lexicographic order for this purpose the binary matrices are uniquely represented by ordered n tuples of integers we have found a necessary and sufficient condition for an arbitrary matrix to be canonical this condition could be the base for realizing recursive algorithm for finding all n m canonical binary matrices and consequently for finding all bipartite graphs up to isomorphism with cardinality of each part equal to n and m