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

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 block pseudospectral method for maxwell s equations a block pseudospectral bps method is proposed as a new way to couple pseudospectral discretizations across interfaces in computations for a linear hyperbolic system the coupling is achieved via discretized derivative matching conditions obtained from the system compared to the standard technique of imposing compatibility conditions based on characteristics of the system the bps method offers better stability and accuracy especially in the case where equation coefficients are discontinuous computational examples for maxwell s equations in nonhomogeneous media demonstrate that bps retains high accuracy over times that are orders of magnitude larger than those for not only low order methods such as yee s but also high order methods such as characteristic based spectral elements

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

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

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

numerical study of time splitting and space time adaptive wavelet scheme for schrodinger equations in this paper we present a time splitting and space time adaptive wavelet scheme for solving schrodinger equations with a small planck constant the scheme is stable and it can overcome high frequency pseudo oscillation of the solution the concrete steps and numerical experiments are also given

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

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

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

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

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

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

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

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

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

quadrature of smooth stochastic processes the integral of a stochastic process is estimated by means of classical quadrature formulae in contrast to certain conventional methods knowledge of covariances is not required and no regularity conditions are assumed explicit error representations and error bounds with respect to thel p norm are established

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

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

non oscillatory spectral element chebyshev method for shock wave calculations a new algorithm based on spectral element discretization and non oscillatory ideas is developed for the solution of hyperbolic partial differential equations a conservative formulation is proposed based on cell averaging and reconstruction procedures that employs a staggered grid of gauss chebyshev and gauss lobatto chebyshev discretizations the non oscillatory reconstruction procedure is based on ideas similar to those proposed by cai et al math comput 52 389 1989 but employs a modified technique which is more robust and simpler in terms of determining the location and strength of a discontinuity it is demonstrated through model problems of linear advection inviscid burgers equation and one dimensional euler system that the proposed algorithm leads to stable non oscillatory accurate results exponential accuracy away from the discontinuity is realized for the inviscid burgers equation example

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

local discriminative learning for pattern recognition abstract local discriminative learning methods approximate a target function a posteriori class probability function directly by partitioning the feature space into a set of local regions and appropriately modeling a simple input output relationship function in each one this paper presents a new method for judiciously partitioning the input feature space in order to accurately represent the target function the method accomplishes this by approximating not only the target function itself but also its derivatives as such the method partitions the input feature space along those dimensions for which the class probability function changes most rapidly thus minimizing bias the efficacy of the method is validated using a variety of simulated and real world data

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

the ode formulation of hyperbolic pdes discretized by the spectral collocation method we formulate systems of hyperbolic partial differential equations pdes with characteristic boundary conditions as pure initial value problems of ordinary differential equations odes this enables a straightforward application of any explicit or implicit method for the time integration of the spatially discretized system we also show how this method relates to the correctional method of treating the boundary points to exemplify we work out the ode formulation for the one dimensional compressible euler equations and a simple two dimensional linear system and conclude with some numerical calculations on these using both explicit and implicit runge kutta methods

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

a direct solver for variable coefficient elliptic pdes discretized via a composite spectral collocation method a numerical method for variable coefficient elliptic problems on two dimensional domains is presented the method is based on high order spectral approximations and is designed for problems with smooth solutions the resulting system of linear equations is solved using a direct solver with o n 1 5 complexity for the pre computation and o nlogn complexity for the solve the fact that the solver is direct is a principal feature of the scheme and makes it particularly well suited to solving problems for which iterative solvers struggle in particular for problems with highly oscillatory solutions numerical examples demonstrate that the scheme is fast and highly accurate for instance using a discretization with 12 points per wavelength a helmholtz problem on a domain of size 100x100 wavelengths was solved to ten correct digits the computation was executed on a standard laptop it involved 1 6m degrees of freedom and required 100 s for the pre computation and 0 3 s for the actual solve

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

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

numerical approximation of quadratic observables of schrodinger type equations in the semi classical limit we apply wigner transform techniques to the analysis of difference methods for schrodinger type equations in the case of a small planck constant in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not numerical test examples are presented to help interpret the theory

bayesian design and analysis of computer experiments use of derivatives in surface prediction this article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain t given values of yo and all of its first derivatives at a set of design sites points in t the intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in t represents a particular configuration of the input parameters it is assumed that the first derivatives are already available e g from a sensitivity analysis or can be produced by the code that implements the model a bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary gaussian stochastic process is used the calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example the issue of experimental design is also discussed in particular the criterion of maximizing the reduction in entropy

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

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 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

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed