a comparison of two model averaging techniques with an application to growth empirics parameter estimation under model uncertainty is a difficult and fundamental issue in econometrics this paper compares the performance of various model averaging techniques in particular it contrasts bayesian model averaging bma currently one of the standard methods used in growth empirics with a new method called weighted average least squares wals the new method has two major advantages over bma its computational burden is trivial and it is based on a transparent definition of prior ignorance the theory is applied to and sheds new light on growth empirics where a high degree of model uncertainty is typically present

a two stage fourth order time accurate discretization for lax wendroff type flow solvers ii high order numerical boundary conditions abstract this paper serves to treat boundary conditions numerically with high order accuracy in order to suit the two stage fourth order finite volume schemes for hyperbolic problems developed in j li and z du 2016 17 as such it is significant when capturing small scale structures near physical boundaries different from previous contributions in literature the current approach constructs a fourth order accurate approximation to boundary conditions by only using the jacobian matrix of the flux function characteristic information instead of its successive differentiation of governing equations leading to tensors of high ranks in the inverse lax wendroff method technically data in several ghost cells are constructed with interpolation so that the interior scheme can be implemented over boundary cells and theoretical boundary condition has to be modified properly at intermediate stages so as to make the two stage scheme over boundary cells fully consistent with that over interior cells this is nonintuitive and highlights the fact that theoretical boundary conditions are only prescribed for continuous partial differential equations pdes while they must be approximated in a consistent way even though they could be exactly valued when the pdes are discretized several numerical examples are provided to illustrate the performance of the current approach when dealing with general boundary conditions

recent progress in unconstrained nonlinear optimization without derivatives we present an introduction to a new class of derivative free methods for unconstrained optimization we start by discussing the motivation for such methods and why they are in high demand by practitioners we then review the past developments in this field before introducing the features that characterize the newer algorithms in the context of a trust region framework we focus on techniques that ensure a suitable geometric quality of the considered models we then outline the class of algorithms based on these techniques as well as their respective merits we finally conclude the paper with a discussion of open questions and perspectives

approximate bayes factors and accounting for model uncertainty in generalised linear models summary ways of obtaining approximate bayes factors for generalised linear models are described based on the laplace method for integrals we propose a new approximation which uses only the output of standard computer programs for estimating generalised linear models this appears to be quite accurate a reference set of proper priors is suggested both to represent the situation where there is not much prior information and to assess the sensitivity of the results to the prior distribution the methods can be used when the dispersion parameter is unknown when there is overdispersion to compare link functions and to compare error distributions and variance functions the methods can be used to implement the bayesian approach to accounting for model uncertainty we describe an application to inference about relative risks in the presence of control factors where model uncertainty is large and important software to implement the methods is available at no cost from statlib

accuracy analysis of explicit runge kutta methods applied to the incompressible navier stokes equations this paper investigates the temporal accuracy of the velocity and pressure when explicit runge kutta methods are applied to the incompressible navier stokes equations it is shown that at least up to and including fourth order the velocity attains the classical order of accuracy without further constraints however in case of a time dependent gradient operator which can appear in case of time varying meshes additional order conditions need to be satisfied to ensure the correct order of accuracy furthermore the pressure is only first order accurate unless additional order conditions are satisfied two new methods that lead to a second order accurate pressure are proposed which are applicable to a certain class of three and four stage methods a special case appears when the boundary conditions for the continuity equation are independent of time since in that case the pressure can be computed to the same accuracy as the velocity field without additional cost relevant computations of decaying vortices and of an actuator disk in a time dependent inflow support the analysis and the proposed methods

on trust region methods for unconstrained minimization without derivatives we consider some algorithms for unconstrained minimization without derivatives that form linear or quadratic models by interpolation to values of the objective function then a new vector of variables is calculated by minimizing the current model within a trust region techniques are described for adjusting the trust region radius and for choosing positions of the interpolation points that maintain not only nonsingularity of the interpolation equations but also the adequacy of the model particular attention is given to quadratic models with diagonal second derivative matrices because numerical experiments show that they are often more efficient than full quadratic models for general objective functions finally some recent research on the updating of full quadratic models is described briefly using fewer interpolation equations than before the resultant freedom is taken up by minimizing the frobenius norm of the change to the second derivative matrix of the model a preliminary version of this method provides some very promising numerical results

on asymptotic normality of limiting density functions with bayesian implications walker 1969 presented a straightforward approach to asymptotic normality of posterior distributions based on independent and identically distributed i i d observations dawid 1970 extended this work by providing a weaker set of conditions to cover some distributions whose ranges depend on the parameters of interest more recently heyde and johnstone 1979 simplified walker s conditions for stochastic processes in this paper we introduce three basic normality conditions for limiting density functions in the bayesian context our approach is data determined and is nonprobabilistic in nature nevertheless the results thus obtained can be easily converted to probabilistic ones when compared with those given by other investigators it is shown that our conditions as a whole have a greater generality and flexibility than those previously assumed either by dawid or heyde and johnstone in particular for the i i d case the commonly assumed wald type uniform boundedness condition for tail behaviours is shown to be unnecessary for asymptotic normality section 3 3 the advantage of our approach is further exemplified by a simple proof of the asymptotic normality of distributions conjugate to the exponential family

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

global convergence of general derivative free trust region algorithms to first and second order critical points in this paper we prove global convergence for first and second order stationary points of a class of derivative free trust region methods for unconstrained optimization these methods are based on the sequential minimization of quadratic or linear models built from evaluating the objective function at sample sets the derivative free models are required to satisfy taylor type bounds but apart from that the analysis is independent of the sampling techniques a number of new issues are addressed including global convergence when acceptance of iterates is based on simple decrease of the objective function trust region radius maintenance at the criticality step and global convergence for second order critical points

higher order temporal integration for the incompressible navier stokes equations in bounded domains author s minion ml saye ri abstract 2018 elsevier inc this paper compares and contrasts higher order semi implicit temporal integration strategies for the incompressible navier stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations particular focus is placed on the imposition of boundary conditions in the semi implicit formulation the accurate treatment of the pressure term and the smoothness of the numerical solution for different formulations the main result presented here is the formulation and numerical validation of a single step semi implicit method called spectral deferred pressure corrections which can in theory obtain arbitrary formal order of accuracy in time numerical results demonstrate up to eighth order accuracy scenarios where optimal temporal accuracy is attained and scenarios where order reduction is observed due to time dependent boundary conditions

sobolev seminorm of quadratic functions with applications to derivative free optimization this paper studies the h 1 sobolev seminorm of quadratic functions the research is motivated by the least norm interpolation that is widely used in derivative free optimization we express the h 1 seminorm of a quadratic function explicitly in terms of the hessian and the gradient when the underlying domain is a ball the seminorm gives new insights into least norm interpolation it clarifies the analytical and geometrical meaning of the objective function in least norm interpolation we employ the seminorm to study the extended symmetric broyden update proposed by powell numerical results show that the new thoery helps improve the performance of the update apart from the theoretical results we propose a new method of comparing derivative free solvers which is more convincing than merely counting the numbers of function evaluations

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

stability and convergence of staggered runge kutta schemes for semilinear wave equations a staggered runge kutta staggered rk scheme is a runge kutta type scheme using a time staggered grid as proposed by ghrist et al in 2000 6 afterwards verwer in two papers investigated the efficiency of a scheme proposed by ghrist et al 6 for linear wave equations we study stability and convergence properties of this scheme for semilinear wave equations in particular we prove convergence of a fully discrete scheme obtained by applying the staggered rk scheme to the mol approximation of the equation

runge kutta methods without order reduction for linear initial boundary value problems it is well known the loss of accuracy when a runge kutta method is used together with the method of lines for the full discretization of an initial boundary value problem we show that this phenomenon called order reduction is caused by wrong boundary values in intermediate stages with a right choice the order reduction can be avoided and the optimal order of convergence in time is achieved we prove this fact for time discretizations of abstract initial boundary value problems based on implicit runge kutta methods moreover we apply these results to the full discretization of parabolic problems by means of galerkin finite element techniques we present some numerical examples in order to confirm that the optimal order is actually achieved

the correct formulation of intermediate boundary conditions for runge kutta time integration of initial boundary value problems a disturbing feature of applying explicit high order runge kutta rk time integrators to initial boundary value problems solved by the method of lines is the loss of accuracy that results from wrong specifications of intermediate stage boundary conditions the solution proposed here is to prescribe analytically those values that would result from applying the rk solver at the boundaries and hence maintain the order of the rk method without a reduction in the step size limit the procedure is described in detail for nonlinear hyperbolic equations and both scalar and vector examples are presented

on the harm that ignoring pretesting can cause in econometrics the same data set is typically used to select the model and to estimate the parameters in the selected model in applied econometrics practice however one typically acts as if the model had been given a priori thus ignoring the fact that the estimators are in fact pretest estimators hence one assumes incorrectly that the estimator is unbiased and that the reported variance conditional on the selected model is equal to its unconditional variance in this paper we nd the unconditional rst and second moments of the pretest estimator in fact of a more general estimator the wals estimator taking full account of the fact that model selection and estimation are an integrated procedure and show that the error in not reporting the correct moments can be large we also show that this error can vary substantially between di2erent model selection procedures finally we ask how the error increases when the number of auxiliary regressors increases

markov chain monte carlo methods for computing bayes factors a comparative review the problem of calculating posterior probabilities for a collection of competing models and associated bayes factors continues to be a formidable challenge for applied bayesian statisticians current approaches that take advantage of modern markov chain monte carlo computing methods include those that attempt to sample over some form of the joint space created by the model indicators and the parameters for each model others that sample over the model space alone and still others that attempt to estimate the marginal likelihood of each model directly because the collection of these is equivalent to the collection of model probabilities themselves we review several methods and compare them in the context of three examples a simple regression example a more challenging hierarchical longitudinal model and a binary data latent variable model we find that the joint model parameter space search methods perform adequately but can be difficult to program and tune whereas the marginal likelihood methods o

regression with imputed covariates a generalized missing indicator approach a common problem in applied regression analysis is that covariate values may be missing for some observations but imputed values may be available this situation generates a trade off between bias and precision the complete cases are often disarmingly few but replacing the missing observations with the imputed values to gain precision may lead to bias in this paper we formalize this trade off by showing that one can augment the regression model with a set of auxiliary variables so as to obtain under weak assumptions about the imputations the same unbiased estimator of the parameters of interest as complete case analysis given this augmented model the bias precision trade off may then be tackled by either model reduction procedures or model averaging methods we illustrate our approach by considering the problem of estimating the relation between income and the body mass index bmi using survey data affected by item non response where the missing values on the main covariates are filled in by imputations

the risk inflation criterion for multiple regression a new criterion is proposed for the evaluation of variable selection procedures in multiple regression this criterion which we call the risk inflation is based on an adjustment to the risk essentially the risk inflation is the maximum increase in risk due to selecting rather than knowing the correct predictors a new variable selection procedure is obtained which in the case of orthogonal predictors substantially improves on aic c p and bic and is close to optimal in contrast to aic c p and bic which use dimensionality penalties of 2 2 and log n respectively this new procedure uses a penalty 2 log p where p is the number of available predictors for the case of nonorthogonal predictors bounds for the optimal penalty are obtained

derivative free restrictively preconditioned conjugate gradient path method without line search technique for solving linear equality constrained optimization abstract in this paper we propose a derivative free restrictively preconditioned conjugate gradient path method without line search technique for solving linear equality constrained optimization this method originates from the classical conjugate gradient method and its restrictively preconditioned variant the preconditioned conjugate gradient path is obtained by solving extended system of the preconditioned equation with restrictively preconditioned conjugate gradient method which is a more robust and effective technique to solve large sparse linear system the global convergence and local superlinear convergence rate of the proposed method are established under some reasonable conditions finally the numerical results are reported to show the effectiveness of the proposed method

least frobenius norm updating of quadratic models that satisfy interpolation conditions quadratic models of objective functions are highly useful in many optimization algorithms they are updated regularly to include new information about the objective function such as the difference between two gradient vectors we consider the case however when each model interpolates some function values so an update is required when a new function value replaces an old one we let the number of interpolation conditions m say be such that there is freedom in each new quadratic model that is taken up by minimizing the frobenius norm of the second derivative matrix of the change to the model this variational problem is expressed as the solution of an m n 1 m n 1 system of linear equations where n is the number of variables of the objective function further the inverse of the matrix of the system provides the coefficients of quadratic lagrange functions of the current interpolation problem a method is presented for updating all these coefficients in m n 2 operations which allows the model to be updated too an extension to the method is also described that suppresses the constant terms of the lagrange functions these techniques have a useful stability property that is investigated in some numerical experiments

an interior affine scaling cubic regularization algorithm for derivative free optimization subject to bound constraints abstract in this paper we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function we first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method at each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique the global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions preliminary numerical results are reported to show the effectiveness of the proposed algorithm

third order implicit explicit runge kutta local discontinuous galerkin methods with suitable boundary treatment for convection diffusion problems with dirichlet boundary conditions abstract to avoid the order reduction when third order implicit explicit runge kutta time discretization is used together with the local discontinuous galerkin ldg spatial discretization for solving convection diffusion problems with time dependent dirichlet boundary conditions we propose a strategy of boundary treatment at each intermediate stage in this paper the proposed strategy can achieve optimal order of accuracy by numerical verification also by suitably setting numerical flux on the boundary in the ldg methods and by establishing an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient and the given boundary conditions we build up the unconditional stability of the corresponding scheme in the sense that the time step is only required to be upper bounded by a suitable positive constant which is independent of the mesh size

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

topology optimization of a reinforced wing box for enhanced roll maneuvers in this paper the aileron reversal speed of a wing torsion box is maximized by reinforcing its upper skin we propose a topology optimization approach using the spectral level set methodology this is based on the level set methods which represent theinterface describing thereinforcements asthe zerolevel setofafunction accordingto the proposed methodology this function is discretized using its fourier coefficients these coefficients become the design variables of the optimization problem assigned to define the reinforcing layout results show that approximately the sameoptimal reinforcement layout isobtained from topologically different initialconfigurations the two underlying ideas of the present work are to propose the spectral level set methodology as a framework for topology optimization given its potential for reducing the design space dimension and considering the possibilities of new custom made materials to readdress the reinforcement perspective in aircraft structures without the weight penalty nomenclature ai bi problem domain boundaries corresponding to componenti ofx at t smoothness parameter av v smoothness parameter a0 lift curve slope bt minimum thickness c positive constant cl derivative of the rolling moment coefficient of the rigid airplane with respect to c y n twist aty produced by a unit moment at

a derivative free nonmonotone line search technique for unconstrained optimization a tolerant derivative free nonmonotone line search technique is proposed and analyzed several consecutive increases in the objective function and also nondescent directions are admitted for unconstrained minimization to exemplify the power of this new line search we describe a direct search algorithm in which the directions are chosen randomly the convergence properties of this random method rely exclusively on the line search technique we present numerical experiments to illustrate the advantages of using a derivative free nonmonotone globalization strategy with approximated gradient type methods and also with the inverse sr1 update that could produce nondescent directions in all cases we use a local variation finite differences approximation to the gradient

a derivative free algorithm for least squares minimization we develop a framework for a class of derivative free algorithms for the least squares minimization problem these algorithms are designed to take advantage of the problem structure by building polynomial interpolation models for each function in the least squares minimization under suitable conditions global convergence of the algorithm is established within a trust region framework promising numerical results indicate the algorithm is both efficient and robust numerical comparisons are made with standard derivative free software packages that do not exploit the special structure of the least squares problem or that use finite differences to approximate the gradients

uobyqa unconstrained optimization by quadratic approximation uobyqa is a new algorithm for general unconstrained optimization calculations that takes account of the curvature of the objective function f say by forming quadratic models by interpolation therefore because no first derivatives are required each model is defined by n 1 n 2 values of f where n is the number of variables and the interpolation points must have the property that no nonzero quadratic polynomial vanishes at all of them a typical iteration of the algorithm generates a new vector of variables widetilde underline x t say either by minimizing the quadratic model subject to a trust region bound or by a procedure that should improve the accuracy of the model then usually f widetilde underline x t is obtained and one of the interpolation points is replaced by widetilde underline x t therefore the paper addresses the initial positions of the interpolation points the adjustment of trust region radii the calculation of widetilde underline x t in the two cases that have been mentioned and the selection of the point to be replaced further uobyqa works with the lagrange functions of the interpolation equations explicitly so their coefficients are updated when an interpolation point is moved the lagrange functions assist the procedure that improves the model and also they provide an estimate of the error of the quadratic approximation to f which allows the algorithm to achieve a fast rate of convergence these features are discussed and a summary of the algorithm is given finally a fortran implementation of uobyqa is applied to several choices of f in order to investigate accuracy robustness in the presence of rounding errors the effects of first derivative discontinuities and the amount of work the numerical results are very promising for n 20 but larger values are problematical because the routine work of an iteration is of fourth order in the number of variables

order reduction and how to avoid it when explicit runge kutta nystrom methods are used to solve linear partial differential equations in this paper we study the order reduction which turns up when explicit runge kutta nystrom methods are used to discretize linear second order hyperbolic equations by means of the method of lines the order observed in practice including its fractional part is obtained it is also proved that the order reduction can be completely avoided taking the boundary values of the intermediate stages of the time semidiscretization the numerical experiments confirm that the optimal order can be reached

assessment and propagation of model uncertainty in most examples of inference and prediction the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model m that formalizes assumptions about how x and y are related m will typically have two parts structural assumptions s such as the form of the link function and the choice of error distribution in a generalized linear model and parameters whose meaning is specific to a given choice of s it is common in statistical theory and practice to acknowledge parametric uncertainty about given a particular assumed structure s it is less common to acknowledge structural uncertainty about s itself a widely used approach involves enlisting the aid of x to specify a plausible single best choice s for s and then proceeding as if s were known to be correct in general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x when miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty in this paper i discuss a bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible by virtue of recent computational advances and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the us space shuttle

consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models generalized linear models are used for regression analysis in a number of cases including categorical responses where the classical assumptions are violated the statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator we present mild general conditions which respectively assure weak or strong consistency or asymptotic normality most of the previous work has been concerned with natural link functions in this case our normality condition though obtained by a different approach is closely related to a condition of haberman 1977a examples show how the general conditions reduce to weak requirements for special exponential families further for regressors with a compact range sufficient conditions are given which do not involve the unknown parameter and are therefore easy to check in practice responses with a bounded range e g categorical responses and stochastic regressors also are treated

model averaging estimation of generalized linear models with imputed covariates a b s t r a c t we address the problem of estimating generalized linear models when some covariate values are missing but imputations are available to fill in the missing values this situation generates a bias precision trade off in the estimation of the model parameters extending the generalized missing indicator method proposed by dardanoni et al 2011 for linear regression we handle this trade off as a problem of model uncertainty using bayesian averaging of classical maximum likelihood estimators baml we also propose a block model averaging strategy that incorporates information on the missing data patterns and is computationally simple an empirical application illustrates our approach

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

cgrs an advanced hybrid method for global optimization of continuous functions closely coupling extended random search and conjugate gradient method abstract a new hybrid method for global optimization of continuous functions is proposed it is a combination of an extended random search method and a descent method random search is used as the global search strategy a newly developed distribution based region control makes use of already detected local minima to refine this search strategy the approach resembles classical step size control in deterministic optimization the descent method is embedded as a local search strategy for the detection of local minima a special realization of this approach is presented in this paper and called cgrs in cgrs the conjugate gradient method is utilized as descent method the proof of global convergence in probability for cgrs is given and extended to other descent methods used in the hybrid optimization approach in order to demonstrate the numerical properties of the approach test sets of multidimensional non convex optimization problems are solved the results are compared to well established hybrid methods for global optimization the new algorithm shows a high success rate with good and adjustable solution precision parameter tuning is not necessary but of course possible the new method proves to be efficient in terms of computational costs

model selection and accounting for model uncertainty in graphical models using occam s window abstract we consider the problem of model selection and accounting for model uncertainty in high dimensional contingency tables motivated by expert system applications the approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic p values leading to the selection of a single model inference is then conditional on the selected model the sampling properties of such a strategy are complex and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest in principle a panacea is provided by the standard bayesian formalism that averages the posterior distributions of the quantity of interest under each of the models weighted by their posterior model probabilities furthermore this approach is optimal in the sense of maximizing predictive ability but this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large often greater than 1

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

an fc based spectral solver for elastodynamic problems in general three dimensional domains this paper presents a spectral numerical algorithm for the solution of elastodynamics problems in general three dimensional domains based on a recently introduced fourier continuation fc methodology for accurate fourier expansion of non periodic functions the proposed approach possesses a number of appealing properties it yields results that are essentially free of dispersion errors it entails mild cfl constraints it runs at a cost that scales linearly with the discretization sizes and it lends itself easily to efficient parallelization in distributed memory computing clusters the proposed algorithm is demonstrated in this paper by means of a number of applications to problems of isotropic elastodynamics that arise in the fields of materials science and seismology these examples suggest that the new approach can yield solutions within a prescribed error tolerance by means of significantly smaller discretizations and shorter computing times than those required by other methods

an explicit fourth order staggered finite difference time domain method for maxwell s equations a new explicit fourth order accurate staggered finite difference time domain fdtd scheme is proposed and applied to electromagnetic wave problems it is fourth order accurate in both space and time conditionally stable and highly efficient with respect to yee s scheme and still retains much of the original simplicity of yee s scheme both extension to perfectly matched layers and modification to deal with dielectric interfaces and perfectly conducting boundaries of the scheme have also been presented numerical examples are shown to illustrate the efficiency of the method