superconvergent explicit two step peer methods we consider explicit two step peer methods for the solution of nonstiff differential systems by an additional condition a subclass of optimally zero stable methods is identified that is superconvergent of order p s 1 where s is the number of stages the new condition allows us to reduce the number of coefficients in a numerical search for good methods we present methods with 4 7 stages which are tested in fortran90 and compared with dopri5 and dop853 the results confirm the high potential of the new class of methods

on spectral multigrid methods for the time dependent navier stokes equations a new splitting scheme is proposed for the numerical solution of the time dependent incompressible navier stokes equations by spectral methods a staggered grid is used for the pressure improved intermediate boundary conditions are employed in the split step for the velocity and spectral multigrid techniques are used for the solution of the implicit equations

a class of implicit peer methods for stiff systems we present a class of s stage implicit two step peer methods for the solution of stiff differential equations using the function values from the previous step in addition to the new function values this allows to increase the order to p s and to ensure zero stability straightforwardly corresponding s stage methods for s 6 of order p s with optimal zero stability are presented and their stability is discussed under special conditions we prove that an optimally zero stable subclass of these methods is superconvergent of order p s 1 for variable step sizes numerical tests and comparison with ode15s show the high potential of this class of implicit peer methods

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

analysis of dual consistency for discontinuous galerkin discretizations of source terms the effects of dual consistency on discontinuous galerkin discretizations of solution and solution gradient dependent source terms are examined two common discretizations are analyzed the standard weighting technique for source terms and the mixed formulation it is shown that if the source term depends on the first derivative of the solution the standard weighting technique leads to a dual inconsistent scheme a straightforward procedure for correcting this dual inconsistency and arriving at a dual consistent discretization is demonstrated the mixed formulation where the solution gradient in the source term is replaced by an additional variable that is solved for simultaneously with the state leads to an asymptotically dual consistent discretization numerical results for a one dimensional test problem confirm that the dual consistent and asymptotically dual consistent schemes achieve higher asymptotic convergence rates with grid refinement than a similar dual inconsistent scheme for both the primal and adjoint solutions as well as a simple functional output

duality based two level error estimation for time dependent pdes application to linear and nonlinear parabolic equations we introduce a duality based two level error estimator for linear and nonlinear time dependent problems the error measure can be a space time norm energy norm final time error or other error related functional the general methodology is developed for an abstract nonlinear parabolic pde and subsequently applied to linear heat and nonlinear cahn hilliard equations the error due to finite element approximations is estimated with a residual weighted approximate dual solution which is computed with two primal approximations at nested levels we prove that the exact error is estimated by our estimator up to higher order remainder terms numerical experiments confirm the theory regarding consistency of the dual based two level estimator we also present a novel space time adaptive strategy to control errors based on the new estimator

essentially nonoscillatory spectral fourier methods for shock wave calculations in this paper we present an essentially nonoscillatory spectral fourier method for the solution of hyperbolic partial differential equations the method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the fourier method the high accuracy away from the shock is enhanced by using filters numerical results confirm that essentially no oscillations develop in the solution also the accuracy of the spectral solution of the inviscid burgers equation is shown to be higher than a fixed order

on the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions abstract we prove the existence of extremal solutions for the third order discontinuous functional nonlinear problem u t f t u u t u t for a e t a b u a a a r l 1 u u u a u b u a 0 l 2 u u a u b u b 0 by using a fixed point theorem after having established some existence results for some auxiliary second order nonlinear problems we observe that together with the discontinuities allowed on the spacial variable u with adequate modifications of technical type analogous results can be obtained when the equation has a second member not necessarily continuous in u

discontinuous polynomial approximations in the theory of one step hybrid and multistep methods for nonlinear ordinary differential equations this paper studies the approximation of the solution of nonlinear ordinary differen tial equations by discontinuous piecewise polynomials of degree k and traces at the nodes of discretization a mesh dependent variational framework underlying this discontinuous ap proximation is derived several families of one step hybrid and multistep schemes are obtained it is shown that the convergence rate in the l2 norm is k 1 the nodal conver gence rate can go up to 2 k 2 depending on the particular scheme under consideration the mesh dependent variational framework introduced here is of special interest in the approxi mation of the solution of optimal control problems governed by differential equations 1 introduction the object of this paper is the study of discontinuous piecewise polynomial approximations to the solution of systems of nonlinear ordinary dif

optimal recursive estimation for networked stochastic uncertain systems with fading measurements and time correlated channel noises abstract this paper is concerned with the optimal linear estimation problem for discrete time stochastic system with time correlated channel noises correlated multiplicative noises and fading measurements the time correlated channel noises are described by a seemingly arma autoregressive moving average model by an equivalent transformation the original system is transformed into the same dimension system with finite step correlated virtual noises and multiple measurement delays the statistical properties of correlation functions among noise state and measurement are analyzed further the optimal linear state estimators including filter predictor and smoother are derived by using an innovation analysis approach a simulation example shows the effectiveness of the proposed algorithms

generalizing global error estimation for ordinary differential equations by using coupled time stepping methods abstract this study introduces new time stepping strategies with built in global error estimators the new methods propagate the defect along with the numerical solution much like solving for the correction or zadunaisky s procedure however the proposed approach allows for overlapped internal computations and therefore represents a generalization of the classical numerical schemes for solving differential equations with global error estimation the resulting algorithms can be effectively represented as general linear methods several explicit self starting schemes akin to runge kutta methods with global error estimation are introduced and the theoretical considerations are illustrated in several examples

a singly diagonally implicit two step peer triple with global error control for stiff ordinary differential equations this paper elaborates a new numerical method for treating stiff ordinary differential equations odes which constitute a basic simulation tool in many areas of study for its efficiency this technique is grounded in singly diagonally implicit two step peer schemes designed recently the latter means that linear systems with the same coefficient matrix are solved to advance a step of the method and to evaluate its global error as well moreover in contrast to many other numerical methods with global error control we monitor and regulate the scaled global error which corresponds better with practitioners needs in other words we present and study one peer triple which is able to solve stiff problems and to ensure that the true error of the derived numerical solution does not exceed the user supplied bound at the same time this implies that local and global error estimation and control mechanisms are also discussed here finally efficiency of our numerical tool is checked in comparison to efficiency

on unscented kalman filtering for state estimation of continuous time nonlinear systems this paper considers the application of the unscented kalman filter ukf to continuous time filtering problems where both the state and measurement processes are modeled as stochastic differential equations the mean and covariance differential equations which result in the continuous time limit of the ukf are derived the continuous discrete ukf is derived as a special case of the continuous time filter when the continuous time prediction equations are combined with the update step of the discrete time ukf the filter equations are also transformed into sigma point differential equations which can be interpreted as matrix square root versions of the filter equations

on asymptotic global error estimation and control of finite difference solutions for semilinear parabolic equations abstract the aim of this paper is to extend the global error estimation and control addressed in lang and verwer siam j sci comput 29 2007 for initial value problems to finite difference solutions of semilinear parabolic partial differential equations the approach presented there is combined with an estimation of the pde spatial truncation error by richardson extrapolation to estimate the overall error in the computed solution approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time asymptotic control in a discrete l 2 norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement numerical examples are used to illustrate the reliability of the estimation and control strategies

analysis and numerical approximation of singular boundary value problems with the p laplacian in fluid mechanics this paper studies a generalization of the cahn hilliard continuum model for multi phase fluids where the classical laplacian has been replaced by a degenerate one i e the so called p laplacian the solution s asymptotic behavior is analyzed at two singular points namely at the origin and at infinity an efficient technique for treating such singular boundary value problems is presented and results of numerical integration are discussed and compared with earlier computed data

adjoint sensitivity analysis for differential algebraic equations algorithms and software an efficient numerical method for sensitivity computation of large scale differential algebraic systems is developed based on the adjoint method issues that are critical for the implementation are addressed complexity analysis and numerical results demonstrate that the adjoint sensitivity method is advantageous over the forward sensitivity method for applications with a large number of sensitivity parameters and few objective functions

existence uniqueness and iterative methods for third order boundary value problems abstract in this paper we shall provide necessary and sufficient conditions for the existence and uniqueness of solutions of third order nonlinear differential equations satisfying three point boundary conditions for the linear case we propose a constructive method which is a variation of the method of chasing for the nonlinear problems sufficient conditions are provided to ensure the convergence of a general class of iterative methods several examples are also included

shock fitted multidomain solution of supersonic flows abstract shock fitting is presented as a way to compute shocks with spectral methods with fitting spectral approximation is performed on smooth portions of the flow and a precise description of a discontinuous shock is possible coupled with spectral domain decomposition one has the ability to resolve efficiently and without stretching transformations boundary layers and other regions in the flow where increased local resolution is needed a supersonic blunt body problem is solved to provide an example of the use of the approach comparisons between computations and experimental measurements are made for body surface quantities shock shape and body normal direction profiles

accurate numerical implementation of the continuous discrete extended kalman filter this paper addresses an accurate and effective implementation of the continuous discrete extended kalman filtering method the technique under discussion is grounded in numerical solution of the moment differential equations to predict the state mean of the stochastic dynamical system and the corresponding error covariance matrix here we apply an efficient embedded runge kutta pair possessing superior stability and many other attractive features including automatic global error control in order to improve performance of the complex computational procedure consisting of the extended kalman filter and the underlying adaptive ode solver thus we introduce a new continuous discrete adaptive extended kalman filter and show its advantage over the standard variant on two test examples in practice this technique allows for much longer sampling intervals without any loss of accuracy and that improves the applied potential of the extended kalman filtering method significantly

eigenvalue for a singular third order three point boundary value problem in this paper the existence of positive solutions for a singular third order three point boundary value problem is investigated by using krasnoselskii s fixed point theorem several sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exists a positive solution are obtained

spectral methods for modeling supersonic chemically reacting flowfields on emploie une methode de runge kutta a deux stades pour integrer les equations en temps et une methode spectrale de chebyshev pour l integration des equations dans l espace comparaison avec une methode de differences finies implicites

existence and uniqueness of solutions for nth order nonlinear two point boundary value problems in this paper we present some general results of existence and uniqueness of solutions for nth order nonlinear two point boundary value problems by using shooting method together with maximum principle and kneser hukahara continuum theorem as applications we give certain concrete sufficient conditions for the existence and uniqueness of solutions of nonlinear two point boundary value problems of nth order lipschitz equations

spectral methods for the euler equations i fourier methods and shock capturing spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals current spectral collociation methods are put into historical context the basic concepts of fourier spectral collocation methods are provided filtering strategies for shock capturing approaches are also presented fourier shock capturing techniques are evaluated using a one dimensional periodic astrophysical nozzle problem

a pseudo spectral method for the simulation of poro elastic seismic wave propagation in 2d polar coordinates using domain decomposition we present a novel numerical approach for the comprehensive flexible and accurate simulation of poro elastic wave propagation in 2d polar coordinates an important application of this method and its extensions will be the modeling of complex seismic wave phenomena in fluid filled boreholes which represents a major and as of yet largely unresolved computational problem in exploration geophysics in view of this we consider a numerical mesh which can be arbitrarily heterogeneous consisting of two or more concentric rings representing the fluid in the center and the surrounding porous medium the spatial discretization is based on a chebyshev expansion in the radial direction and a fourier expansion in the azimuthal direction and a runge kutta integration scheme for the time evolution a domain decomposition method is used to match the fluid solid boundary conditions based on the method of characteristics this multi domain approach allows for significant reductions of the number of grid points in the azimuthal direction for the inner grid domain and thus for corresponding increases of the time step and enhancements of computational efficiency the viability and accuracy of the proposed method has been rigorously tested and verified through comparisons with analytical solutions as well as with the results obtained with a corresponding previously published and independently benchmarked solution for 2d cartesian coordinates finally the proposed numerical solution also satisfies the reciprocity theorem which indicates that the inherent singularity associated with the origin of the polar coordinate system is adequately handled

doubly quasi consistent parallel explicit peer methods with built in global error estimation recently kulikov presented the idea of double quasi consistency which facilitates global error estimation and control considerably more precisely a local error control implemented in such methods plays a part of global error control at the same time however kulikov studied only nordsieck formulas and proved that there exists no doubly quasi consistent scheme among those methods here we prove that the class of doubly quasi consistent formulas is not empty and present the first example of such sort this scheme belongs to the family of superconvergent explicit two step peer methods constructed by weiner schmitt podhaisky and jebens we present a sample of s stage doubly quasi consistent parallel explicit peer methods of order s 1 when s 3 the notion of embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi consistent peer method and hence its global error at the same time numerical examples of this paper confirm clearly that the usual local error control implemented in doubly quasi consistent numerical integration techniques is capable of producing numerical solutions for user supplied accuracy conditions in automatic mode

approximation of weak adjoints by reverse automatic differentiation of bdf methods we shed light on the relation between the discrete adjoints of multistep backward differentiation formula bdf methods and the solution of the adjoint differential equation to this end we develop a functional analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations we devise a petrov galerkin finite element fe interpretation of the bdf method and its discrete adjoint scheme obtained by reverse internal numerical differentiation we show how the fe approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation we also show convergence of the discrete adjoints to the classical adjoints on the inner time interval finally we give numerical results for non adaptive and fully adaptive bdf schemes the presented framework opens the way to carry over techniques on global error estimation from fe methods to bdf methods

global error estimation based on the tolerance proportionality for some adaptive runge kutta codes modern codes for the numerical solution of initial value problems ivps in odes are based in adaptive methods that for a user supplied tolerance d attempt to advance the integration selecting the size of each step so that some measure of the local error is email protected although this policy does not ensure that the global errors are under the prescribed tolerance after the early studies of stetter considerations concerning a theory for ode solvers in r burlisch r d grigorieff j schroder eds numerical treatment of differential equations proceedings of oberwolfach 1976 lecture notes in mathematics vol 631 springer berlin 1978 pp 188 200 tolerance proportionality in ode codes in r marz ed proceedings of the second conference on numerical treatment of ordinary differential equations humbold university berlin 1980 pp 109 123 and the extensions of higham global error versus tolerance for explicit runge kutta methods ima j numer anal 11 1991 457 480 the tolerance proportionality of adaptive ode solvers j comput appl math 45 1993 227 236 the reliability of standard local error control algorithms for initial value ordinary differential equations in proceedings the quality of numerical software assessment and enhancement ifip series springer berlin 1997 it has been proved that in many existing explicit runge kutta codes the global errors behave asymptotically as some rational power of d this step size policy for a given ivp determines at each grid point t n a new step size h n 1 h t n d so that h t d is a continuous function of t in this paper a study of the tolerance proportionality property under a discontinuous step size policy that does not allow to change the size of the step if the step size ratio between two consecutive steps is close to unity is carried out this theory is applied to obtain global error estimations in a few problems that have been solved with the code gauss2 s gonzalez pinto r rojas bello gauss2 a fortran 90 code for second order initial value problems based on an adaptive two stage runge kutta gauss method with this discontinuous step size policy

adjoint consistency analysis of discontinuous galerkin discretizations this paper is concerned with the adjoint consistency of discontinuous galerkin dg discretizations adjoint consistency in addition to consistency is the key requirement for dg discretizations to be of optimal order in l 2 as well as measured in terms of target functionals we provide a general framework for analyzing the adjoint consistency of dg discretizations which is also useful for the derivation of adjoint consistent methods this analysis will be performed for the dg discretizations of the linear advection equation the interior penalty dg method for elliptic problems and the dg discretization of the compressible euler equations this framework is then used to derive an adjoint consistent dg discretization of the compressible navier stokes equations numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions

the tolerance proportionality of adaptive ode solvers abstract modern software for solving ordinary differential equation ode initial value problems requires the user to specify the ode and choose a value for the error tolerance the software can be thought of as a black box with a dial turning the dial changes the accuracy and expense of the integration process it is therefore of interest to know how the global error varies with the error tolerance in this work we look at explicit runge kutta methods and show that with any standard error control method and ignoring higher order terms the global error in the numerical solution behaves like a known rational power of the error tolerance this generalises earlier work of stetter who found sufficient conditions for the global error to be linear in the tolerance we also display the order of the next highest term we then analyse continuous runge kutta schemes and show what order of interpolation is necessary and sufficient for the continuous approximation to inherit the tolerance proportionality of the discrete formula finally we extend the results to the case of ode systems with constant delays thereby generalising some previous results of the author

the dynamical behavior of the discontinuous galerkin method and related difference schemes we study the dynamical behavior of the discontinuous galerkin finite element method for initial value problems in ordinary differential equations we make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system we show that under certain conditions the discontinuous galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system we also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method

nirk based accurate continuous discrete extended kalman filters for estimating continuous time stochastic target tracking models abstract this paper presents three state estimators grounded in the variable stepsize gauss and lobatto type nested implicit runge kutta nirk formulas of orders 4 and 6 and designed for treating continuous time stochastic systems arisen in radar tracking our filters are built within the extended kalman filtering ekf framework and based on accurate numerical integrations of the corresponding moment differential equations mdes automatic local and global error regulation mechanisms implemented in these methods allow the committed discretization error to be under control and made negligible in automatic mode the latter raises the state estimation accuracy of the constructed filters significantly this also leads to the advanced notion of accurate continuous discrete extended kalman filtering acd ekf developed by kulikov and kulikova in 2013 2016 our novel methods are constructed within the same approach but possess the improved accuracy and efficiency in comparison to their predecessors due to both more effective error control mechanisms implemented for integrating mdes and more accurate iterations used for treating arisen nonlinear equations in the revised filters numerical experiments with the updated state estimators and their comparison to the cited earlier designed acd ekfs are fulfilled in severe conditions of tackling a seven dimensional radar tracking problem where an aircraft executes a coordinated turn in matlab this examination suggests that the novel state estimation algorithms outperform their predecessors and possess a promising potential for solving target tracking tasks in real world applications

cubature kalman filtering for continuous discrete systems theory and simulations in this paper we extend the cubature kalman filter ckf to deal with nonlinear state space models of the continuous discrete kind to be consistent with the literature the resulting nonlinear filter is referred to as the continuous discrete cubature kalman filter cd ckf we use the ito taylor expansion of order 1 5 to transform the process equation modeled in the form of stochastic ordinary differential equations into a set of stochastic difference equations building on this transformation and assuming that all conditional densities are gaussian distributed the solution to the bayesian filter reduces to the problem of how to compute gaussian weighted integrals to numerically compute the integrals we use the third degree cubature rule for a reliable implementation of the cd ckf in a finite word length machine it is structurally modified to propagate the square roots of the covariance matrices the reliability and accuracy of the square root version of the cd ckf are tested in a case study that involves the use of a radar problem of practical significance the problem considered herein is challenging in the context of radar in two respects high dimensionality of the state and increasing degree of nonlinearity the results presented herein indicate that the cd ckf markedly outperforms existing continuous discrete filters

new third and fourth order singly diagonally implicit two step peer triples with local and global error controls for solving stiff ordinary differential equations in this paper we present new singly diagonally implicit two step peer triples equipped with local and global error controls for providing preassigned accuracies of numerical integration of stiff ordinary differential equations odes in automatic mode recently kulikov and weiner 2015 reported an efficient numerical integration tool of order 2 which solves accurately many difficult stiff odes including large scale systems obtained from semidiscretization of partial differential equations pdes corresponding to user supplied requests moreover the cited method is not only accurate but it is also more efficient than the well known matlab code ode23s with local error control here we further extend the published technique and construct variable stepsize singly diagonally implicit two step peer triples of the higher orders 3 and 4 our numerical experiments suggest that these triples are suitable for treating stiff problems with prescribed accuracy conditions in addition performance of the presented methods can be comparable to that of the built in stiff matlab code ode15s with local error control which is considered to be a benchmark means for solving stiff odes by many practitioners for some test problems

a 1d exact treatment of shock waves within spectral methods in plane geometry abstract we present a very exact numerical technique for solving 1d euler equations coupled with the transport equations for the entropy and the chemical abundances with or without shock formation two moving grids are used before and after the shock formation quantities are expanded on both sides of the matching point in chebychev polynomials series after the shock is formed rankine hugoniot conditions are used to determine the velocity of the shock and the matching conditions across the shock typical results are presented

existence of solutions for a class of third order nonlinear boundary value problems abstract in this paper we are concerned with the following third order ordinary differential equation x t f t x t x t x t 0 0 with the nonlinear boundary conditions x 0 0 g x 0 x 0 a h x 1 x 1 b where a b r f 0 1 r 3 r is continuous g h r 2 r are continuous the existence result is given by using a priori estimate nagumo condition upper and lower solutions and leray schauder degree and we give an example to demonstrate our result

efficient goal oriented global error estimators for bdf methods using discrete adjoints in this paper we develop new goal oriented global error estimators for variable multistep backward differentiation formulae bdf methods these estimators use for the first time in the context of multistep methods discrete adjoints computed by adjoint differentiation of the nominal integration scheme the derivation is based on the recently developed petrov galerkin finite element formulation of bdf methods and their discrete adjoint schemes and makes use of the dual weighted residual methodology defect integrals or local truncation errors are used as local error quantities we prove the asymptotic correctness and optimal convergence of the novel estimators for the one step bdf method for multistep bdf methods using a selfstarting procedure we show that the estimator with defect integrals converges but suboptimal and that its effectivity index converges to an offset value on the other hand the estimator with local truncation errors is again asymptotically correct and of optimal order we confirm these results numerically with a real world example from chemical engineering we give promising numerical results for the estimation accuracy in variable bdf type methods with changing orders and stepsizes finally we give a first use of the novel estimators for goal oriented global error control of a challenging stiff test problem

a novel fixed point iteration method for the solution of third order boundary value problems in this work a new alternative uniformly convergent iterative scheme is presented and applied for the solution of an extended class of linear and nonlinear third order boundary value problems that arise in physical applications the method is based on embedding green s functions into well known fixed point iterations including picard s and krasnoselskii mann s schemes convergence of the numerical method is proved by manipulating the contraction principle the effectiveness of the proposed approach is established by implementing it on several numerical examples including linear and nonlinear third order boundary value problems the results show highly accurate approximations when compared to exact and existing numerical solutions

existence result for a third order ode with nonlinear boundary conditions in presence of a sign type nagumo control abstract in this work we provide an existence and location result for the third order nonlinear differential equation u t f t u t u t u t where f a b r 3 r is a continuous function and two types of boundary conditions u a a u b u b 0 u a b or u a a u a u a 0 u b c with r 2 r continuous functions monotonous in the second variable and a b c r we assume that f satisfies a sign type nagumo condition which allows an asymmetric unbounded behaviour on the nonlinearity the arguments used concern leray schauder degree theory and lower and upper solutions technique

global error estimation and the backward differentiation formulas the way in which global error estimates might be used to control global error is discussed various global error estimation techniques are surveyed with special attention to defect correction in the discrete realm or difference correction and to those techniques that use a continuous approximation to the solution the problems of stiffness and rapidly varying stepsize is discussed for global error estimation for the bdfs and an unusual solution proposed

on global error estimation and control for initial value problems this paper addresses global error estimation and control for initial value problems for ordinary differential equations the focus lies on a comparison between a novel approach based on the adjoint method combined with a small sample statistical initialization and the classical approach based on the first variational equation control is achieved through tolerance proportionality both approaches are found to work well and to enable estimation and control in a reliable manner

on quasi consistent integration by nordsieck methods in this paper we study advantages of numerical integration by quasi consistent nordsieck formulas all quasi consistent numerical methods possess at least one important property for practical use which has not attracted attention yet i e the global error of a quasi consistent method has the same order as its local error this means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms in other words the global error control can be as cheap as the local error control in the methods under discussion here we apply the above mentioned idea to nordsieck adams moulton methods which are known to be quasi consistent moreover some nordsieck adams moulton methods are even super quasi consistent the latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods in turn this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control we also introduce implicitly extended nordsieck methods such that in some sense they form pairs of embedded formulas with their source nordsieck methods this facilitates the local error control in quasi consistent nordsieck schemes numerical examples presented in this paper confirm clearly the power of quasi consistent integration in practice

adaptive space time finite element methods for parabolic optimization problems in this paper we derive a posteriori error estimates for space time finite element discretizations of parabolic optimization problems the provided error estimates assess the discretization error with respect to a given quantity of interest and separate the influences of different parts of the discretization time space and control discretization this allows us to set up an efficient adaptive algorithm which successively improves the accuracy of the computed solution by construction of locally refined meshes for time and space discretizations

standardized pseudospectral formulation of the inviscid supsersonic blunt body problem a highly accurate pseudospectral numerical approximation to the generalized coordinate nonconservative form of the euler equations is implemented for supersonic flow over an axisymmetric blunt body geometry shock fitting is employed to maintain global accuracy and minimize the corrupting influence of numerical viscosity the variables in the euler equations as well as the physical grid coordinates are collocated via lagrange interpolating polynomials and the problem is then cast in the standard form of a large system of ordinary differential equations dx d q x which can be solved using standard solution techniques that do not require an explicit criteria for the minimum time step code verification is performed by demonstrating through a series of grid refinement tests that the error in the approximation to a taylor maccoll solution converges to 10 12 grid refinement tests for flow over a blunt body show convergence of the numerical error also to 10 12 the code is validated for supersonic flow over a blunt body by comparison with the modified newtonian approximation for the surface pressure distribution and empirical predictions for the shock shape the ability of the method to capture unsteady flow phenomena is demonstrated on the problem of a planar acoustic wave interacting with an attached shock

multi adaptive galerkin methods for odes i we present multi adaptive versions of the standard continuous and discontinuous galerkin methods for odes taking adaptivity one step further we allow for individual time steps order and quadrature so that in particular each individual component has its own time step sequence this paper contains a description of the methods an analysis of their basic properties and a posteriori error analysis in the accompanying paper a logg siam j sci comput submitted we present adaptive algorithms for time stepping and global error control based on the results of the current paper

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

doubly quasi consistent fixed stepsize numerical integration of stiff ordinary differential equations with implicit two step peer methods abstract the problem of accurate and efficient numerical integration of ordinary differential equations odes is a key issue of computation science which has a valued practical impact in various topics of research in this respect effective global error estimation and control strategies are of particular theoretical and practical significance they allow many applied dynamic systems to be treated numerically for user supplied accuracy conditions in automatic mode the property of double quasi consistency plays an outstanding role in the mentioned topic because the global or true error of a doubly quasi consistent method is asymptotically equal to its local error therefore this entails quite a nontrivial implication that the true error of numerical integration can be cheaply evaluated and regulated within a single solution run the latter was considered to be impossible for decades the property of double quasi consistency goes back to 2009 when kulikov studied it in the first time but failed to find doubly quasi consistent numerical methods among nordsieck formulas later on kulikov and weiner proved the existence of doubly quasi consistent numerical schemes among superconvergent explicit two step parallel peer methods and examined their practical efficiency in comparison to explicit matlab ode solvers in 2010 the focus of the present research is on accurate numerical integration formulas for treating stiff odes which often arise in practice and for which explicit methods are shown to be ineffective here we make the first step towards an accurate and efficient numerical solution of stiff odes and prove existence of implicit doubly quasi consistent formulas we fulfill our investigation of double quasi consistency within the family of fixed stepsize implicit two step peer schemes and construct two methods of convergence orders 3 and 4 which possess excellent stability properties our theoretical results are confirmed with numerical integration of both nonstiff and stiff test problems in this paper

spectral methods for the euler equations the blunt body problem revisited we solve the blunt body problem using shock fitting and a chehyshev spectral collocation method careful attention is paid to the boundary and shock acceleration equations

thirteen ways to estimate global error various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations particularly ordinary differential equations are classified described and compared for most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

analysis of error control strategies for continuous runge kutta methods there has been considerable recent progress in the analysis and development of interpolation schemes that can be associated with discrete runge kutta methods with the availability of these schemes it can now be asked that a numerical method provide a continuous approximation to the solution thispaper rather than view such a continuous method as an interpolant superimposed on a standard discrete method considers how the interpolant and its associated defect can be effectively used in the underlying error and stepsize control mechanism in particular four error control strategies are considered that can be used in methods based on runge kutta formula pairs with interpolants an asymptotic and a nonasymptotic analysis of each strategy are presented it is shown that a strategy based on direct defect control can provide significant advantages over existing strategies with only a modest increase in cost

a two dimensional time dependent chebyshev method of collocation for the study of astrophysical flows abstract we present a two dimensional time dependent collocation chebyshev spectral method as a tool for the solution of certain astrophysical flows in this paper we solve the navier stokes equations in spherical coordinates the problem of the propagation of an initial perturbation in an atmosphere of a compact star is used to demonstrate the method and its power in particular viscosity and rotation are included

numerical investigations on global error estimation for ordinary differential equations four techniques of global error estimation which are richardson extrapolation rs zadunaisky s technique zd solving for the correction sc and integration of principal error equation ipee have been compared in different integration codes dopri5 dvode dstep theoretical aspects concerning their implementations and their orders are first given second a comparison of them based on a large number of tests is presented in terms of cost and precision sc is a method of choice for one step methods it is much more precise and less costly than rs and leads to the same precision as zd for half its cost ipee can provide the order of the error for a cheap cost in codes based on one step methods in multistep codes only rs and ipee have been implemented since they are the only ones whose theoretical justification has been extended to this case there rs still provides a more reliable estimation than ipee however as these techniques are based on variations of the global error irrespective of the numerical method used they fail to provide any more usefull information once the numerical method has reached its limit of accuracy due to the finite arithmetic

existence and uniqueness of solutions for third order nonlinear boundary value problems in this paper we present some general results of existence and uniqueness of solutions of nonlinear two point boundary value problems for third order nonlinear differential equations by using the shooting method as applications we give certain concrete sufficient conditions for the existence and uniqueness

a variational principle for adaptive approximation of ordinary differential equations a variational principle inspired by optimal control yields a simple derivation of an error representation global error local error weight for general approximation of functions of solutions to ordinary differential equations this error representation is then approximated by a sum of computable error indicators to obtain a useful global error indicator for adaptive mesh refinements a uniqueness formulation is provided for desirable error representations of adaptive algorithms

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

unstructured grid method for stress wave propagation in elastic media a new numerical method named unstructured grid method is presented to calculate two dimensional stress wave propagation problems in elastic media this method is based on the dynamic equilibrium equations of the investigated lumps that are formed among the auxiliary unstructured grids the approach to the solution is obtained by calculating the node displacements and the central point stresses of the spatial grids alternately the stability criterion of the method is given and tested numerically analysis of dispersion for numerical phase velocity is studied the rule is provided to determine the spatial step size of discretization mesh so as to ensure the numerical accuracy of the unstructured grid method numerical results are compared with the available exact solutions and with the solutions of central difference method for demonstrating the good accuracy of the new method the unstructured grid method possesses higher calculating speed than the central difference method finally the dynamic behavior of a square plate with a hole under an impact loading is researched

global error estimation with runge kutta triples abstract the applications of runge kutta rk interpolation to global error estimation using the zadunaisky and related techniques are considered it is shown that the pseudo problem can be based on dense output values within any one step and reliable global error estimates can be obtained at the integration mesh points by using special rk formulae some special formulae of orders 2 6 are presented together with numerical results

variable stepsize interpolating explicit parallel peer methods with inherent global error control in this paper we present variable stepsize explicit parallel peer methods grounded in the interpolation idea approximation stability and convergence are studied in detail in particular we prove that some interpolating peer methods are stable on any variable mesh in practice double quasi consistency is utilized to introduce an efficient global error estimation formula in the numerical methods under discussion the main advantage of these new adaptive schemes is the fact that the leading terms of their local and true errors coincide thus controlling the local error of such methods by cheap standard techniques automatically regulates their global error as well numerical experiments of this paper support theoretical results presented below and illustrate how the new global error control concept works in practice we also conduct a comparison with explicit ode solvers in matlab

implicit boundary equations for conservative navier stokes equations abstract co existence of the physical and numerical boundary conditions makes implicit boundary treatment a particularly difficult problem in modern cfd simulations previous studies adopted space time extrapolation or specially designed partial differential equations on the boundaries that are different from those of interior points they are often formulated in terms of primitive variables and are very challenging for complicated boundary types to be converted to and implemented in the conservative variables that are preferred in numerical simulations more importantly different boundary equations or different extrapolation techniques may compromise the stability accuracy or convergence rate of the a stable schemes that are developed for interior points a new methodology for implicit boundary treatment is proposed in this study by introducing a simple correction matrix t a set of generalized equations q t i t r are developed in terms of conservative variables it is applicable for both the interior domain t 0 and the boundaries t 0 it is in a partial differential equation form satisfies the boundary conditions accurately independent of the time and spatial discretizations any one sided schemes can be used on the boundaries but still maintain the upwind property implicit solution techniques are made significantly easy to implement using for example the data parallel lower upper relation method and the newton method combined with the gmres method for subsidiary iterations numerical experiments show that the proposed methodology produces stable simulations for very large cfl numbers and preserve the imposed boundary values accurately

on the zero stability of the variable order variable stepsize bdf formulas this paper contains bounds for the mesh size ratiosh j 1 h j of nonequidistant grids such that any combination of the widely used backward differentiation formulas with ordersl 1 l m m 3 4 5 performs zerostable

on noncharacteristic boundary conditions for discrete hyperbolic initial boundary value problems abstract the stability properties of some discrete hyperbolic initial boundary value problems involving two dependent variables and noncharacteristic boundary conditions are examined approximations based on the lax wendroff leap frog and crank nicolson methods are considered the most accurate approximations are found to be the least stable the results for the model problems can be used in the stability analysis of larger problems where characteristics occur singly or in pairs in particular discretizations of the shallow water and gas dynamics equations which involve boundary conditions written in natural variables are considered some computations demonstrating the stability results are discussed an appendix discussing numerical stability verification techniques is included

cubature kalman filters in this paper we present a new nonlinear filter for high dimensional state estimation which we have named the cubature kalman filter ckf the heart of the ckf is a spherical radial cubature rule which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear bayesian filter specifically we derive a third degree spherical radial cubature rule that provides a set of cubature points scaling linearly with the state vector dimension the ckf may therefore provide a systematic solution for high dimensional nonlinear filtering problems the paper also includes the derivation of a square root version of the ckf for improved numerical stability the ckf is tested experimentally in two nonlinear state estimation problems in the first problem the proposed cubature rule is used to compute the second order statistics of a nonlinearly transformed gaussian random variable the second problem addresses the use of the ckf for tracking a maneuvering aircraft the results of both experiments demonstrate the improved performance of the ckf over conventional nonlinear filters

global error estimation and control in linearly implicit parallel two step peer w methods the class of linearly implicit parallel two step peer w methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations those schemes allow for parallelism across the method that is an important feature for implementation on modern computational devices most importantly all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration this property results in the fact that no order reduction occurs when they are applied to very stiff problems in this paper we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user supplied accuracy requirement in automatic mode an algorithm of such global error control and other technical particulars are also discussed here numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions including nonstiff stiff very stiff and large scale differential equations a comparison with the well known stiff solver rodas is also shown

local and global error estimation and control within explicit two step peer triples this paper elaborates a global error estimation and control mechanism in explicit two step peer methods these recently designed methods exhibit their high efficiency even in comparison to the best explicit runge kutta pairs more precisely we form here triples of the so called superconvergent explicit peer schemes and show that they are cheap and able to achieve preassigned accuracy conditions in automatic mode for comparison we present also numerical data derived by built in explicit matlab ode solvers implemented with only local error control especially we point out that a scaled global error is computed and regulated in this paper in contrast to the earlier published results where the absolute values of the global error have been utilized

an accuracy analysis of boundary conditions for the forced shallow water equations the accuracy of numerical boundary conditions for the linearized one dimensional shallow water equations is determined by extending the dispersion fourier analysis originally presented by c k chu and a sereny j comput phys 15 476 1974 accuracy is measured by calculating steady state amplitudes of incoming and outgoing numerical waves at a boundary as functions of a forcing frequency radiating closed and forced radiating boundary conditions are studied in combination with two numerical schemes the richardson sielecki explicit finite difference scheme and the galerkin finite element method with piecewise linear basis functions and crank nicolson time stepping contamination of the finite element solution with short waves is seen to vary with the choice of extraneous boundary conditions an example of l n trefethen s j comput phys 49 199 1983 interpretation of the b gustafsson h kreiss and a sundstrom math comput 26 649 1972 instability is also given for the finite element method

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

tracking discontinuities in hyperbolic conservation laws with spectral accuracy it is well known that the spectral solutions of conservation laws have the attractive distinguishing property of infinite order convergence also called spectral accuracy when they are smooth e g c canuto m y hussaini a quarteroni t a zang spectral methods for fluid dynamics springer verlag heidelberg 1988 j p boyd chebyshev and fourier spectral methods second ed dover new york 2001 c canuto m y hussaini a quarteroni t a zang spectral methods fundamentals in single domains springer verlag berlin heidelberg 2006 if a discontinuity or a shock is present in the solution this advantage is lost there have been attempts to recover exponential convergence in such cases with rather limited success the aim of this paper is to propose a discontinuous spectral element method coupled with a level set procedure which tracks discontinuities in the solution of nonlinear hyperbolic conservation laws with spectral convergence in space spectral convergence is demonstrated in the case of the inviscid burgers equation and the one dimensional euler equations

existence and uniqueness for two point boundary value problems boundary value problems associated with y f x y y for 0 x 1 are considered using techniques based on the shooting method conditions are given on f x y y which guarantee the existence on 0 1 of solutions of some initial value problems working within the class of such solutions conditions are then given on nonlinear boundary conditions of the form g y 0 y 0 0 h y 0 y 0 y 1 y 1 0 which guarantee the existence of a unique solution of the resulting boundary value problem

spectral methods for the euler equations ii chebyshev methods and shock fitting on exprime chaque composante de la vitesse dans les equations d euler par une somme de polynomes de tchebycheff qui constitue une fonction d interpolation on utilise les expressions spectrales ainsi obtenues pour calculer plusieurs ecoulements gazeux bidimensionnels par la methode d ajustement d ondes de choc selon cette methode les conditions initiales correspondent a une onde de choc plane normale a un ecoulement supersonique et on determine la forme veritable de l onde a l aide des formules de rankine hugoniot qui expriment les conditions aux limites

improved boundary conditions for viscous reacting compressible flows previous studies on physical boundary conditions for flame boundary interactions of an ideal multicomponent compressible gas have neglected reactive source terms in their boundary condition treatments by combining analyses of incompletely parabolic systems with those based on the hyperbolic euler equations a rational set of boundary conditions is determined to address this shortcoming accompanying these conditions is a procedure for implementation into a maltidimensional code in the limits of zero reaction rate or one species the boundary conditions reduce in a predictable way to cases found in the literature application is made to premixed and nonpremixed flames in one and two dimensions to establish efficacy inclusion of source terms in boundary conditions derived from characteristic analysis is essential to avoid unphysical generation of pressure and velocity gradients as well as flow reversals minor deficiencies in the boundary conditions are attributed primarily to the diffusive terms imposing vanishing diffusive boundary normal flux gradients works better than imposing vanishing fluxes but neither is entirely satisfactory

solvability of some third order boundary value problems with asymmetric unbounded nonlinearities abstract in this paper we present existence and location results for the third order separated boundary value problems u t f t u t u t u t with the boundary conditions u a a u a b u b c or u a a c 1 u a c 2 u a b c 3 u b c 4 u b c with c 1 c 2 c 3 c 4 r and a b c r we assume f a b r 3 r is a continuous function satisfying one sided nagumo type condition which allows an asymmetric unbounded behaviour the arguments used concern leray schauder degree and lower and upper solution techniques

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations