the eigenvalues of second order spectral differentiation matrices the eigenvalues of the pseudospectral second derivative matrix with homogeneous dirichlet boundary conditions are important in many applications of spectral methods this paper investigates some of their properties numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately but the errors in the remaining ones are large it is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid in particular it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a chebyshev distribution of nodes and six points per wavelength for a uniform distribution these results are in agreement with the observed fractions of accurate eigenvalues by using the characteristic polynomial a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2 or better the effect of filtering on the eigenvalues is studied numerically

heat fluctuation distribution from non equilibrium states a non equilibrium state is described by parameters alpha j aj which are ensemble averages of dynamical phase functions aj gamma by using a projection operator technique of zwanzig 1961 one can derive from the classical liouville equation an integro differential equation for g vj the amplitude for the probability that the aj have respective values vj these previously derived results are applied to the heat flux aj in a fluid and an approximate differential equation derived for g v j where j aj the first moment of this equation yields the catteneo vernotte equation 1958 with a set of auxiliary assumptions which are also employed in solving the equation for g the author obtains g g 0 1 g v j where g 0 kappa pi mu 3 2 exp mu v j 2 kappa is the einstein distribution and g is a sum of products of scalar combinations of j v j and del t and laguerre polynomials ln 12 z2 where z identical to mu kappa 1 2 v j the einstein distribution g 0 gives correctly nu 2 to o j2 when del t 0 but not in the presence of a temperature gradient

a pseudospectral chebychev method for the 2d wave equation with domain stretching and absorbing boundary conditions in this paper we develop a method for the simulation of wave propagation on artificially bounded domains the acoustic wave equation is solved at all points away from the boundaries by a pseudospectral chebychev method absorption at the boundaries is obtained by applying one way wave equations at the boundaries without the use of damping layers the theoretical reflection coefficient for the method is compared to theoretical estimates of reflection coefficients for a fourier model of the problem these estimates are confirmed by numerical results modification of the method by a transformation of the grid to allow for better resolution at the center of the grid reduces the maximum eigenvalues of the differential operator consequently for stability the maximum timestep iso 1 n as compared too 1 n2 for the standard chebychev method therefore the chebychev method can be implemented with efficiency comparable to that of the fourier method moreover numerical results presented demonstrate the superior performance of the new method

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

generalization of thermodynamic potentials including information in this work we analyze the connection between information theory and thermodynamics three different potentials are presented all related to the kullback leibler divergence these potentials represent three different conditions of energy and information exchange between a reservoir and a subsystem the first potential describes a situation where the two systems are exchanging energy without any external control the second potential considers the situation where the reservoir can get information through the state of another secondary system about the internal state of the subsystem this information can then be used by a maxwell demon to increase the free energy of the subsystem in this second scenario it is assumed that the energy cost of the information probe and the maxwell demon are at the expense of the reservoir finally a third case is considered where the information obtained through the secondary system is used not only to manipulate an internal maxwell demon but to perform directly work over the subsystem this last case involves a direct modification of the hamiltonian of the subsystem

spectral methods based on prolate spheroidal wave functions for hyperbolic pdes we examine the merits of using prolate spheroidal wave functions pswfs as basis functions when solving hyperbolic pdes using pseudospectral methods the relevant approximation theory is reviewed and some new approximation results in sobolev spaces are established an optimal choice of the band limit parameter for pswfs is derived for single mode functions our conclusion is that one might gain from using the pswfs over the traditional chebyshev or legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

adaptive pseudospectral solution of a diffuse interface model a diffuse interface type model using an energy based variational formulation with a free energy that is a function of the density and its gradients is presented all of the boundary terms are retained and related to external surface forces which can be of particular interest when considering the fluid fluid solid region the numerical solution of these types of problems can be troublesome if a thin transition layer is desired here chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied a mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived in order to achieve high precision for problems exhibiting thin transition layers a modified version of the arc length monitor function is proposed which yields a sufficiently smooth coordinate transformation at every step of the continuation process a fixed number of iterations is implemented so that the equidistribution equations are not solved completely at each step which saves a considerable amount of computational effort numerical results for the static phase field model exhibiting thin transition layers are presented

multiplicity adjustments in testing for bioequivalence bioequivalence of two drugs is usually demonstrated by rejecting two one sided null hypotheses using the two one sided tests for pharmacokinetic parameters area under the concentration time curve auc and maximum concentration cmax by virtue of the intersection union test there is no need for multiplicity adjustment in testing the two one sided null hypotheses within each parameter however the decision rule for bioequivalence often requires equivalence to be achieved simultaneously on both parameters that contain four one sided null hypotheses together without adjusting for multiplicity the family wise error rate fwer could fail to be controlled at the nominal type i error rate the multiplicity issue for bioequivalence in this regard is scarcely discussed in the literature to address this issue we propose two approaches including a closed test procedure that controls fwer for the simultaneous auc and cmax bioequivalence and requires no adjustment of the type i error and an alpha adaptive sequential testing aast that controls fwer by pre specifying the significance level on auc 1 and obtaining it for cmax 2 adaptively after testing of auc while both methods control fwer the closed test requires testing of eight intersection null hypotheses each at and aast is at times accomplished through a slight deduction in 1 and no deduction in 2 relative to the latter considers equivalence reached in auc a higher importance than that in cmax illustrated with published data the two approaches although operate differently can lead to the same substantive conclusion and are better than a traditional method like bonferroni adjustment copyright 2014 john wiley sons ltd

on a non equilibrium partition function for heat conduction the authors provide a physical interpretation for the lagrange multipliers associated with heat conduction in an information theoretical description of non equilibrium systems such an identification is based on the generalised gibbs equation of extended irreversible thermodynamics

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

non linear heat conduction in dense fluids in a dense fluid with very large heat flow j the thermal conductivity lambda lambda 0 lambda 2j2 an estimate of lambda 2 lambda 0 is made for hard spheres with mass and diameter appropriate to ar and density just below the gas solid transition this estimate indicates that fourier s law will hold in the form appropriate to low j if j or approximately 109 w m 2 and that the j dependence of pressure and internal energy is of the same order as that of lambda the calculation utilises an earlier derivation from the classical liouville equation which expresses lambda at high density in terms of the thermodynamic pressure the latter can be calculated with the aid of reciprocity and integrability conditions of extended non equilibrium thermodynamics which treats j as a state variable

information theoretic extended entropy for steady heat condition a phase space distribution for a far from equilibrium steady state in which thermodynamic temperature t and heat flux j are specified is calculated by maximising the information theoretic entropy when this distribution is used to calculate free energy f the latter is obtained as a function of the distribution moduli which in turn are calculated from a consistency condition as functions of j and t the coefficient of the o j2 term in f can be expressed in terms of the radial distribution function g2 and three particle distribution g3 a numerical estimate is made for a hard sphere model of liquid ar at 87 k and high density using the percus yevick solution for g2 and the kirkwood superposition for g3 the estimate confirms the conclusion of an earlier theory that non linear effects at liquid density are negligible except possibly for unrealistically large j fluctuation theory is invoked to make an estimate of the negligibly small o j4 term in f

the shallow water equations in lagrangian coordinates recent advances in the collection of lagrangian data from the ocean and results about the well posedness of the primitive equations have led to a renewed interest in solving flow equations in lagrangian coordinates we do not take the view that solving in lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted rather the grid in lagrangian coordinates represents the initial position of particles and it does not change with time we apply numerical methods traditionally used to solve differential equations in eulerian coordinates to solve the shallow water equations in lagrangian coordinates the difficulty with solving in lagrangian coordinates is that the transformation from eulerian coordinates results in solving a highly nonlinear partial differential equation the non linearity is mainly due to the jacobian of the coordinate transformation which is a precise record of how the particles are rotated and stretched the inverse jacobian must be calculated thus lagrangian coordinates cannot be used in instances where the jacobian vanishes for linear spatial flows we give an explicit formula for the jacobian and describe the two situations where the lagrangian shallow water equations cannot be used because either the jacobian vanishes or the shallow water assumption is violated we also prove that linear in space steady state solutions of the lagrangian shallow water equations have jacobian equal to one in the situations where the shallow water equations can be solved in lagrangian coordinates accurate numerical solutions are found with finite differences the chebyshev pseudospectral method and the fourth order runge kutta method the numerical results shown here emphasize the need for high order temporal approximations for long time integrations

new quadrature formulas from conformal maps gauss and clenshaw curtis quadrature like legendre and chebyshev spectral methods make use of grids strongly clustered at boundaries from the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal nevertheless such methods may waste a factor of pi 2 with respect to each space dimension we propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight sided domain the new methods are compared with related ideas of bakhvalov kosloff and tal ezer rokhlin and alpert and others an advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands for example one of the formulas presented is proved to converge 50 faster than gauss quadrature for functions analytic in an varepsilon neighborhood of 1 1

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

generation of pseudospectral differentiation matrices i we present a simple method for computing n times n pseudospectral differentiation matrices of order p in cal o pn 2 operations for the case of quasi polynomial approximation the algorithm is based on fornberg s finite difference algorithm and is numerically stable a fortran implementation is included a necessary and sufficient condition for d p d p 1 is also given

regular article spectral collocation time domain modeling of diffractive optical elements a spectral collocation multi domain scheme is developed for the accurate and efficient time domain solution of maxwell s equations within multi layered diffractive optical elements special attention is being paid to the modeling of out of plane waveguide couplers emphasis is given to the proper construction of high order schemes with the ability to handle very general problems of considerable geometric and material complexity central questions regarding efficient absorbing boundary conditions and time stepping issues are also addressed the efficacy of the overall scheme for the time domain modeling of electrically large and computationally challenging problems is illustrated by solving a number of plane as well as non plane waveguide problems

optimal blended spectral element operators for acoustic wave modeling spectral element methods based on high order polynomials are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media however to retain computational efficiency very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided in the present work we devise an approach for reducing such errors by considering modified discrete wave operators we analyze consistent and lumped operators together with blended operators weighted averages of consistent and lumped operators furthermore using the operator blending approach and a novel dispersion analysis method we develop optimal spectral element operators that have increased numerical accuracy without resorting to very high order operators the new operators are faster and computationally more efficient than consistent operators our approach is based on the tensor product decomposition of the element matrices into 1d f

applications of reciprocity to non linear extended thermodynamics kinetic equations abstract reciprocity relations and integrability conditions can be used in non linear extended irreversible thermodynamics to calculate forces and phenomenological coefficients uniquely in terms of parameters in kinetic equations derived from a model one assumes a set of n even variables i and corresponding i with an equation from the model for i independent thermodynamic forces terms are assumed known comparison of powers of in the model equation with corresponding powers in the canonical onsager casimir formalism yields equations to determine all quantities in the latter save the higher dependence of entropy which is affected by uncertainties in measurement by introducing a phenomenological ansatz one can relate these higher entropy terms to corresponding terms in the model equation a numerical illustration is made for a relaxing scalar structural parameter appropriate to a simple liquid

generation of higher order pseudospectral integration matrices a new explicit expression of the higher order pseudospectral integration matrices is presented using an explicit formula for computing iterated integrals of chebyshev polynomials applications to initial value problems boundary value problems linear integral and integro differential equations are presented the present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods

stability of a chebychev pseudospectral solution of the wave equation with absorbing boundaries abstract stability of the pseudospectral chebychev collocation solution of the two dimensional acoustic wave problem with absorbing boundary conditions is investigated the continuous one dimensional problem with one absorbing boundary and one dirichlet boundary has previously been shown to be far from normal consequently the spectrum of that problem says little about the stability behavior of the solution our analysis proves that the discrete formulation with dirichlet boundaries at all boundaries is near normal and hence the formulation with absorbing boundaries at all boundaries either for one dimensional or two dimensional wave propagation is not far from normal the near normality follows from the near normality of the second order derivative pseudospectral differential operator further the nearness to normality is independent of the boundary discretization stability limits on the timestep are however dependent on the boundary operator with an explicit euler method having the most restrictive condition the crank nicolson implementation has a stability limit the same as the dirichlet formulation furthermore in this case the restriction scales by 1 2 in moving from one dimension to two dimensions exactly as in the central finite difference approximation numerical results confirm the predicted values on allowable timesteps obtained from a spectral analysis for both chebychev and modified chebychev implementations we conclude that the spectrum of the evolution operator is informative for predicting the behavior of the numerical solution

nonlinear reciprocity and the maximum entropy formalism abstract the maximum entropy formalism calculates a phase space distribution p x which maximizes the information theoretic entropy subject to specified values of internal energy and state variables i and i i 1 n the latter are averages calculated from p of dynamical functions ai and ai ilai respectively where l is the liouville operator from p one extracts a fluctuation distribution g a v for the values of the a and a a kinetic equation for the fluctuation distribution has been derived by grabert using projection operators appropriate to a system in thermal contact with its surroundings and g is proposed as an ansatz for its solution from the first moments of the kinetic equation after introduction therein of g one extracts phenomenological equations for ai and hi whose terms can be grouped to exhibit onsager symmetry even when nonlinear terms are included the groupings can be affected exactly but not uniquely

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach