pseudospectral method based on prolate spheroidal wave functions for frequency domain electromagnetic simulations we apply prolate spheroidal wave functions of order zero as basis functions in the pseudospectral method for frequency domain electromagnetic simulation problems like the traditional pseudospectral frequency domain psfd methods based on chebyshev and legendre polynomial series the prolate psfd method yields exponential order of accuracy in terms of the number of samples utilized per wavelength the prolate expansion is superior to the chebyshev and legendre polynomial series by a factor of spl pi 2 in addition the prolate psfd method employs a more uniform spatial grid achieving better resolution near the center of the domain

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

numerical algorithms for eigenvalue assignment by constant and dynamic output feedback algorithms are proposed for eigenvalue assignment eva by constant as well as dynamic output feedback the main algorithm is developed for single input multioutput systems and the results are then extended to multiinput multioutput systems in computing the feedback use is made of the fact that the closed loop eigenvalues can almost always be assigned arbitrarily close to the desired locations in the complex plane provided the system satisfies the condition m p n where m p and n are respectively the number of inputs outputs and states of the system the eva problem has been treated as a converse of the algebraic eigenvalue problem the proposed algorithms are based on the implicitly shifted qr algorithm for solving the algebraic eigenvalue problem the performance of the algorithms is illustrated by several numerical examples

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

numerical study of pseudospectral methods in shock wave applications abstract in this paper we use shock capturing spectral methods to simulate compressible flows in the presence of shock waves three applications are considered the first is an interaction of a one dimensional shock with an entropy wave the second case deals with interactions between shock wave and an entropy wave in two space dimensions and the third case is two dimensional shock vortex interactions the first two applications are found in the study of turbulence in high speed flow the last application is a key element in understanding the acoustic dynamics in aeroacoustic and in design of supersonic jet in aerodynamics the purpose of this study is to show the feasibility of simulating shocked flow with spectral methods the numerical methods involved are the chebyshev and fourier collocation methods the euler equations of gas dynamics is discretized by pseudospectral collocation methods in space and a nonlinearly stable third order runge kutta methods in time the fluxes are evaluated pointwise directly and not by the cell averaging technique a significant reduction in cpu time and storage usage are achieved by incorporating several well established numerical techniques such as grid transformation and filtering into the spectral algorithm the results of this study indicate that spectral method is well suited not only for smooth problems but also for those with discontinuity

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

a computational algorithm for pole assignment of linear single input systems an efficient computational algorithm for pole assignment of linear multiinput systems is proposed a preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form the gain matrix elements are then found by orthogonal transformation of the closed loop system matrix into upper quasi triangular form whose diagonal blocks correspond to the desired poles the algorithm is numerically stable the computed gain matrix being exact for a system with slightly perturbed matrices it works equally well with real and complex distinct and multiple poles and is applicable to ill conditioned and high order problems

high order point value enhanced finite volume method for two dimensional hyperbolic equations on unstructured meshes abstract presently an approximate delta function adf is defined in multi dimensional space in the form of a finite order polynomial that holds identical integral properties to the dirac delta function when used in conjunction with a finite order polynomial integrand spanning a finite domain by using this adf over a two dimensional unstructured mesh a compact high order point value enhanced finite volume method pfv is constructed the corresponding scheme is capable of storing and updating cell averaged values inside each element as well as the unknown quantities and their derivatives when needed on vertex points the sharing of vertex information with surrounding elements reduces the number of degrees of freedom relative to other compact and high order methods of comparable convergence rate to ensure conservation cell averaged values are updated using an identical approach to the one traditionally followed in the finite volume method to verify the performance of the pfv scheme at successive orders its accuracy and stability are vetted using benchmark problems associated with the two dimensional wave and euler equations encompassing a total of six standard test cases

an algorithm for the multiinput eigenvalue assignment problem a very simple and inexpensive algorithm is presented for pole placement in the multiinput case the algorithm consists of orthogonal reduction to a block hessenberg form and a simple linear recursion it yields a matrix f such that a bf has any specified set of eigenvalues whenever the system is controllable it is extremely easy to program on a computer the algorithm is not a robust pole placement algorithm but appears to give comparable results in most well conditioned cases at a fraction of the cost it is a direct noniterative algorithm and no eigenvalues or singular values are computed the algorithm does not need any complex arithmetic even when complex conjugate eigenvalues need to be assigned

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

optimal decomposition of the domain in spectral methods for wave like phenomena a strategy for determining the optimal number of grid points and subdomains in a spectral method with domain decomposition on a serial computer is presented the rapidly growing computational cost for large numbers of grid points in each subdomain is balanced against the exponential convergence for spectral approximation of smooth functions and the optimum is found as the number of grid points and subdomains that gives the minimal computational cost for a given accuracy the typical length scale of the problem is found to influence the number of subdomains but not the number of grid points within each subdomain

globally constraint preserving fr dg scheme for maxwell s equations at all orders abstract computational electrodynamics ced the numerical solution of maxwell s equations plays an incredibly important role in several problems in science and engineering high accuracy solutions are desired and the discontinuous galerkin dg method is one of the better ways of delivering high accuracy in numerical ced maxwell s equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable balsara and kappeli 2019 presented a von neumann stability analysis of globally constraint preserving dg schemes for ced up to fourth order that paper was focused on developing the theory and documenting the superior dissipation and dispersion of dgtd schemes in media with constant permittivity and permeability in this paper we present working dgtd schemes for ced that go up to fifth order of accuracy and analyze their performance when permittivity and permeability vary strongly in space our dgtd schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh our first finding is that at fourth and higher orders of accuracy one has to evolve some zone centered modes in addition to the face centered modes it is well known that the limiting step in dg schemes causes a reduction of the optimal accuracy of the scheme though the schemes still retain their formal order of accuracy with weno type limiters in this paper we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the dg scheme this very favorable second finding ensures that dgtd schemes retain optimal accuracy even in the presence of large spatial variations in permittivity and permeability we also study the conservation of electromagnetic energy in these problems our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space as long as the conductivity is zero

analysis of an hp nonconforming discontinuous galerkin spectral element method for wave propagation we analyze the consistency stability and convergence of an hp discontinuous galerkin spectral element method of kopriva j comput phys 128 1996 pp 475 488 and kopriva woodruff and hussaini internat j numer methods engrg 53 2002 pp 105 122 the analysis is carried out simultaneously for acoustic elastic coupled elastic acoustic and electromagnetic wave propagation our analytical results are developed for both conforming and nonconforming approximations on hexahedral meshes using either exact integration with legendre gauss quadrature or inexact integration with legendre gauss lobatto quadrature a mortar based nonconforming approximation is developed to treat both h and p nonconforming meshes simultaneously the mortar approach is constructed in such a way that consistency stability and convergence analyses for nonconforming approximations follow the conforming counterparts with minimal modifications in particular it casts hp nonconforming interfaces into equiv

a discrete splitting finite element method for numerical simulations of incompressible navier stokes flows the presence of the pressure and the convection terms in incompressible navier stokes equations makes their numerical simulation a challenging task the indefinite system as a consequence of the absence of the pressure in continuity equation is ill conditioned this difficulty has been overcome by various splitting techniques but these techniques incur the ambiguity of numerical boundary conditions for the pressure as well as for the intermediate velocity whenever introduced we present a new and straightforward discrete splitting technique which never resorts to numerical boundary conditions the non linear convection term can be treated by four different approaches and here we present a new linear implicit time scheme these two new techniques are implemented with a finite element method and numerical verifications are made copyright 2005 john wiley sons ltd

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

parallel and large scale matrix computations in control some ideas abstract the design and analysis of time invariant linear control systems give rise to a variety of interesting linear algebra problems numerically effective methods now exist for several of these problems however algorithms for large scale computations and efficient parallel algorithms for these problems are virtually nonexistent in this paper we propose several efficient general purpose parallel algorithms for single input and multiinput eigenvalue assignment problems a desirable feature of these algorithms is that they are composed of simple linear algebraic operations such as matrix vector multiplication solution of a linear system and computations of the eigensystem and singular values of a symmetric matrix for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms the proposed algorithms thus have potential for implementations on some existing and future parallel processors we also propose a numerical method for the sylvester matrix equation arising in the construction of a luenberger observer the method does not need reduction to condensed forms and is thus suitable for large and sparse matrices the method also exhibits a certain parallelism

numerical simulation of transitional flow past a wing using flux reconstruction method in this paper the capabilities of a high order flux reconstruction scheme for the implicit large eddy simulations of a transitional flow at a low reynolds number of 60 000 around the sd7003 wing are examined this is the case posed in the first and second international workshops on the high order computational fluid dynamics method held in 2012 and 2013 the correction function used in the flux reconstruction scheme is the radau polynomial which is equivalent to the discontinuous galerkin scheme a time accurate implicit lower upper symmetric gauss seidel solution algorithm for the application of the flux reconstruction scheme to complex unsteady flows is developed and it is found to be able to produce comparable results to the explicit runge kutta scheme while achieving better computational efficiency simulations are carried out at 4 and 8 deg with polynomials of degree p 1 2 and 3 resulting in second third and fourth order accurate flux reconstruction schemes respectively two structured he

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

numerical calculation of incomplete gamma functions by the trapezoidal rule the trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function a x in the regiona 0 since this application of the rule is not standard a careful investigation of the remainder terms using the euler maclaurin formula is carried out the outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with surprising accuracy in the stated region

high order flux correction finite difference schemes for strand grids a novel high order method combining unstructured flux correction along body surfaces and high order finite differences normal to surfaces is formulated for unsteady viscous flows on strand grids the flux correction algorithm is applied in each unstructured layer of the strand grid and the layers are then coupled together via a source term containing derivatives in the strand direction strand direction derivatives are approximated to high order via summation by parts operators for first derivatives and second derivatives with variable coefficients we show how this procedure allows for the proper truncation error canceling properties required for the flux correction scheme the resulting scheme possesses third order design accuracy but often exhibits fourth order accuracy when higher order derivatives are employed in the strand direction especially for highly viscous flows we prove discrete conservation for the new scheme and time stability in the absence of the flux correction terms results in two dimensions are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second order algorithms

technical communique complete parametric approach for eigenstructure assignment in a class of second order linear systems this note deals with eigenstructure assignment in the second order linear system email protected email protected cq bu using the proportional plus derivative feedback controller u k 0q k email protected under the controllability condition of the matrix pair ab very simple general and complete parametric expressions in direct closed forms for both the closed loop eigenvector matrix and the feedback gains are established in terms of the closed loop eigenvalues and a group of parameter vectors the main computations are two sets of elementary matrix transformations which can be replaced by a series of singular value decompositions when the closed loop eigenvalues are chosen a priori the approach utilizes directly the original system data a b and c and involves manipulations on only n dimensional matrices an example illustrates the effect of the proposed approach

an optimal runge kutta method for steady state solutions of hyperbolic systems an optimal two stage runge kutta time integration scheme is derived to compute steady state approximations to hyperbolic equations the analysis is general in the sense that it requires only that the eigenvalues of the derivative matrix lie in the right half of the complex plane thus it is applicable to spatial discretizations which do not have uniformly spaced points as examples the method is applied to two hyperbolic problems which have been discretized in space by chebyshev spectral collocation

an algorithm to assign eigenvalues in a hessenberg matrix single input case we propose a conceptually simple algorithm to assign eigenvalues in a hessenberg matrix the method is based on the evaluation of a simple recursive relation and it is extremely easy to program on a computer numerical experiments compare favorably to those of other methods recently proposed for the problem

impact of number representation for high order implicit large eddy simulations high order numerical methods for unstructured grids combine the superior accuracy of high order spectral or finite difference methods with the geometric flexibility of low order finite volume or fi

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

short note a comparison of the discontinuous galerkin and spectral difference method on triangulations using pkd polynomials this paper gives a first numerical comparison of the discontinuous galerkin dg and spectral difference method sd under almost equivalent conditions for smooth test cases

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

linear and numerical linear algebra in control theory some research problems abstract this paper presents an informal survey of theoretical and computational aspects of three linear algebra problems arising in control theory namely the sylvester observer matrix equation the matrix eigenvalue assignment problem and the stability and stabilization of large second order systems difficulties are highlighted recent developments are mentioned some research problems in these areas which should be of interest to both linear algebraists and control theorists are described

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

parametric eigenstructure assignment in second order descriptor linear systems this note considers eigenstructure assignment in second order descriptor linear systems via proportional plus derivative feedback it is shown that the problem is closely related with a type of so called second order sylvester matrix equations through establishing two general parametric solutions to this type of matrix equations two complete parametric methods for the proposed eigenstructure assignment problem are presented both methods give simple complete parametric expressions for the feedback gains and the closed loop eigenvector matrices the first one mainly depends on a series of singular value decompositions and is thus numerically simple and reliable the second one utilizes the right factorization of the system and allows the closed loop eigenvalues to be set undetermined and sought via certain optimization procedures an example shows the effect of the proposed approaches

spectral element method for linear fan tone noise radiation a numerical method for prediction of acoustic spinning mode radiation from turbofan inlets is presented sound propagation is modeled by the linearized mass conservation equation for irrotational flows and solved in the frequency domain the mean flow through the inlet is obtained as a solution of the full potential equation both the mean flow and the acoustic problem are approximated by galerkin projection in spectral element spaces of continuous piecewise polynomials defined on the same grid the gauss chebyshev lobatto points within the elements are generated via transfinite interpolation and cad projection procedures embedded within the code the linear algebraic systems obtained are then solved using either direct or sparse iterative solvers based on the message passing interface standard for interprocessor communication the singularity appearing in the acoustic integrals on the symmetry axis is treated by the use of a collocation operator based on the gauss chebyshev instead of the gauss chebyshev lobatto points to eliminate reflections from the radiation boundaries a novel frequency domain formulation of the matched layer technique wherein waves entering the layer are exponentially damped is proposed the overall computing procedure is first validated on a tone radiation problem from a semi infinite cylinder and then applied to an experimental jt15d turbofan inlet setup

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

revisiting the spectral analysis for high order spectral discontinuous methods the spectral analysis is a basic tool to characterise the behaviour of any convection scheme by nature the solution projected onto the fourier basis enables to estimate the dissipation and the dispersion associated with the spatial discretisation of the hyperbolic linear problem in this paper we wish to revisit such analysis focusing attention on two key points the first point concerns the effects of time integration on the spectral analysis it is shown with standard high order finite difference schemes dedicated to aeroacoustics that the time integration has an effect on the required number of points per wavelength the situation depends on the choice of the coupled schemes one for time integration one for space derivative and one for the filter and here the compact scheme with its eighth order filter seems to have a better spectral accuracy than the considered dispersion relation preserving scheme with its associated filter especially in terms of dissipation secondly such a coupled space time approach is applied to the new class of high order spectral discontinuous approaches focusing especially on the spectral difference method a new way to address the specific spectral behaviour of the scheme is introduced first for wavenumbers in 0 0 following the matrix power method for wavenumbers above an aliasing phenomenon always occurs but it is possible to understand and to control the aliasing of the signal it is shown that aliasing depends on the polynomial degree and on the number of time steps a new way to define dissipation and dispersion is introduced and applied to wavenumbers larger than since the new criteria recover the previous results for wavenumbers below the new proposed approach is an extension of all the previous ones dealing with dissipation and dispersion errors at last since the standard finite difference schemes can serve as reference solution for their capability in aeroacoustics it is shown that the spectral difference method is as accurate as or even more accurate than the considered finite difference schemes

pole assignment via sylvester s equation it is shown that the pole assignment problem can be reduced to solving the linear matrix equations ax xa bg fx g successively for x and then f for almost any choice of g the result is a new pole assignment procedure and proof of the pole assignment theorem that should play an important role in both theoretical and practical applications

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

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

parametric control to a type of quasi linear second order systems via output feedback abstractthis paper considers the design of output feedback control for a type of quasi linear second order systems with the time varying coefficient matrices containing the state variables and a time varying parameter vector based on the solution to a type of second order generalised sylvester matrix equations general complete parameterisation of a quasi linear output feedback controller is established with respect to the state variables the time varying parameter vector the constant closed loop system and another two groups of arbitrary parameters and also for the left and right closed loop eigenvectors matrices with the proposed parametric output feedback control the closed loop system can be transformed into a constant linear system with desired eigenstructure finally simulation results are provided to illustrate the convenience and effectiveness of application in the general spacecraft rendezvous problem

a point value enhanced finite volume method based on approximate delta functions abstract we revisit the concept of an approximate delta function adf introduced by huynh 2011 1 in the form of a finite order polynomial that holds identical integral properties to the dirac delta function when used in conjunction with a finite order polynomial integrand over a finite domain we show that the use of generic adf polynomials can be effective at recovering and generalizing several high order methods including taylor based and nodal based discontinuous galerkin methods as well as the correction procedure via reconstruction based on the adf concept we then proceed to formulate a point value enhanced finite volume pfv method which stores and updates the cell averaged values inside each element as well as the unknown quantities and if needed their derivatives on nodal points the sharing of nodal information with surrounding elements saves the number of degrees of freedom compared to other compact methods at the same order to ensure conservation cell averaged values are updated using an identical approach to that adopted in the finite volume method here the updating of nodal values and their derivatives is achieved through an adf concept that leverages all of the elements within the domain of integration that share the same nodal point the resulting scheme is shown to be very stable at successively increasing orders both accuracy and stability of the pfv method are verified using a fourier analysis and through applications to the linear wave and nonlinear burgers equations in one dimensional space

neurodynamics based robust pole assignment for high order descriptor systems in this paper a neurodynamic optimization approach is proposed for synthesizing high order descriptor linear systems with state feedback control via robust pole assignment with a new robustness measure serving as the objective function the robust eigenstructure assignment problem is formulated as a pseudoconvex optimization problem a neurodynamic optimization approach is applied and shown to be capable of maximizing the robust stability margin for high order singular systems with guaranteed optimality and exact pole assignment two numerical examples and vehicle vibration control application are discussed to substantiate the efficacy of the proposed approach

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

spectral element formulations on non conforming grids a comparative study of pointwise matching and integral projection methods pointwise matching pm and integral projection ip methods are two widely used techniques to extend the classical weak formulations to include non conforming grids we present spectral element formulations on polynomial p type and geometric h type non conforming grids using both the pm also known as the constrained approximation and ip also known as the mortar element methods we systematically compare the convergence characteristics of pm and ip methods for diffusion convection and convection diffusion equations consistency errors due to the non conforming formulations of the diffusion equation result in convergence problems for the pm method using the maximum rule both non conforming formulations for the unsteady convection operator result in eigenvalue spectrum with positive real values causing convergence problems due to the consistency errors however small physical diffusion in the convection diffusion equation eliminates these problems resulting in spectral convergence for both methods encouraged by this we present spectral element formulations for incompressible navier stokes equations using pm and ip methods on p type and h type non conforming grids and demonstrate spectral convergence for unsteady and steady test cases results for two dimensional lid driven cavity flow at re 1000 are also presented

feedback stabilization of a second order system a nonmodal approach a novel approach for feedback stabilization of a second order model is proposed specifically two nonmodal algorithms are described and it is shown mathematically that under some mild assumptions on the damping matrix the feedback matrix obtained by each algorithm indeed stabilizes a closed loop system the first algorithm requires the solution of a symmetric positive definite linear system and the inversion of a small matrix a remarkable feature of this algorithm is that it does not require knowledge of the stiffness and damping for implementation which makes it very feasible for practical application the second algorithm requires the solution of a small linear least squares problem and an estimate of the stability index which can be obtained just by computing the extremal eigenvalues of the data matrices which are symmetric for almost all practical applications these minimal computational requirements make the proposed algorithms suitable for practical implementations even for large and sparse systems using the state of the art techniques of matrix computations this is in sharp contrast with the traditional modal approach ordinarily used by practicing engineers which requires the solution of a quadratic eigenvalue problem or equivalently the solution of a 2n x 2n generalized eigenvalue problem and therefore may not be practical for very large and sparse problems besides the proposed algorithms the paper also contains some mathematical results on the bounds of the eigenvalues of a second order pencil which may be useful for investigating stability of second order systems and are of independent interest

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

block pseudospectral methods for maxwell s equations ii two dimensional discontinuous coefficient case block pseudospectral bps methods are examined for maxwell s equations in two dimensional inhomogeneous media for the case of a rectangular strip with a straight line interface blocks may be coupled via fictitious points or a generalization of characteristic outflow conditions the bps methods generalize to curvilinear strips described by a change of variables such strips can conform to interfaces while overlapping with a high order free space grid to form a composite grid method numerical experiments on strips with dielectrics lossy materials perfect conductors and absorbing layers indicate that the two coupling methods are comparable and accurate with just 3 4 points per wavelength full composite examples are included to demonstrate high accuracy and geometric flexibility

on consistency and rate of convergence of flux reconstruction for time dependent problems this study is directed at a rigorous characterization of the consistency and convergence of discontinuous finite element schemes formulated using flux reconstruction fr we show that the fr formulation is consistent for linear advection and converges to the exact solution for any scheme that is stable in the von neumann sense the numerical solution for a scheme of polynomial order p is composed of p 1p 1 eigenmodes of which one and exactly one is physical such that it exhibits the analytical dispersion behavior in the limit of asymptotic grid resolution the remaining p modes are spurious such that the fraction of energy received by them from the initial condition vanishes in the asymptotic limit on grid refinement the rate of convergence of the numerical solution is a function of time starting from a short time rate at t 0 t 0 associated with interpolation and asymptotically approaching a long time rate as t t associated with numerical differentiation both these rates can be inferred directly from the eigensystem of the numerical derivative operator we verify these analytical expectations using simple experiments in 1 d and 2 d

numerically robust pole assignment for second order systems we propose two new methods for solution of the eigenvalue assignment problem associated with the second order control system global hsize 30pc specifically the methods construct feedback matrices f 1 and f 2 such that the closed loop quadratic pencil has a desired set of eigenvalues and the associated eigenvectors are well conditioned method 1 is a modification of the singular value decomposition based method proposed by juang and maghami which is a second order adaptation of the well known robust eigenvalue assignment method by kautsky et al for first order systems method 2 is an extension of the recent non modal approach of datta and rincon for feedback stabilization of second order systems robustness to numerical round off errors is achieved by minimizing the condition numbers of the eigenvectors of the closed loop second order pencil control robustness to large plant uncertainty will not be explicitly considered in this paper numerical results for both the two methods are favourable a compara

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

the lobatto cell robust explicit higher order fdtd that handles inhomogeneous media to improve numerical dispersion and second order convergence behavior of the traditional yee finite difference time domain fdtd algorithm various higher order finite difference implementations have been proposed many have difficulty handling interfaces between distinct dielectric media or metallic sheets because the high order taylor expansions on which these codes are based are invalidated by field or derivative discontinuities one sided extensions extrapolations are a common approach to remedy interface discontinuities here another approach based upon whitney form mass lumped finite elements is pursued to develop the ldquolobatto cell rdquo a high order replacement for the yee cell with appropriate continuity behavior at contrasting media interfaces the lobatto cell retains many robust features of the yee method discrete conservation laws a guarantee of conditional stability and straightforward explicit updating but provides higher order interpolation and dispersion error convergence a notable drawback of the method is larger modeling granularity staircasing which is managed here by combining the lobatto cell method with preexisting robust subgridding techniques

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 new flux splitting scheme a new flux splitting scheme is proposed the scheme is remarkably simple and yet its accuracy rivals and in some cases surpasses that of roe s solver in the euler and navier stokes solutions carried out in this study the scheme is robust and converges as fast as the roe splitting we propose an appropriately defined cell face advection mach number using values from the two straddling cells via associated characteristic speeds this interface mach number is then used to determine the upwind extrapolation for the convective quantities accordingly the name of the scheme is coined as the advection upstream splitting method ausm we also introduce a new pressure splitting which is shown to behave successfully yielding much smoother results than other existing pressure splittings of particular interest is the supersonic blunt body problem in which the roe scheme gives anomalous solutions the ausm produces correct solutions without difficulty for a wide range of flow conditions as well as grids

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

a new metaheuristic for numerical function optimization vortex search algorithm abstract in this study a new single solution based metaheuristic namely the vortex search vs algorithm is proposed to perform numerical function optimization the proposed vs algorithm is inspired from the vortex pattern created by the vortical flow of the stirred fluids to provide a good balance between the explorative and exploitative behavior of a search the proposed method models its search behavior as a vortex pattern by using an adaptive step size adjustment scheme the proposed vs algorithm is tested over 50 benchmark mathematical functions and the results are compared to both the single solution based simulated annealing sa and pattern search ps and population based particle swarm optimization pso2011 and artificial bee colony abc algorithms a wilcoxon signed rank test is performed to measure the pair wise statistical performances of the algorithms the results of which indicate that the proposed vs algorithm outperforms the sa ps and abc algorithms while being competitive with the pso2011 algorithm

convergence analysis of shock capturing and shock fitting solutions on unstructured grids a new shock fitting technique for unstructured two and three dimensional meshes has been recently proposed and developed by the authors in the present paper both global and local a posteriori grid convergence analysis is used to quantitatively measure the discretization error and order of convergence of the numerical solutions obtained using this new unstructured shock fitting technique specifically the analysis has considered the numerical solutions of two different flows characterized by the presence of strong shocks a transonic source flow and an hypersonic flow past a circular cylinder it is shown that the shock fitting technique allows to compute numerical solutions that converge both pointwise and in a global sense with an observed order of accuracy that is very close to the design order of the spatial discretization scheme and with very small discretization errors

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

connections between the filtered discontinuous galerkin method and the flux reconstruction approach to high order discretizations the purpose of this paper is to provide new insights on the connections that exist between the discontinuous galerkin method dg the flux reconstruction method fr and the recently identified energy stable flux reconstruction method esfr when solving time dependent conservation laws all these schemes appear to be quite similar and it is important to understand how they are related in this paper we first review results on the stability of the discontinuous galerkin method and extend it to the filtered discontinuous galerkin method we then consider the flux reconstruction approach and show its connections with dg in particular we show how the energy stable flux reconstruction method introduced by vincent et al is equivalent to a filtered dg method hence giving a new proof of its stability also it allows the use of the method without having to know the special form of the flux correction polynomials finally we underline some fundamental differences that exist between fr and dg

a novel strategy for deriving high order stable boundary closures based on global conservation i basic formulas abstract it is well known that for initial boundary value problems one may need to decrease the order of boundary closures to ensure strict stability also called time stability of finite difference schemes for high order schemes this often reduces the global convergence rate significantly to cure this issue we derive high order accurate boundary closures that satisfy a global conservation property we show that the boundary closures cannot be both high order accurate and conservative in the l 2 inner product if the grid is equidistant therefore we derive particular sets of nonuniform solution points and quadrature weights near the domain boundaries the new schemes are high order accurate on the modified grids and conservative in the derived quadrature rules although we have no theoretical stability proof numerical eigenvalue analysis suggests that the new schemes are strictly stable up to eleventh order global accuracy for scalar linear advection equations numerical experiments with one and two dimensional linear equations corroborate the design orders of the schemes

a hybrid multilevel method for high order discretization of the euler equations on unstructured meshes higher order discretization has not been widely successful in industrial applications to compressible flow simulation among several reasons for this one may identify the lack of tailor suited best practice relaxation techniques that compare favorably to highly tuned lower order methods such as finite volume schemes in this paper we investigate solution algorithms in conjunction with high order spectral difference discretization for the euler equations using such techniques as multigrid and matrix free implicit relaxation methods in particular we present a novel hybrid multilevel relaxation method that combines optionally matrix free implicit relaxation techniques with explicit multistage smoothing using geometric multigrid furthermore we discuss efficient implementation of these concepts using such tools as automatic differentiation

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

an algorithm for pole assignment of time invariant linear systems an algorithm is suggested for the computation of a linear state feedback for a single input system such that the resultant closed loop system has specified eigenvalues the algorithm is more efficient than many comparable techniques and has some desirable numerical properties it is closely related to the qr algorithm for the eigenproblem and extends to descriptor systems