a chebyshev spectral method for gas transients in pipelines a chebyshev pseudospectral method is proposed for slow transients simulation of gas transportation systems the scheme is theoretically investigated comparisons with the lax wendroff scheme prove that the chebyshev method is more efficient in terms of storage and computer time infinite order accuracy for smooth solutions is predicted by the theory and observed in the experiments

preconditioned minimal residual methods for chebyshev spectral calculations abstract the problem of preconditioning the pseudospectral chebyshev approximation of an elliptic operator is considered the numerical sensitivity to variations of the coefficients of the operator are investigated for two classes of preconditioning matrices one arising from finite differences the other from finite elements the preconditioned system is solved by a conjugate gradient type method and by a dufort frankel method with dynamical parameters the methods are compared on some test problems with the richardson method 13 and with the minimal residual richardson method 21

finite element preconditioning for spectral multigrid methods abstract the good performance of finite element preconditioning for the algebraic spectral systems is demonstrated different types of multigrid strategies are compared the most efficient strategy consists of a pure richardson relaxation preconditioned by one v cycle of a finite element multigrid method the good smoothing properties of line relaxation become obvious the computational work for the various methods is estimated

multigrid methods for combined finite difference and fourier problems abstract multigrid methods for combined finite difference and fourier pseudospectral problems are considered applications to two and three dimensional elliptic systems are given suitable relaxation schemes for isotropic and anisotropic problems are presented numerical results demonstrate the efficiency of these methods

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

the spectrum of the chebyshev collocation operator for the heat equation it is shown that the eigenvalues of the pseudospectral chebyshev second derivative operator with separated boundary conditions are real negative and distinct an application to the full potential equation is also presented

a chebyshev collocation algorithm for the solution of advection diffusion equations an original method to precondition chebyshev collocated advection diffusion operators is proposed the preconditioner based on an upwind finite difference approximation is tested on one and two dimensional problems the preconditioner behaves efficiently for advection dominated cases and produces rates of convergence independent of the number of nodes finally an ad hoc designed multi domain strategy is introduced

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

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

thermal convection in vertical cylinders a method based on potentials of velocity abstract any solenoidal vector field can be written in terms of two scalar fields solving the navier stokes equation for these two so called potentials is a widely used method although difficulties have been reported in using this method to describe non cartesian confined flows in particular cylindrical geometries difficulties arise because the potentials are coupled at the boundaries but also because of possible numerical instabilities associated with the high order of the equations this paper shows how all these difficulties can be circumvented firstly by properly deriving the required boundary conditions but also through an adequate discretization which is the final requirement the use of spectral methods together with a reduction of order for the axisymmetric modes is sufficient for this purpose in this paper as a test a velocity potentials method is used to solve the stability problem for the onset of convection in a fluid confined in a cylindrical container

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

on the boundary treatment in spectral methods for hyperbolic systems abstract spectral methods have been successfully applied to the simulation of slow transients in gas transportation networks implicit time advancing techniques are naturally suggested by the nature of the problem the aim of this paper is to clarify the correct treatment of the boundary conditions in order to avoid any stability restriction originated by the boundaries the beam and warming and the lerat schemes are unconditionally linearly stable when used with a chebyshev pseudospectral method engineering accuracy for a gas transportation problem is achieved at courant numbers up to 100

a spectral multigrid method for the stokes problem in streamfunction formulation abstract suitable spectral multigrid components for the stokes problem in streamfunction formulation are presented we split the stokes problem into a system of two equations with the laplace operator we discretize by means of a collocation or pseudo spectral method preconditioning with finite differences is employed numerical results for the spectral multigrid method are presented

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

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

on the homflypt skein module of s 1 x s 2 let k be a subring of the field of rational functions in x v s which contains x pm 1 v pm 1 s pm 1 if m is an oriented 3 manifold let s m denote the homflypt skein module of m over k this is the free k module generated by isotopy classes of framed oriented links in m quotiented by the homflypt skein relations 1 x 1 l xl s s 1 l 0 2 l with a positiv e twist xv 1 l 3 l sqcup o frac v v 1 s s 1 l where o is the unknot we give two bases for the relative homflypt skein module of the solid torus with 2 points in the boundary the first basis is related to the basis of s s 1 times d 2 given by j hoste and m kidwell and also v turaev the second basis is related to a young idempotent basis for s s 1 times d 2 based on the work of a aiston h morton and c blanchet we prove that if the elements s 2n 1 for n a nonzero integer and the elements s 2m v 2 for any integer m are invertible ink then s s 1 times s 2 k torsion module oplus k here the free part is generated by the empty link phi in addition if the elements s 2m v 4 for m an integer are invertible in k then s s 1 times s 2 has no torsion we also obtain some results for more general k

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

the spectral viscosity method applied to simulation of waves in a stratified atmosphere a spectral collection method is used to solve an atmospheric gravity wave simulation problem modelled by the non linear inviscid two dimensional euler equations including gravity the non linear transfer of energy to higher frequencies will destroy the long time calculations if no energy dissipation mechanism is introduced in the numerical method the spectral viscosity method is applied for this purpose and we discuss implementation aspects effects on the resulting system of ordinary differential equations tuning of parameters and evaluation of results an implementation is found that works well also for larger simulations and guidelines for the application of the method are given

on numerical boundary treatment of hyperbolic systems for finite difference and finite element methods it is well known that the stability of the initial boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment in fact we show that for any boundary treatment one can construct systems for which the boundary treatment on the natural variables is not stable our main theorem is that one can introduce a treatment of the boundaries for which the stability of the system follows immediately from the stability for a scalar equation this is accomplished by operating on the characteristic variables for those quantities that are not specified on the boundary in a working code this can be accomplished by adding a correction term to the existing boundary algorithm the analysis and computational results are presented for both finite differences and semi discrete galerkin methods

spectral methods for exterior elliptic problems this paper deals with spectral approximations for exterior elliptic problems in two dimensions as in the conventional finite difference or finite element methods it is found that the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions we introduce a spectral boundary treatment at infinity which is compatible with the infinite order interior spectral scheme computational results are presented to demonstrate the spectral accuracy attainable although we deal with a simple laplace problem throughout the paper our analysis covers more complex and general cases

dichromatic link invariants we investigate the skein theory of oriented dichromatic links in s3 we define a new chromatic skein invariant for a special class of dichromatic links this invariant generalizes both the two variable alexander polynomial and the twisted alexander polynomial alternatively one may view this new 1 2 invariant as an invariant of oriented monochromatic links in s x d and as such it is the exact analog of the twisted alexander polynomial we discuss basic properties of this new invariant and applications to link interchangeability for the full class of dichromatic links we show that there does not exist a chromatic skein invariant which is a mutual extension of both the two variable alexander polynomial and the twisted alexander polynomial

spectral methods for high order equations abstract solving navier stokes equations by means of scalar potentials leads to higher order equations with low order boundary conditions this approach usually involves numerical instabilities that do not appear using the primitive navier stokes equations such difficulties can be avoided with an explicit evaluation of some of the high order derivatives at the boundary we present three numerical techniques based on this method and on spectral expansions we apply them to the linear stability analysis of thermal convection in a cylinder

model assisted inference for treatment effects using regularized calibrated estimation with high dimensional data consider the problem of estimating average treatment effects when a large number of covariates are used to adjust for possible confounding through outcome regression and propensity score models we develop new methods and theory to obtain not only doubly robust point estimators for average treatment effects which remain consistent if either the propensity score model or the outcome regression model is correctly specified but also model assisted confidence intervals which are valid when the propensity score model is correctly specified but the outcome model may be misspecified with a linear outcome model the confidence intervals are doubly robust that is being also valid when the outcome model is correctly specified but the propensity score model may be misspecified our methods involve regularized calibrated estimators with lasso penalties but carefully chosen loss functions for fitting propensity score and outcome regression models we provide high dimensional analysis to establish the desired properties of our methods under comparable sparsity conditions to previous results which give valid confidence intervals when both the propensity score and outcome models are correctly specified we present simulation studies and an empirical application which demonstrate advantages of the proposed methods compared with related methods based on regularized maximum likelihood estimation the methods are implemented in the r package mathtt rcal

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

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

boundary conditions in chebyshev and legendre methods we discuss two different ways of treating non dirichlet boundary conditions in chebyshev and legendre collocation methods for second order differential problems an error analysis is provided the effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights

konvergenzaussagen fur kollokationsverfahren bei elliptischen randwertaufgaben we examine the convergence of collocation with polynomials for elliptic problems the general theory is imbedded in thel p theory from which known results of existence and regularity are adopted convergence results are deduced in the cases of periodic dirichlet and mixed boundary conditions it becomes obvious that the order of convergence essentially depends on the smoothness of the solution