differentiable interpolants for high order runge kutta methods for a particular family of pairs of explicit runge kutta methods of orders p 1 and p sets of efficient continuously differentiable interpolants of several orders up to p are characters zed algorithmically in terms of several arbitrary parameters the approach can be applied in an obvious way to yield interpolants for other types of explicit runge kutta methods as well as families of other types of explicit and implicit methods derivative evaluations required for each pair of methods are reused and additional derivative evaluations are selected in an attempt to minimize the total number of stages required the analysis provides a lower bound on the number of stages required and indicates for example why twelve stages are required to provide interpolants for eight stage pairs of orders 5 and 6 in contrast to the eighteen stages used to obtain a known interpolant of order 7 for a pair of methods of orders 6 and 7 only sixteen stages are required by the proposed derivation details for interpola

a nonstandard euler scheme for y g y y f y y 0 we introduce a nonstandard euler scheme for solving the differential equation y g y y f y y 0 which has the same linear stability properties as the differential equation and is conservative when g 0 the method is based on a physically motivated reduction of the equation to a system of two first order equations and the use of lie group integrators the method is demonstrated on a few examples and compared to a standard matlab adaptive solver

a simple step size selection algorithm for ode codes a very simple way of selecting the step size when solving an initial problem for a system of ordinary differential equations odes is examined it is shown to control the true error and the stability of the integration in a gross measure it is shown to be as efficient as control of the local error numerical results illustrate the analysis and explore the robustness of the approach

error analysis of a specialized numerical method for mathematical models from neuroscience abstract the exponential euler method is a nonstandard approximation scheme that was developed specifically for the hodgkin huxley differential equation models that arise in neuroscience and was one of the discretization schemes used in the neural systems package called genesis in this article we show the scheme is first order accurate develop a second order accurate extension and suggest ways the method can be used to compute approximations to certain time dependent partial differential equations we also apply the scheme to an integro differential equation model of neuronal activity and furnish sample computations that illustrate our theoretical results

variable stepsize implementation of multistep methods for y f x y y the falkner method is a multistep scheme intended for the numerical solution of second order initial value problems where the first derivative does appear explicitly in this paper we develop a procedure to obtain k step falkner methods explicit and implicit in their variable step size versions providing recurrence formulas to compute the coefficients efficiently considering a pair of explicit and implicit formulae these may be implemented in predictor corrector mode

linearized θ methods part ii reaction diffusion equations abstract second order accurate in space partially linearized triangular and diagonal methods for reaction diffusion equations which employ either a standard or a delta formulation are developed and applied to both the study of a system of one dimensional reaction diffusion equations with algebraic nonlinear reaction terms and the propagation of a one dimensional confined laminar flame these methods require the solution of tridiagonal matrices for each dependent variable and either uncouple or sequentially couple the dependent variables at each time step depending on whether they are diagonally or triangularly linearized techniques respectively partially linearized diagonal methods yield larger errors than partially linearized triangular techniques and the accuracy of the latter depends on the time step standard or delta formulation implicitness parameter and the order in which the equations are solved fully and partially linearized operator splitting methods for reaction diffusion equations are also developed the latter provide explicit expressions for the solution of the reaction operator

piecewise homotopy methods for nonlinear ordinary differential equations abstract piecewise homotopy perturbation methods are developed for the solution of nonlinear ordinary differential equations these methods are based on the introduction of an artificial or book keeping parameter and the expansion of the solution in a power series of this parameter and provide analytical solutions in open intervals which are smooth everywhere three piecewise adaptive homotopy perturbation methods based on the use of either a fixed number of approximants and a variable step size a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size are presented and applied to eight nonlinear ordinary differential equations it is shown that piecewise adaptive homotopy perturbation methods predict essentially the same solutions as matlab s variable step variable order solvers and variable order transition matrix techniques provided that five term approximations of the decomposition method are applied to both the displacement and the velocity it is also shown that piecewise homotopy perturbation techniques that use three term approximations to both the displacement and the velocity provide essentially the same results as those obtained with a second order accurate time linearization technique when the same step is employed in both schemes

piecewise linearized methods for initial value problems piecewise linearized methods for the solution of initial value problems in ordinary differential equations are developed by approximating the right hand sides of the equations by means of a taylor polynomial of degree one the resulting approximation can be integrated analytically to obtain the solution in each interval and yields the exact solution for linear problems three adaptive methods based on the norm of the jacobian matrix maintaining constant the value of the approximation errors incurred by the linearization of the right hand sides of the ordinary differential equations and richardson s extrapolation are developed numerical experiments with some nonstiff first and second order ordinary differential equations indicate that the accuracy of piecewise linearized methods is in general superior to those of the explicit modified second order accurate euler method and the implicit trapezoidal rule but lower than that of the explicit fourth order accurate runge kutta technique it is also shown that piecewise linearized methods do not exhibit computational i e spurious modes for the relaxation oscillations of the van der pol oscillator and for those systems of equations which satisfy certain conservation principles conserve more accurately the invariants than the trapezoidal rule an error bound for piecewise linearized methods is provided for ordinary differential equations whose right hand sides satisfy certain lipschitz conditions

nonstandard finite difference equations for odes and 1 d pdes based on piecewise linearization a method for the solution of initial and boundary value problems in nonlinear ordinary differential equations and for one dimensional partial differential equations which provides c 1 solutions is presented the method is based on the linearization of the differential equation in intervals which contain only two grid points and provides three point nonstandard finite difference equations for the nodal amplitudes the method is applied to steady reaction diffusion equations two point singularly perturbed boundary value problems and the steady burgers equation and compared with standard finite difference and finite element formulations for one dimensional partial differential equations the temporal derivatives are first discretized and the resulting ordinary differential equation accounts for both the temporal and spatial stiffnesses and is solved by means of piecewise linearization since the linearization includes a jacobian matrix it may be easily employed to refine the mesh where steep gradients occur

linearized methods for ordinary differential equations the conservation properties singularities and nonlinear dynamics of both time linearized methods which provide piecewise analytical solutions and keep the independent variable continuous and implicit linearized techniques which are based on discretization and linearization are analyzed in this paper it is shown that both methods are implicit and provide explicit maps but they do not preserve the energy in conservative systems it is also shown that time linearized methods preserve both the fixed points and the linear stability of the original ordinary differential equation whereas linearized techniques do preserve the fixed points and the linear stability of attractors but the stability of the repellers depends on the time step and the implicitness parameter the results clearly indicate that the linearized techniques which more faithfully reproduce the nonlinear dynamics of the original ordinary differential equation are second order accurate in time

finite analytic numerical method for unsteady two dimensional navier stokes equations abstract the main purpose of this paper is to develop a finite analytic fa numerical solution for unsteady two dimensional navier stokes equations the fa method utilizes the analytic solution in a small local element to formulate the algebraic representation of partial differential equations in this study the combination of linear and exponential functions that satisfy the governing equation is adopted as the boundary function thereby improving the accuracy of the finite analytic solution two flows one a starting cavity flow and the other a vortex shedding flow behind a rectangular block are solved by the fa method the starting square cavity flow is solved for reynolds numbers of 400 1000 and 2000 to show the accuracy and stability of the fa solution the fa solution for flow over a rectangular block h h 4 predicts the strouhal number for reynolds numbers of 100 and 500 to be 0 156 and 0 125 details of the flow patterns are given in addition to streamlines and vorticity distribution rest streamlines are given to illustrate the vortex motion downstream of the block

a linearly implicit predictor corrector method for reaction diffusion equations abstract a novel linearly implicit predictor corrector scheme is developed for the numerical solution of reaction diffusion equations iterative processes are avoided by treating the nonlinear reaction terms explicitly while maintaining superior accuracy and stability properties compared to the well known methods and linearly implicit runge kutta methods the proposed method allows the opportunity of solving large systems of reaction diffusion equations by alleviating the necessity of solving the accompanying large linear systems of algebraic equations due to the natural parallelism which surfaces across the system numerical results confirm the enhanced stability accuracy and efficiency of the method when applied to reaction diffusion equations arising in biochemistry and population ecology

linearization techniques for singular initial value problems of ordinary differential equations linearization methods for singular initial value problems in second order ordinary differential equations are presented these methods result in linear constant coefficients ordinary differential equations which can be integrated analytically thus yielding piecewise analytical solutions and globally smooth solutions the accuracy of these methods is assessed by comparisons with exact and asymptotic solutions of homogeneous and non homogeneous linear and nonlinear lane emden equations it is shown that linearization methods provide accurate solutions even near the singularity or the zeros of the solution in fact it is shown that linearization methods provide more accurate solutions than methods based on perturbation methods it is also shown that the accuracy of these techniques depends on the nonlinearity of the ordinary differential equations and may not be a monotonic function of the step size

on vergnaud s time integration method for autocatalytic cure rate equations a hybrid numerical analytical scheme for time integration of the cure rate equation governing the autocatalytic cure reaction of epoxy resins is presented the method is an extension of vergnaud s approach for the integration of heterocatalytic cure reactions of epoxy resins it is applicable to first order semi linear ordinary differential equations with separated variables where the integration of the dependent variable can be obtained analytically numerical quadrature is used for the integration of the independent variable which appears in a time dependent integral of an arrhenius term containing temperature consistency and l stability of the method are proved and some error estimates are provided several numerical results validate the high accuracy of the proposed integration scheme

application of the natural stress formulation for solving unsteady viscoelastic contraction flows abstract we present a numerical scheme for a previously unexploited formulation of the equations for unsteady viscoelastic flow the formulation aligns the polymer stress along particle paths streamlines utilising the characteristic curves associated with the hyperbolic part of the constitutive equations we illustrate the approach for the oldroyd b model in the benchmark 4 1 contraction for moderate elasticity numbers we show that the scheme is able to accurately capture the re entrant corner singularity for the polymer stresses and the pressure the latter variable being inaccurately determined by schemes using the traditional formulation in terms of cartesian polymer stresses a space step restriction for stability is derived which can be numerically limiting in certain recirculation regions this contrasts with the equivalent space step restriction for the formulation in cartesian stresses which is limiting in flow regions of high velocity gradients for example at sharp corners in contraction flows

a higher order local linearization method for solving ordinary differential equations the local linearization ll method for the integration of ordinary differential equations is an explicit one step method that has a number of suitable dynamical properties however a major drawback of the ll integrator is that its order of convergence is only two the present paper overcomes this limitation by introducing a new class of numerical integrators called the llt method that is based on the addition of a correction term to the ll approximation in this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the ll method in particular it is proved that the llt method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence the performance of the introduced method is further illustrated through computer simulations

high order hierarchical a and l stable integration methods numerical integration of stiff first order systems of differential equations is considered it is shown how a c continuous polynomial time discretization in conjunction with a weighted residual method can be used to derive methods corresponding to the diagonal and first subdiagonal pade approximants in this manner a stable schemes of order 2k and l stable schemes of order 2k l are obtained at least for k 4 the methods are hierarchical in the sense that a scheme with a given accuracy embraces the equations of all lower order methods that have the same stability type

time dependent algorithms for the simulation of viscoelastic flows with spectral element methods applications and stability this paper presents the development of spectral element methods to simulate unsteady flows of viscoelastic fluids using a closed form differential constitutive equation the generation and decay poiseuille planar flows are considered as benchmark problems to test the abilities of our computational method to deal with truly time dependent flows satisfactory results converging toward steady state regimes have been obtained for the flow through a four to one planar abrupt contraction with unsteady algorithms time dependent simulations of viscoelastic flows are prone to numerical instabilities even for simple geometrical configurations possible methods to improve the numerical stability of the computational algorithms are discussed in view of the results carried out with numerical simulations for the flows through a straight channel and the four to one contraction

linearized factorization techniques for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four approximate factorization techniques which do not account for the approximate factorization errors are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

shear induced chaos in nonlinear maxwell model fluids a generalized model for the behavior of the stress tensor in non newtonian fluids is investigated for spatially homogeneous plane couette flow showing a variety of nonlinear responses and deterministic chaos mapping of chaotic solutions is achieved through the largest lyapunov exponent for the two main parameters the shear rate and the temperature and or density bifurcation diagrams and stability analysis are used to reveal some of the rich dynamics that can be found suggested mechanisms for stability loss in these complex fluids include hopf saddle node and period doubling bifurcations

linearization techniques for singularly perturbed initial value problems of ordinary differential equations linearization methods for singularly perturbed initial value problems in ordinary differential equations based on taylor series expansions and piecewise analytical solutions are presented if only the linear terms are retained it is shown that exponentially fitted techniques are obtained whereas if quadratic terms are kept then riccati s equations result both linear and quadratic linearization methods are shown to be nonuniformly convergent in the small perturbation parameter and the former are more accurate than the latter in equally spaced grids linearization methods are also used with three different adaptive mesh refinement methods that account for the magnitude of the jacobian the variation of the solution and the resolution of the initial layer and it is shown that mesh refinement methods based on the variation of the solution can provide very accurate solutions although they may yield large errors at the outer edge of the initial layer if the grid spacing increases rapidly it is also shown that for values of the perturbation parameter on the order of unity quadratic linearization methods are more accurate than linearization methods and can account for the presence of internal layers it is also shown that linearization methods on grids whose spacing obeys a geometric progression are not robust and may suffer from a loss in the order of convergence compared with adaptive methods that resolve the boundary layer or adapt the grid according to the numerical solution

a bias correction method for indirect identification of closed loop systems a novel least squares like method is developed to deal with the bias problem in indirect identification of closed loop systems with coloured disturbances it is shown that combined with a bias correction procedure this method can give rise to consistent parameter estimates by fully and efficiently using the given information about the regulator simulation results are also included for illustration

numerical study for differential constitutive equations with polymer melts by using a hybrid finite element volume method this paper covers the numerical investigation of complex flow tube tooling cable coating using the single equation representation of the pom pom sxpp model here the hybrid scheme consists of a taylor galerkin finite element discretisation and a cell vertex fluctuation distribution finite volume approach is employed for this method the momentum continuity equations have been solved by a fractional staged taylor galerkin pressure correction procedure and invoke a cell vertex fluctuation distribution finite volume scheme for the constitutive law stress response and pressure drop across the flow are investigated with respect to variation in weissenberg number and solvent fraction investigation of complex flow tube tooling cable coating using the single equation representation of the pom pom sxpp model influence of variation in weissenberg number on stress response and pressure drop across the flow evaluate free surface movement by using the time dependent prediction method which is termed phan thien dh dt scheme

piecewise linearized methods for oscillators with fractional power nonlinearities a piecewise linearization method based on the taylor series expansion of nonlinear ordinary differential equation with respect to time displacement and velocity is developed for the study of one degree of freedom nonlinear oscillators with smooth and fractional power nonlinearities the method provides smooth solutions and explicit nonstandard finite difference expressions for the displacement and velocity and is exact for constant coefficients linear ordinary differential equations with linear time dependent forcings the method is applied to ten oscillators with fractional power nonlinearities and its results are compared with those of harmonic balance techniques ritz procedure and numerical solutions based on nonstandard finite difference methods in terms of displacement velocity energy and angular frequency when available it is shown that the linearization method presented here provides slightly more less accurate results than those of nonstandard methods for fractional powers greater smaller than one and both techniques are more accurate than those that freeze the nonlinearities at the previous time step or linearize the nonlinear terms with respect to only the displacement or the velocity for the examples and time steps considered in this paper the linearization method has been found to be more accurate than the harmonic balance and ritz procedures

a stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

linearization methods for reaction diffusion equations 1 d problems three types of linearized finite difference methods are presented the first type is based on a formulation which provides discrete solutions in both space and time it is shown that this type is a linearly implicit rosenbrock s or a w method depending on whether the full or an appropriate jacobian is employed the second one requires the solution of two point linear ordinary differential equations and provides either piecewise continuous or piecewise differentiable solutions in space and discrete in time the third type is based on the discretization of the spatial coordinate and provides a system of linear ordinary differential equations in time which can be integrated analytically therefore the third type of linearized finite difference methods provides continuous solutions in time and discrete in space and may be transformed into one of the first type by discretizing the time variable

an indirect method for transfer function estimation from closed loop data abstract an indirect method is introduced that is able to estimate consistently the transfer function of a linear plant on the basis of data obtained from closed loop experiments even in the situation where the model of the noise disturbance on the data is not accurate moreover the method allows approximate identification of the open loop plant with an explicit and tunable expression for the bias distribution of the resulting model

testing viscoelastic numerical schemes using the oldroyd b fluid in newtonian kinematics abstract we focus here on using a newtonian velocity field to evaluate numerical schemes for two different formulations of viscoelastic flow the two distinct formulations we consider correspond to either using a fixed basis for the elastic stress or one that uses the flow directions or streamlines the former is the traditional cartesian stress formulation whilst the later may be referred to as the natural stress formulation of the equations we choose the oldroyd b fluid and three benchmarks in computational rheology the 4 1 contraction flow the stick slip and cross slot problems in the context of the contraction flow fixing the kinematics as newtonian actually gives a larger stress singularity at the re entrant corner the matched asymptotics of which are presented here numerical results for temporal and spatial convergence of the two formulations are compared first in this decoupled velocity and elastic stress situation to assess the performance of the two approaches this may be regarded as an intermediate test case before proceeding to the far more difficult fully coupled velocity and stress situation we also present comparison results between numerics and asymptotics for the stick slip problem finally the natural stress formulation is used to investigate the cross slot problem again in a newtonian velocity field

parametric excitation of computational mode of the leapfrog scheme applied to the van der pol equation the leapfrog scheme is applied to the van der pol equation when the amplitude of oscillation of the physical mode exceeds a critical value the computational mode is parametrically excited by the physical mode the growth of the computational mode interrupts the integration based on the leapfrog scheme the critical amplitude of the physical mode is determined by the linear stability analysis and the parametric excitation theory the runge kutta smoother eliminating the computational mode enables the longtime integration based on the leapfrog scheme

a globally consistent nonlinear least squares estimator for identification of nonlinear rational systems this paper considers identification of nonlinear rational systems defined as the ratio of two nonlinear functions of past inputs and outputs despite its long history a globally consistent identification algorithm remains illusive this paper proposes a globally convergent identification algorithm for such nonlinear rational systems to the best of our knowledge this is the first globally convergent algorithm for the nonlinear rational systems the technique employed is a two step estimator though two step estimators are known to produce consistent nonlinear least squares estimates if a n consistent estimate can be determined in the first step how to find such a n consistent estimate in the first step for nonlinear rational systems is nontrivial and is not answered by any two step estimators the technical contribution of the paper is to develop a globally consistent estimator for nonlinear rational systems in the first step this is achieved by involving model transformation bias analysis noise variance estimation and bias compensation in the paper two simulation examples and a practical example are provided to verify the good performance of the proposed two step estimator

low reynolds number turbulence in nonlinear maxwell model fluids a generalized nonlinear maxwell model which had previously been analyzed for plane couette geometry is here applied to a lid driven cavity flow the full three dimensional hydrodynamical problem is treated numerically depending on the relevant model parameters both smooth laminar and low reynolds number turbulent flows are found strikingly similar to the experimentally observed elastic turbulence phenomena in polymer solutions representative results of the calculated flow patterns as well as measures for the turbulent nature of the flow are presented graphically

variational multi scale stabilized formulations for the stationary three field incompressible viscoelastic flow problem abstract in this paper three field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows these methods allow one to use equal interpolation for the problem unknowns u p elastic deviatoric stress velocity pressure and to stabilize dominant convective terms starting from residual based stabilized formulations the proposed method introduces a term by term stabilization which is shown to have a superior behavior when there are stress singularities a general discontinuity capturing technique for the elastic stress component is also proposed which allows one to eliminate the local oscillations that can appear when the weissenberg number we is high and the fluid flow finds an abrupt change in the geometry the formulations are tested in the classical 4 1 planar contraction benchmark up to we 5 in the inertial case with reynolds number re 1 and up to we 6 5 in the quasi non inertial case with re 0 01 the standard oldroyd b constitutive model is used for the rheological behavior and linear and quadratic elements for the spatial approximation

s stability of piecewise linearized and linearized θ methods the s stability of piecewise linearized and linearized q methods is analyzed and it is shown that piecewise linearized methods are not s stable although they are a stable while linearized q methods are s stable if and only if 1 2 q 1 it is also shown that for prothero and robinson s problems the piecewise linearized method yields accurate results even for large values of the stepsize while for stiff differential equations the piecewise linearized method yields more accurate solutions than s stable techniques like linearized and standard implicit q methods with q 1 2 even for large values of the stepsize

linearized θ methods i ordinary differential equations fully linearized methods for autonomous and non autonomous ordinary differential equations are derived by approximating the non linear terms by means of the first degree polynomials which result from taylor s series expansions these methods are implicit but result in explicit solutions a stable consistent and convergent however they may be very demanding in terms of both computer time and storage because the matrix to be inverted is in general dense the accuracy of fully linearized methods is comparable to that of the standard implicit iterative methods and deteriorates as the value of is decreased from 0 5 for which both and fully linearized methods are second order accurate partially linearized methods based on the partial linearization of non linear terms have also been developed these methods result in diagonal or triangular matrices which may be easily solved by substitution their accuracy however is lower than that of fully linearized methods

linearization methods for reaction diffusion equations multidimensional problems four types of linearization methods for the numerical solution of multidimensional reaction diffusion equations are presented the first two types are based on the discretization of the time variable time linearization and approximate factorization the first type also discretizes the spatial coordinates results in block tridiagonal matrices and provides discrete solutions in space and time the second type employs space linearization yields linear ordinary differential equations in space and produces either piecewise continuous or piecewise differentiable solutions the third type is based on the discretization of the spatial coordinates and time linearization and yields continuous solutions in time the fourth type uses time and space linearization and results in a multidimensional linear elliptic equation whose solution by means of separation of variables provides continuous approximations in space and discrete in time the fourth type also yields nine point finite difference expressions compared with the five point ones of the first three types

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

exponential methods for one dimensional reaction diffusion equations abstract non iterative exponential time linearization and iterative exponential quasilinearization techniques for one dimensional reaction diffusion equations based on the discretization of the time derivative the freezing of the coefficients of the resulting linear ordinary differential equations and the piecewise analytical solution of these ordinary differential equations are presented and applied to the nagumo and fisher equations these techniques yield three point finite difference expressions that depend in an exponential manner on either the diffusion reaction and transient terms or the diffusion and reaction terms it is shown that first order accurate in time exponential time linearization and exponential quasilinearization methods which account for reaction and diffusion processes in the ordinary differential operator are more robust than standard second order accurate techniques even for very large time steps and grid sizes and yield results in excellent accord with those obtained by means of non standard finite difference schemes and nodal integral formulations for the nagumo and fisher equations unlike non standard finite difference methods the exponential schemes presented in this paper do not require any knowledge of the exact solution of the differential equation and can be easily derived in a rather systematic manner compared with nodal integral formulations the exponential techniques presented here only require the solution of second order ordinary differential equations

comparison of some instrumental variable methods consistency and accuracy aspects a general analysis of various instrumental variable methods is given the investigation includes methods for open loop operation as well as methods designed for closed loop operation the instrumental variables are formed as different combinations of inputs delayed inputs delayed outputs filtered inputs and external setpoint variations necessary conditions as well as sufficient conditions for consistency are explicitly derived and discussed the paper also includes some simple counter examples to general consistency for some common instrumental variable methods also asymptotic accuracy properties are reviewed it is e g studied how the use of filters will influence the covariance matrix of the parameter estimates

continuous explicit runge kutta methods of order 5 a continuous explicit runge kutta cerk method provides a continuous approximation to an initial value problem such a method may be obtained by appending additional stages to a discrete method or alternatively by solving the appropriate order conditions directly owren and zennaro have shown for order 5 that the latter approach yields some cerk methods that require fewer derivative evaluations than methods obtained by appending stages in contrast continuous methods of order 6 that require the minimum number of stages can be obtained by appending additional stages to certain discrete methods this article begins a study to understand why this occurs by making no assumptions to simplify solution of the order conditions the existence of other types of cerk methods of order 5 is established while methods of the new families may not be as good for implementation as the owren zennaro methods the structure is expected to lead to a better understanding of how to construct families of methods of higher order

bias correction in least squares identification abstract the least squares method in system identification leads generally to biased parameter estimates a conceptually simple modification is to estimate the bias and to compute compensated parameter estimates when white output noise is the only disturbance this principle compensated least squares cls can readily be used to obtain consistent estimates the main purpose of the paper is to investigate the accuracy of the parameter estimates obtained when the cls method is used it is proved that the estimates are asymptotically gaussian distributed an explicit expression for the covariance matrix is given it is also shown that a commonly used instrumental variable method gives asymptotically better accuracy than the compensated least squares method

instrumental variable methods for closed loop system identification in this paper several instrumental variable iv and instrumental variable related methods for closed loop system identification are considered and set in an extended iv framework extended iv methods require the appropriate choice of particular design variables as the number and type of instrumental signals data prefiltering and the choice of an appropriate norm of the extended iv criterion the optimal iv estimator achieves minimum variance but requires the exact knowledge of the noise model for the closed loop situation several iv methods are put in an extended iv framework and characterized by different choices of design variables their variance properties are considered and illustrated with a simulation example

identification of closed loop systems with low order controllers a bias correction method without noise modelling was recently proposed for indirect identification of closed loop systems a key assumption used in deriving this method required that the order of the controller is not less than that of the open loop plant a significant feature of the results presented in this paper is the removal of this assumption to enable the method to be applicable to unbiased identification of closed loop systems with low order controllers

numerical aspects in modeling high deborah number flow and elastic instability investigating highly nonlinear viscoelastic flow in 2d domain we explore problem as well as property possibly inherent in the streamline upwinding technique supg and then present various results of elastic instability the mathematically stable leonov model written in tensor logarithmic formulation is employed in the framework of finite element method for spatial discretization of several representative problem domains for enhancement of computation speed decoupled integration scheme is applied for shear thinning and boger type fluids from the analysis of 4 1 contraction flow at low and moderate values of the deborah number de the solution with supg method does not show noticeable difference from the one by the computation without upwinding on the other hand in the flow regime of high de especially in the state of elastic instability the supg significantly distorts the flow field and the result differs considerably from the solution acquired straightforwardly when the strength of elastic flow and thus the nonlinearity further increase the computational scheme with upwinding fails to converge and evolutionary solution does not become available any more all this result suggests that extreme care has to be taken on occasions where upwinding is applied and one has to first of all prove validity of this algorithm in the case of high nonlinearity on the contrary the straightforward computation with no upwinding can efficiently model representative phenomena of elastic instability in such benchmark problems as 4 1 contraction flow flow over a circular cylinder and flow over asymmetric array of cylinders asymmetry of the flow field occurring in the symmetric domain enhanced spatial and temporal fluctuation of dynamic variables and flow effects caused by extension hardening are properly described in this study

first second and third order fractional step methods for the three field viscoelastic flow problem in this paper three different fractional step methods are designed for the three field viscoelastic flow problem whose variables are velocity pressure and elastic stress the starting point of our methods is the same as for classical pressure segregation algorithms used in the newtonian incompressible navier stokes problem these methods can be understood as an inexact lu block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level the final schemes allow one to solve the problem in a fully decoupled form where each equation for velocity pressure and elastic stress is solved separately the first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation whereas the key point for the second order scheme is a first order extrapolation of these variables the third order fractional step method relies on yosida s scheme referring to the spatial discretization either the galerkin method or a stabilized finite element formulation can be used we describe the fractional step methods first assuming the former and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work this discretization in space shows very good stability permitting in particular the use of equal interpolation for all variables

estimation of spatial derivatives and identification of continuous spatio temporal dynamical systems a new approach for the estimation of spatial derivatives and the identification of a class of continuous spatio temporal dynamical systems from experimental data is presented in this study the proposed identification approach is a combination of implicit adams integration and an orthogonal forward regression algorithm ofr in which the operators are expanded using polynomials as basis functions the noisy experimental data are de noised by using biorthogonal spline wavelet filters and the spatial derivatives are estimated using a multiresolution analysis method finally a bootstrap method is applied to refine the identified parameters from the ofr algorithm the resulting identified models of the spatio temporal evolution form a system of partial differential equations examples are provided to demonstrate the efficiency of the proposed method

linearly implicit methods for the nonlinear schrodinger equation in nonhomogeneous media the nonlinear schrodinger equation in one dimensional cartesian coordinates is studied numerically by means of three linearly implicit finite difference methods that result in block tridiagonal matrices at each time level as a function of the damping pumping and changes in the refraction index the numerical experiments performed to determine the effects of the time step spatial step size and implicitness parameters of the dispersion and nonlinear terms show that the most accurate results are obtained when both the dispersion and nonlinear terms are treated by means of a second order accurate trapezoidal discretization and full linearization of the nonlinear terms it is also shown that if either the dispersion or the nonlinear terms are treated with a first order accurate approximation substantial errors are found and these errors are largest when the nonlinear terms are discretized by means of a first order difference and do not decrease substantially as the number of grid points is increased it has been observed that when solitons move from a medium to another one characterized by a higher refraction index the amplitude and speed of the soliton increase whereas its width decreases upon approaching the interface the amplitude of the soliton was found to increase downstream while some radiation was observed upstream both the increase in amplitude and radiation were found to increase with the change in the refraction index the effect of the distance where the index of refraction changes was found to be small if this distance is smaller than the width of the soliton for solitons propagating from a medium to another one of lower refraction index the amplitude and speed of the soliton decrease whereas its width increases the amplitude of the soliton increases downstream and spreads towards the boundary where it is reflected when the index of refraction decreases below a threshold it was found that the soliton may become trapped at the interface the results presented in this paper indicate that theories based on treating solitons as particles perturbation methods inverse scattering and variational formulations can provide accurate results only when the change in the refraction index is sufficiently small

a numerical method for the computation of the lyapunov exponents of nonlinear ordinary differential equations in this paper an alternative method to compute the lyapunov exponents of dynamical systems described by ordinary differential equations odes is introduced the lyapunov exponents are computed in terms of the solutions of two piecewise linear odes that approximate respectively the solutions of the original ode and its associated variational equation this approach is strongly connected with the local linearization ll method for odes and its major advantage is that these piecewise linear odes might be exactly integrated in a non simultaneous way the performance of the method is illustrated with a numerical example

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

routes to spatiotemporal chaos in the rheology of nematogenic fluids with a view to understanding the rheochaos observed in recent experiments in a variety of orientable fluids we study numerically the equations of motion of the spatiotemporal evolution of the traceless symmetric order parameter of a sheared nematogenic fluid in particular we establish by decisive numerical tests that the irregular oscillatory behavior seen in a region of parameter space where the nematic is not stably flow aligning is in fact spatiotemporal chaos we outline the dynamical phase diagram of the model and study the route to the chaotic state we find that spatiotemporal chaos in this system sets in via a regime of spatiotemporal intermittency with a power law distribution of the widths of laminar regions as in h chate and p manneville phys rev lett 58 112 1987 further the evolution of the histogram of band sizes shows a growing length scale as one moves from the chaotic towards the flow aligned phase finally we suggest possible experiments in which one can observe the intriguing behaviors discussed here

on diffusive methods and exponentially fitted techniques liouville s transformations are employed to reduce a one dimensional time dependent convection diffusion reaction operator to a diffusion reaction one which upon a control volume second order accurate discretization results in the same finite difference methods as those of exponentially fitted techniques two variants of these techniques are considered depending on the continuity and or smoothness of the analytical albeit approximate solution to the ordinary differential equations that result upon discretization of the time variable time linearization and piecewise spatial linearization of linear equations it is shown that these exponential techniques include those of allen and southwell il in and el mistikawy and werle for steady linear convection diffusion operators and are linearly stable and monotonic it is also shown that exponentially fitted techniques account for the characteristic times of reaction diffusion and residence and the time step employed in the discretization of the time variable can use adaptive refinement techniques and can account for the convection diffusion and reaction the convection and diffusion the reaction and diffusion and the diffusion processes however the latter three require iterative techniques to find the numerical solution

the numerical prediction of planar viscoelastic contraction flows using the pom pom model and higher order finite volume schemes this study investigates the numerical solution of viscoelastic flows using two contrasting high order finite volume schemes we extend our earlier work for poiseuille flow in a planar channel and the single equation form of the extended pom pom sxpp model m aboubacar j p aguayo p m phillips t n phillips h r tamaddon jahromi b a snigerev m f webster modelling pom pom type models with high order finite volume schemes j non newtonian fluid mech 126 2005 207 220 to determine steady state solutions for planar 4 1 sharp contraction flows the numerical techniques employed are time stepping algorithms one of hybrid finite element volume type the other of pure finite volume form the pure finite volume scheme is a staggered grid cell centred scheme based on area weighting and a semi lagrangian formulation this may be implemented on structured or unstructured rectangular grids utilising backtracking along the solution characteristics in time for the hybrid scheme we solve the momentum continuity equations by a fractional staged taylor galerkin pressure correction procedure and invoke a cell vertex finite volume scheme for the constitutive law a comparison of the two finite volume approaches is presented concentrating upon the new features posed by the pom pom class of models in this context of non smooth flows here the dominant feature of larger shear and extension in the entry zone influences both stress and stretch so that larger stretch develops around the re entrant corner zone as weissenberg number increases whilst correspondingly stress levels decline

re entrant corner flows of oldroyd b fluids the method of matched asymptotic expansions is used to construct solutions for the planar steady flow of oldroyd b fluids around re entrant corners of angles 1 2 1 two types of similarity solu

a piecewise linearized algorithm based on the krylov subspace for solving stiff odes numerical methods for solving ordinary differential equations odes have received considerable attention in recent years in this paper a piecewise linearized algorithm based on krylov subspaces for solving initial value problems ivps is proposed matlab versions for autonomous and non autonomous odes of this algorithm have been implemented these implementations have been compared with other piecewise linearized algorithms based on pade approximants recently developed by the authors of this paper comparing both precisions and computational costs in equal conditions four case studies have been used in the tests that come from stiff biology and chemical kinetics problems experimental results show the advantages of the proposed algorithms especially when the dimension is increased in stiff problems

linearly implicit approximate factorization exponential methods for multi dimensional reaction diffusion equations abstract linearly implicit approximate factorization methods are used to reduce the solution of multi dimensional reaction diffusion equations to a sequence of two point boundary value problems in ordinary differential equations which are solved analytically in a piecewise manner it is shown that depending on the allocation of the transient diffusion and reaction terms to the second order ordinary differential operators several families of piecewise exponential methods can be obtained when only diffusion is included a quadratic analytical solution is obtained whereas if transient and diffusion diffusion and reaction and transient diffusion and reaction terms are included piecewise exponential methods result it is shown that exponential methods that only account for reaction and diffusion in the differential operator may result in ill conditioned systems of algebraic equation whereas if diffusion transient and reaction are included ill conditioning is avoided if the time step is sufficiently small it is also shown that the most robust exponential method is that that accounts for transient and diffusion terms in the differential operator and this method provides solutions in good accord with those of a linearly implicit approximate factorization technique which employs a three point second order accurate discretization of the diffusion terms it is also shown that a hybrid exponential method that solves the transient reaction and diffusion terms at odd time levels and the transient and diffusion terms at even time levels also provides very accurate solutions

iterative and non iterative full and approximate factorization methods for multidimensional reaction diffusion equations a variety of second order accurate in both space and time full and approximate factorization methods for the numerical solution of two dimensional reaction diffusion equations is presented these methods may use time linearization and yield linearly implicit techniques and one dimensional operators in each direction it is shown that if the factorization errors are neglected linearly implicit approximate factorization methods provide uncoupled equations whereas if these errors are considered the equations are coupled and must be solved iteratively it is also shown that the allocation of the reaction and diffusion terms to the one dimensional operators plays a paramount role in determining the accuracy of approximate factorization methods and preserving the symmetry of the original differential problem iterative full and approximate factorization methods that do require iterations are also presented and for the problem considered here these methods are shown to converge in about two iterations and provide solutions in agreement with those obtained with linearly implicit full and approximate factorization techniques

implicit compact linearized θ methods with factorization for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of implicit three point compact linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four compact approximate factorization techniques which do not account for the approximate factorization errors and which involve three point stencils for each one dimensional operator are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized compact implicit method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

piecewise quasilinearization techniques for singularly perturbed volterra integro differential equations abstract three families of iterative quasilinear methods for the numerical solution of singularly perturbed volterra integro differential equations are presented two of these methods are based on the piecewise integration of the differential equations and the use of quadrature rules while the third one employs a locally frozen approximation for the coefficients and right hand side of the differential equation it is shown that these quasilinear methods are first order accurate for step sizes equal to or larger than the initial layer thickness and are less accurate than exponentially fitted techniques whose coefficients are selected so that the finite difference equations have truncation errors of second and third order even though they exhibit only first order accuracy for initial layer thicknesses equal to or smaller than the step size it is also shown that the quasilinear methods presented in this paper are of exponential type and provide more accurate results than the backward euler and trapezoidal methods for step sizes larger than the initial layer thickness and the quasilinear frozen method is less accurate than quasilinear techniques that make use of an integrating factor and numerical quadratures

non standard explicit integration algorithms based on linearization for nonlinear dynamic response analysis non standard finite difference methods for initial value problems of second order ordinary differential equations based on piecewise linearization are developed linearization methods provide piecewise analytical solutions which are globally differentiable result in explicit finite difference algorithms and are exact for constant coefficients equations with a right hand side which depends linearly on time the accuracy of these methods is assessed by obtaining the solutions of several conservative and dissipative stiff and non stiff regular and chaotic problems and comparing them with those of the matlab ode45 solver state transition matrix algorithms and non standard euler techniques

finite volume solution of the two dimensional euler equations on a regular triangular mesh the two dimensional euler equations have been solved on a triangular grid by a multigrid scheme using the finite volume approach by careful construction of the dissipative terms the scheme is designed to be secondorder accurate in space provided the grid is smooth except in the vicinity of shocks where it behaves as firstorder accurate in its present form the accuracy and convergence rate of the triangle code are comparable to that of the quadrilateral mesh code of jameson

high order finite volume methods for viscoelastic flow problems in this paper accurate and stable finite volume schemes for solving viscoelastic flow problems are presented two contrasting finite volume schemes are described a hybrid cell vertex scheme and a pure cell centred counterpart both schemes employ a time splitting algorithm to evolve the solution through time towards steady state in the case of the hybrid scheme a semi implicit formulation is employed in the momentum equation based on the taylor galerkin approach with a pressure correction step to enforce incompressibility the basis of the pure finite volume approach is a backward euler scheme with a semi lagrangian step to treat the convection terms in the momentum and constitutive equations two distinct finite volume schemes are presented for solving the systems of partial differential equations describing the flow of viscoelastic fluids the schemes are constructed to be second order accurate in space the issue of stability is also addressed with respect to the treatment of convection numerical examples are presented illustrating the performance of these schemes on some steady and transient problems that possess analytical solutions

piecewise linearized and linearized θ methods for ordinary and partial differential equations lnitial and boundary value problems appear frequently in many branches of physics in this paper several numerical methods based on linearization techniques for solving these prob lems are reviewed first piecewise linearized methods and linearized 0 methods are considered for the solution of initial value problems in ordinary differential equations second piecewise linearized techniques for two point boundary value problems in ordinary differential equations are developed and used in conjunction with a shooting method in order to overcome the lack of convergence asso ciated with shooting piecewise linearized methods which provide piecewise analytical solutions and yield nonstandard finite difference schemes axe presented third methods of lines in either space or time for the solution of one dimensional convection reaction diffusion problems that transform the original problem into an initial or boundary value one are reviewed methods of lines in time that result in boundary value problems at each time step can be solved by means of the techniques described here whereas methods of lines in space that yield initial value problems and employ ei ther piecewise linearized techniques or linearized 0 methods in time are also developed finally for multidimensional problems approximate factorization methods are first used to transform the multi dimensional problem into a sequence of one dimensional ones which are then solved by means of the linearized and piecewise linearized methods presented here 2003 elsevier science ltd all rights reserved

the sine gordon equation in the finite line a rather complete numerical study of the perturbed and nonperturbed sine gordon equation in one dimensional cartesian coordinates and finite domains is presented the numerical study is based on five three point linearly implicit finite difference methods of different spatial accuracy for the unperturbed sge the accuracy of these five methods is found to be very sensitive to the implicitness parameter less sensitive to the spatial order of accuracy and almost no sensitive to the time step for second order accurate temporal discretizations the largest errors of the five methods were found to occur at the front of the soliton solution of the unperturbed sge but the techniques were very accurate for long time computations of the unperturbed sge in both infinite domains and in finite domains upon many collisions of the kinks and antikinks with the boundaries despite the fact that they do not preserve a discrete energy it is shown that the largest values of the kinetic and total energies and the smallest values of the strain and potential energies occur when solitons and antisolitons strike on the boundaries subject to homogeneous neumann conditions and the four energies remain constant between collisions with the boundaries for the malomed s perturbation one of the fronts of the soliton soliton doublet and soliton antisoliton doublet is trapped at one boundary and the other one undergoes a change in amplitude and its front describes a curved trajectory the soliton antisoliton breather oscillates in a damped manner and slowly drifts towards one boundary the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet of the unperturbed sge are robust under kivshar and malomed s perturbations whereas the soliton antisoliton breather loses its integrity becomes a kink antikink pair and then yields a chaotic behaviour in both space and time small amplifications proportional to the first order temporal derivative of the amplitude do not alter substantially the initial dynamics of the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet but their amplitudes exhibit oscillatory behaviour of increasing amplitude at large times however small amplifications cause first a transition from the soliton antisoliton breather to the kink antikink solution a linearized stability analysis of the spatially homogeneous solutions of the unperturbed sge is performed and analytical solutions are obtained by means of the method of separation of variables for both homogeneous dirichlet and neumann boundary conditions and the results show that the solution corresponding to a zero amplitude is oscillatory in both space and time whereas that corresponding to an amplitude equal to p may exhibit exponential or linear growth depending on the length of the domain if either of these growths is present it is shown that the evolution of small initial perturbations may result in homoclinic crossings and chaotic behaviour in both space and time in the absence of perturbations in the sge and the existence of an attractor when perturbations of the malomed s type are present in either case the solution may exhibit certain spatial symmetries or antisymmetries at large times finally the results presented in this paper indicate that the accurate computation of homoclinic orbits of the sge is not only controlled by the spatial accuracy of the numerical method but also by its energy preserving characteristics and perhaps by the number of grid points used in the numerical technique

simulation for viscoelastic flow by a finite volume element method stability of a second order finite element finite volume fe fv hybrid scheme is investigated on the basis of flows with increasing weissenberg number fes are used to discretise the balances of mass and momentum for the stress equation a fv method is used based on the recent development with fluctuation distribution schemes for pure convection problems examples considered include a start up channel flow flow past a cylinder and the non smooth 4 1 contraction flow for an oldroyd b fluid a considerable gain in efficiency per time step can be obtained compared to an alternative pure fe implementation a distribution based on the flux terms is unstable for higher weissenberg numbers and this is also true for a distribution based on source terms alone the instability is identified as being caused by the interaction of the balance equations and stress equation a combination of distribution schemes based on flux and source terms however gives a considerable improvement to the hybrid fe fv implementation with respect to limiting weissenberg number attenuation the hybrid scheme is more stable than the pure fe alternative for the smooth flow past a cylinder but less so for the non smooth contraction flow the influence of additional strain rate stabilisation techniques is also analysed and found to be beneficial

hybrid atomistic continuum method for the simulation of dense fluid flows we present a hybrid atomistic continuum method for multiscale simulations of dense fluids in this method the atomistic part is described using a molecular dynamics description while the continuum flow is described by a finite volume discretization of the incompressible navier stokes equations the two descriptions are combined in a domain decomposition formulation using the schwarz alternating method a novel method has been proposed in order to impose non periodic velocity boundary conditions from the continuum to the atomistic domain based on an effective boundary potential consistent body forces a particle insertion algorithm and specular walls the extraction of velocity boundary conditions for the continuum from the atomistic domain is formulated by taking into account the associated statistical errors the advantages and drawbacks of the proposed schwarz decomposition method as compared to related flux based schemes are discussed the efficiency and applicability of the method is demonstrated by considering hybrid and full molecular dynamics simulations of the flow of a lennard jones fluid past a carbon nanotube

high order integration of smooth dynamical systems theory and numerical experiments this paper describes a new class of algorithms for integrating linear second order equations and those containing smooth non linearities the algorithms are based on a combination of ideas from standard newmark integration methods and extrapolation techniques for the algorithm to work the underlying newmark method must be stable second order accurate and produce asymptotic error expansions for response quantities containing only even ordered terms it is proved that setting the newmark parameter equal to 1 2 gives a desirable asymptotic expansion irrespective of the setting for numerical experiments are conducted for one linear and two non linear applications

a numerical study of the kernel conformation transformation for transient viscoelastic fluid flows this work presents a numerical application of a generic conformation tensor transformation for simulating highly elastic flows of non newtonian fluids typically observed in computational rheology in the kernel conformation framework 14 the conformation tensor constitutive law for a viscoelastic fluid is transformed introducing a generic tensor transformation function the numerical stability of the application of the kernel conformation for highly elastic flows is ultimately related with the specific kernel function used in the matrix transformation but also to the existence of singularities introduced either by flow geometry or by the characteristics of the constitutive equation in this work we implement this methodology in a free surface marker and cell discretization methodology implemented in a finite differences method the main contributions of this work are two fold on one hand we demonstrate the accuracy of this kernel conformation formulation using a finite differences method and free surfaces on the other hand we assess the numerical efficiency of specific kernel functions at high weissenberg number flows the numerical study considers different viscoelastic fluid flow problems including the poiseuille flow in a channel the lid driven cavity flow and the die swell free surface flow the numerical results demonstrate the adequacy of this methodology for high weissenberg number flows using the oldroyd b model we analyze a generic conformation tensor transformation for simulating the hwnp the numerical scheme is constructed in the context of finite differences the numerical study considers transient benchmarks in computational rheology complex flows are solved including the die swell free surface problem

a finite volume scheme with shock fitting for the steady euler equations abstract analysis of two alternative finite volume formulations in respect of accuracy on non uniform meshes and number of spurious modes leads to a preference for the more compact cell vertex scheme over the cell centre scheme the resulting equations are solved iteratively by using a lax wendroff procedure as a smoother for a multigrid algorithm then application of boundary conditions in a natural way leads rapidly to all individual residuals being driven close to zero except at shocks at shocks the residuals should not be zero and a shock fitting procedure is introduced to avoid this inconsistency sharp accurate solutions on a relatively coarse mesh are obtained for a channel low problem in which the zierep singularity is displayed and for the naca 0012 aerofoil

a bias correction method for closed loop identification of linear parameter varying systems abstract due to safety constraints and unstable open loop dynamics system identification of many real world processes often requiresgathering data from closed loop experiments in this paper we present a bias correction scheme for closed loop identification of linear parameter varying input output lpv io models which aims at correcting the bias caused by the correlation between the input signal exciting the process and output noise the proposed identification algorithm provides a consistent estimate of the open loop model parameters when both the output signal and the scheduling variable are corrupted by measurement noise the effectiveness of the proposed methodology is tested in two simulation case studies

intermediate boundary conditions in operator splitting techniques and linearization methods the intermediate boundary conditions for the solution of linear one dimensional reaction diffusion equations have been determined analytically for the case that the reaction and diffusion operators are solved once each in each time step these boundary conditions have been used to solve systems of nonlinear one dimensional reaction diffusion equations by means of linearized q methods and time linearized techniques which are based on the linearization of the nonlinear algebraic and differential respectively equations of the reaction operator both techniques provide analytical solutions to the reaction operator although in discrete and continuous forms respectively since the linearization of reaction operators may result in dense jacobian matrices diagonally and triangularly linearized techniques which uncouple or couple in a sequential manner respectively the dependent variables are proposed it is shown that the accuracy of time linearized methods is higher than that of linearized q techniques whereas the accuracy of both linearization methods deteriorates as the coupling between dependent variables is weakened

the relative efficiency of alternative defect control schemes for high order continuous runge kutta formulas the development of continuous runge kutta formulas for nonstiff problems has been the subject of several investigations in recent years in enright siam j numer anal 26 1989 pp 588 599 the author proposed a new error and stepsize control strategy based on controlling the size of the defect of the underlying continuous solution that is particularly appropriate for numerical methods based on this class of formulas in this investigation the author presents an analysis of the relative efficiency of existing continuous runge kutta methods the author introduces theoretical measures useful in evaluating the potential of such methods and apply these measures to several methods of orders 4 through 8 the author also presents numerical results to illustrate that the theoretical measures of potential are consistent with the observed performance of nonstiff problems over a wide range of error tolerances

a nonlinear weighted least squares finite element method for the oldroyd b viscoelastic flow this paper studies a nonlinear weighted least squares finite element method for the steady oldroyd b viscoelastic flow our least squares functional involves the l 2 norm of the residuals of each equation multiplied by a proper weight the weights include mass conservation constant mesh dependent weight and a nonlinear weighting function we prove that the least squares approximation converges to the solutions of the linearized versions of the viscoelastic fluid model at the best possible rate and then present the planar channel flow problem illustrating our theoretical results results of the least squares approach indicate that with carefully chosen nonlinear weighting functions and linear basis functions the numerical solution exhibits a second order convergence rate for velocity and viscous stress and superlinear convergence rate in polymeric stress and pressure the method is applied to the 4 to 1 planar contraction problem the effects of weissenberg numbers are also presented in the work

brief on the relation between a bias eliminated least squares bels and an iv estimator in closed loop identification a bias correction method for closed loop identification introduced in the literature as the bias eliminated least squares bels method zheng feng automatica 31 1995 1019 is shown to be equivalent to a basic instrumental variable estimator applied to a predictor for the closed loop system this predictor is a function of the plant parameters and the known controller corresponding to the related method using a least squares criterion the method is referred to as the tailor made iv method for closed loop identification the indicated equivalence greatly facilitates the understanding and the analysis of the bels method

indirect closed loop identification by optimal instrumental variable method recently a new bias compensating least squares bcls method was proposed for the identification of a closed loop system with high order controller the major feature of this method is that it can achieve consistent parameter estimation without modelling the coloured noises acting on the system this paper studies the connection between the bcls method and the instrumental variable iv family it is shown that the bcls method is a kind of weighted instrumental variable wiv method whose results do not depend upon the order of the controller

a unified framework for bias compensation based methods in correlated noise case this technical note presents a unified framework for bias compensation principle bcp based methods applied for identification of linear systems subject to correlated noise by introducing a non singular matrix and an auxiliary vector uncorrelated with the noise the unified framework is established since there are rich possibilities of the choices of the introduced matrix and vector the proposed unified framework is very flexible it can be verified that the existing bcp based methods are special cases of the achieved result it also shows that the unified framework can be used for deriving new or simplified versions of the bcp type methods