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 simulation of supersonic reactive flows we present in this paper numerical simulations of reactive flows interacting with shock waves we argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long term integrations of complicated flows even in the presence of shock waves a spectral code is described in detail and the theory that leads to each of its components is explained results of interactions of hydrogen jets with shock waves are presented and analyzed and comparisons with eno finite difference schemes are carried out

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

filtering in legendre spectral methods we discuss the impact of modal filtering in legendre spectral methods both on accuracy and stability for the former we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity computational results confirm that less strict necessary conditions appear to be adequate we proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability the results are illustrated by computational experiments

asymptotic convergence criteria of solutions of delayed functional differential equations abstract the convergence of all solutions of the scalar equation x f t x t is studied under the main assumption that every constant is its solution the criterion and the sufficient conditions for convergence are proved for the considered classes of equations it is among others proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution as applications criteria for some classes of linear equations as well as for nonlinear equations are obtained some comparisons with already known results are given too

comments on filter based stabilization of spectral element methods p fischer and j mullen 3 have recently proposed a stabilization technique for the solution of the unsteady navier stokes equations with the spectral element method it is based on interpolations in physical space given the values of a function u on a gauss lobatto legendre gll mesh with n 1 d nodes per element where d is the space dimension and n is the degree of the polynomial approximation in each direction in each element one uses the polynomial interpolant to compute u at the n d nodes gll mesh so that one obtains a new polynomial approximation the degree of which in each direction is then n 1 combined with a relaxation method this filtering of the highest frequencies is applied at each time step an important advantage of the technique is that interelement continuity and boundary conditions are preserved its efficiency for high reynolds number flows was demonstrated from results of numerical simulations in the present note we point out that this technique finds a simple interpretation in legendre spectral space and that this interpretation remains true when a chebyshev polynomial approximation is used although the result is easy to obtain it is not a priori obvious so that to our knowledge this viewpoint has not yet been clearly stated moreover we emphasize the link only briefly mentioned in the conclusion of 3 to the filtering procedure suggested in 2 and finally we consider the case of the fourier spectral method by extension to the trigonometric polynomials without loss of generality we can restrict ourselves to the one dimensional situation let 1 1 pn n n the space of the polynomials of maximum degree n defined on li n i 0 the set of the n 1 legendre polynomials of degree i and n i n i 0 the set of the gll nodes associated with pn i e those that solve 1 x2 l n x 0

asymptotic representation of solutions of equationẏ t β t y t y t τ t abstract in this paper the equation t t y t y t t is considered where c i 1 r i 1 t 1 r 0 t t is an increasing function on i 1 t t i 1 0 c i 1 r results about asymptotic structure and behavior of the solutions of this equation are proved a connection with the equation t c t x t t and the asymptotic representation of its solutions is obtained

the divergence equation in rough spaces abstract we aim at extending the existence theory for the equation div v f in a bounded or exterior domain with homogeneous dirichlet boundary conditions to a class of solutions which need not have a trace at the boundary typically the weak solutions that we shall consider will belong to some besov space b p q s with s 1 1 p 1 p after generalizing the notion of a solution for this equation we propose an explicit construction by means of the classical bogovski formula this construction enables us to keep track of a marginal information about the trace of solutions in particular it ensures that the trace is zero if f is smooth enough we expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics

legendre pseudospectral viscosity method for nonlinear conservation laws in this paper the legendre spectral viscosity sv method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied the authors prove that by adding a small amount of sv bounded solutions of the legendre sv method converge to the exact scalar entropy solution the convergence proof is based on compensated compactness arguments and therefore applies to certain 2 times 2 systems finally numerical experiments for scalar as well as the one dimensional system of gas dynamics equations are presented which confirm the convergence of the legendre sv method moreover these numerical experiments indicate that by post processing the sv approximation one can recover the entropy solution within spectral accuracy

de aliasing on non uniform grids algorithms and applications we present de aliasing rules to be used when evaluating non linear terms with polynomial spectral methods on non uniform grids analogous to the de aliasing rules used in fourier spectral methods they are based upon the idea of super collocation followed by a galerkin projection of the non linear terms we demonstrate through numerical simulation that both accuracy and stability can be greatly enhanced through the use of this approach we begin by deriving from the numerical quadrature rules used by galerkin type projection methods the number of quadrature points and weights needed for quadratic and cubic non linearities we then present a systematic study of the effects of super collocation when using both a continuous galerkin and a discontinuous galerkin method to solve the one dimensional viscous burgers equation we conclude by examining three direct numerical simulation flow examples incompressible turbulent flow in a triangular duct incompressible turbulent flow in a channel at re 395 and compressible flow past a pitching airfoil at re 45 000

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

exponential solutions of equation ẏ t β t y t δ y t τ abstract asymptotic behaviour of solutions of first order differential equation with two deviating arguments of the form y t t y t y t is discussed for t a criterion for representing solutions in exponential form is proved as consequences inequalities for such solutions are given connections with known results are discussed and a sufficient condition for existence of unbounded solutions generalizing previous ones is derived an illustrative example is considered too

on asymptotically autonomous retarded functional differential equations the aim of the present paper is to investigate the asymptotic behavior of the solutions of the asymptotically autonomous retarded functional differential equation t lxt m t xt we obtain a convergence result without assuming that m t is in lp 1 p moreover we give bounds on the rate of convergence as well

l2 perturbation of a linear delay differential equation abstract in this paper we present an asymptotic formula for the solutions of the nonautonomous delay differential equation t a 0 a t x t b 0 b t x t assuming that it is an l 2 perturbation of the autonomous equation t a 0 x t b 0 x t our theorem is an essential generalization of the corresponding result in the classical book by r bellman and k l cooke the proof is based on a new result on the asymptotic constancy of the solutions of a linear delay equation

the hartman wintner theorem for functional differential equations abstract in this paper we present a generalization of the hartman wintner theorem about the asymptotic behavior of the solutions of linear ordinary differential equations to functional differential equations

asymptotic integration of linear differential equations abstract we establish a new result on the asymptotic integration of linear differential systems of the form y r y where is diagonal and r is small more specifically we allow for different scales of asymptotic behaviour of the individual matrix elements of and r we also discuss an application to a self adjoint fourth order equation

idempotent filtering in spectral and spectral element methods the comparison of numerical results for implicit explicit and fully explicit runge kutta time integration methods for a nozzle flow problem shows that filtering can significantly degrade the accuracy of the numerical solution for long time integration problems we demonstrate analytically and numerically that filtering in time errors become additive for u n x t k dt u n x t u n x t when nonidempotent filters are used and suggest the development and implementation of idempotent filters

an asymptotic theory for nonlinear functional differential equations abstract the hartman wintner theorem on asymptotic integration is established for a certain class of functional differential equations with nonlinear perturbations by looking them as abstract ordinary differential equations results for delay differential equations are included in this study consequences and applications are also shown

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

stability of pseudospectral and finite difference methods for variable coefficient problems it is shown that pseudospectral approximation to a special class of variable coefficient one dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain computer experiments indicate similar results are valid for more general problems similarly computer results indicate that the leapfrog finite difference scheme is stable even though the wave speed changes sign within the domain however both schemes can be asymptotically unstable in time when a fixed spatial mesh is used

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows