a class of bidiagonal schemes for solving the euler equations a class of implicit bidiagonal schemes is developed for the conservative euler equations the schemes are based on a predictor corrector approach for the characteristic variables adapted according to the sign of the characteristic speeds of information they are second order accurate in space and time and have intrinsic dissipative properties a new fully implicit boundary treatment is developed for this class of schemes allowing high cfl numbers in all regions of the flow domain results are presented for the quasi one dimensional flow in a nozzle n recent years substantial progress has been made in the development of numerical methods for solving conservative systems of hyperbolic equations such as the euler equations describing inviscid fluid flow m one of the main achievements has been the development of implicit methods based on tridiagonal solution procedures 2 the use of explicit impl icit artificial viscosity5 and of implicit boundary conditions6 7 has made these algorithms very robust and efficient recently an implicit method based on a bidiagonal solution technique has been introduced by maccormack 1 the first part of this paper is concerned with further developments related to the bidiagonal solution technique a general class of second order accurate bidiagonal schemes is proposed containing a class of implicit schemes and one of semiexplicit schemes it is shown that these schemes are in trinsically dissipative so that no artificial viscosity is necessary for linear stability the second part of the paper introduces an implicit boundary treatment for these bidiagonal schemes that preserves the stability properties of the interior scheme in deed while a number of successful boundary procedures have been proposed for explicit and tridiagonal implicit schemes this stability remained an unsolved problem in earlier ap plications of bidiagonal techniques 8 9 numerical experiments for the euler flow in a nozzle confirm the unconditional linear stability allowing the use of very high cfl numbers

an implicit bidiagonal numerical method for solving the navier stokes equations in recent years much progress has been made in solving fluid dynamical problems using finite difference methods solving inviscid compressible problems in two and three dimensions has become almost routine with many suitable methods explicit or implicit available the problem of compressible viscous flows in complicated geometries remains however a major challenge here fine mesh spacing in the viscous flow regions makes the explicit methods with their simple boundary conditions extremely costly existing implicit methods can make use of large time steps but require inversions of large block tridiagonal matrices a method recently developed by maccormack eliminates this disadvantage by introducing a predictor corrector scheme requiring the inversion of only block bidiagonal matrices it is the aim of present work to extend this method to allow solution of viscous compressible problems in general coordinates for arbitrary two dimensional geometries

characteristic boundary conditions for the two step taylor galerkin fem a general framework for implementing numerical boundary conditions based upon the rigorous application of characteristic theory has been developed for the two step taylor galerkin fem scheme the method consists of solving the compatibility equations for the temporal change in characteristic variables using the first step of the two step taylor galerkin fem this application of characteristic boundary conditions is consistent with the spatial and temporal discretization of the two step taylor galerkin fem it is ideal for domains discretized with linear unstructured finite elements as time and space extrapolation from interior elements is not required boundary conditions are constructed from the characteristic solution for solid wall and symmetry boundaries flow exits and flow inlets two simulations are shown highlighting the performance of these boundary conditions

grid generation techniques in computational fluid dynamics revue des differentes methodes on montre que les regions tridimensionnelles sont mieux traitees par fragmentation du champ en sous regions avec generation de grilles dans chaque sous region interet des grilles adaptatives couplees a des solutions physiques

passivity preserving model reduction via interpolation of spectral zeros abstract an algorithm is developed for passivity preserving model reduction of linear time invariant systems implementation schemes are described for both medium scale dense and large scale sparse applications the algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros these interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain block matrix which has the spectral zeros as its spectrum explicit interpolation is avoided and passivity of the reduced model is established instead through satisfaction of the necessary conditions of the positive real lemma it is also shown that this procedure indirectly solves the associated controllability and observability riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations from these a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable

finite volume solutions of convection diffusion test problems the cell vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics more recently one of us has proposed an extension for viscous flows the purpose of the present paper is two fold first we have applied this scheme to a well known convection diffusion model problem involving flow round a 180 degrees bend which highlights some of the issues concerning the application of the boundary conditions in such cell based schemes the results are remarkably good when the boundary conditions are applied in an appropriate manner in our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one dimensional problem our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra convergence phenomena enjoyed by the proposed methods 25 refs 9 figs 1 tab

balanced parametrization of classes of linear systems canonical forms and parametrizations are presented for several sets of minimal systems of given dimension asymptotically stable systems allpass systems bounded real systems positive real systems minimum phase systems and the class of all minimal systems the approach is based on balancing techniques for these classes of systems applications of these results to hankel operators and model reduction are discussed

a penalty function free line search sqp method for nonlinear programming we propose a penalty function free non monotone line search method for nonlinear optimization problems with equality and inequality constraints this method yields global convergence without using a penalty function or a filter each step is required to satisfy a decrease condition for the constraint violation as well as that for the objective function under some reasonable conditions the proposed mechanism for accepting steps also combines the non monotone technique on the decrease condition for the constraint violation which leads to flexibility and an acceptance behavior comparable with filter based methods furthermore it is shown that the proposed method can avoid the maratos effect if the search directions are improved by second order corrections soc so locally superlinear convergence is achieved we also present some numerical results which confirm the robustness and efficiency of our approach

non oscillatory and non diffusive solution of convection problems by the iteratively reweighted least squares finite element method in this paper we first introduce the least squares finite element method lsfem for two dimensional steady state pure convection problems with smooth solutions and compare the lsfem with other finite element methods we prove that the lsfem has the same stability estimate as the original equation i e the lsfem has better control of the streamline derivative than the streamline upwinding petrov galerkin method numerical convergence rates are given to show that the lsfem is almost optimal then we use this lsfem as a framework to develop an iteratively reweighted least squares finite element method irlsfem to obtain non oscillatory and non diffusive solutions for problems with contact discontinuities this new method produces a highly accurate numerical solution that has a sharp discontinuity in one element a number of examples solved by using triangular and bilinear elements are presented to show that the method can convect contact discontinuities without error

application of time iterative schemes to incompressible flow the computation of steady incompressible flows by an euler implicit algorithm is studied using both the incompressible equations and the low mach number compressible equations the incompressible equations are handled by adding an artificial time derivative to the continuity equation this allows both the pressure and velocity to be obtained implicitly in one dimensional problems both systems converge rapidly even at low mach numbers where the eigenvalues are very stiff in two dimensions where approximate factorization is required the presence of stiff eigenvalues is highly detrimental stiffness can be avoided in the incompressible equations by selecting an appropriate pseudo mach number this insures reliable convergence and results in an efficient incompressible flow algorithm in the case of compressible equations the mach number cannot be chosen arbitrarily and the contamination introduced by approximate factorization must be removed a matrix preconditioning factor that accomplishes this is developed and demonstrated with this modification the convergence rate is the same as in the incompressible case and is independent of mach number rapid convergence is observed at mach numbers as low as 6 05

numerical boundary condition procedures for euler solvers a numerical boundary condition procedure for euler solvers is presented the procedure is based on a variation of the method of characteristi cs due to kentzer a second order a ccurate numerical algorithm using this procedure and the maccormack explicit finite difference method is presented results are presented for several two dimensional and three dimensional inviscid nozzle flowfields solutions obtained by the present method agree well with solutions obtained by method of characteristi cs algorithms and experimental data

eliminating gibbs phenomena a non linear petrov galerkin method for the convection diffusion reaction equation abstract in this article we consider the numerical approximation of the convection diffusion reaction equation one of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection dominated case can lead to non physical oscillations in the numerical approximation often referred to as gibbs phenomena the idea of this article is to consider the approximation problem as a residual minimization in dual norms in l q type sobolev spaces with 1 q we then apply a non standard non linear petrov galerkin discretization that is applicable to reflexive banach spaces such that the space itself and its dual are strictly convex similar to discontinuous petrov galerkin methods this method is based on minimizing the residual in a dual norm replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non linear inexact mixed method this generalizes the petrov galerkin framework developed in the context of discontinuous petrov galerkin methods to more general banach spaces for the convection diffusion reaction equation this yields a generalization of a similar approach from the l 2 setting to the l q setting a key advantage of considering a more general banach space setting is that in certain cases the oscillations in the numerical approximation vanish as q tends to 1 as we will demonstrate using a few simple numerical examples

surface boundary conditions for the numerical solution of the euler equations we consider the implementation of boundary conditions at solid walls in inviscid euler solutions by upwind finite volume methods we review some current methods for the implementation of surface boundary conditions and examine their behavior for the problem of an oblique shock reflecting off a planar surface we show the importance of characteristic boundary conditions for this problem and introduce a method of applying the classical flux difference splitting of roe as a characteristic boundary condition consideration of the equivalent problem of the intersection of two equal and opposite oblique shocks was very illuminating on the role of surface boundary conditions for an inviscid flow and led to the introduction of two new boundary condition procedures denoted as the symmetry technique and the curvature corrected symmetry technique examples of the effects of the various surface boundary conditions considered are presented for the supersonic blunt body problem and the subcritical compressible flow over a circular cylinder dramatic advantages of the curvature corrected symmetry technique over the other methods are shown with regard to numerical entropy generation total pressure loss drag and grid convergence

application of the implicit maccormack scheme to the parabolized navier stokes equations application de la methode de differences finies implicite de maccormack a la resolution des equations bidimensionnelles parabolisees de navier stokes discussion des avantages et inconvenients de la methode comparaison avec le schema de beam warming en ce qui concerne la precision la stabilite le temps de calcul la place en memoire et la facilite d implementation application a un ecoulement laminaire hypersonique sur un coin de compression

space marching method for calculating steady supersonic flows on a grid adapted to the solution a noniterative implicit space marching method based on an adaptive grid approach is developed for solving the 2d steady state euler equations describing supersonic gas flows without streamwise separation a grid adapted to the solution is generated by using a variational method optimizing such important properties of a grid as smoothness orthogonality and grid cell volume variation simultaneously the weight function in the integral of adaptation is constructed so that the grid lines are accumulated near strong gradients and curvatures of the solution curve a special treatment of the boundary points is proposed to determine the location of nodes near slope discontinuities of the boundary mesh line the noniterative implicit space marching method of y c vigneronet al is applied to solve the euler equations written in an arbitrary curvilinear system of coordinates the streamwise and transverse derivatives in the governing equations are approximated by the second order upwind richardson and second order symmetric tvd schemes respectively implicit boundary condition procedure based on the theory of characteristics for hyperbolic systems of equations is employed numerical calculations show that the resolution of strong gradient flow fields can significantly be improved by using the grid adapted to the solution while the number of grid nodes is the same

a finite element technique for solving first order pdes in lp an approximation technique for solving first order pdes in l p omega 1 le p infty is proposed the method is a generalization of the least squares method to non hilbertian settings a priori and a posteriori error estimates are proven numerical tests in l 1 omega show that this type of technique can handle discontinuities without resorting to limiting procedures

line search filter inexact secant methods for nonlinear equality constrained optimization we present inexact secant methods in association with line search filter technique for solving nonlinear equality constrained optimization for large scale applications it is expensive to get an exact search direction and hence we use an inexact method that finds an approximate solution satisfying some appropriate conditions the global convergence of the proposed algorithm is established by using line search filter technique the second order correction step is used to overcome the maratos effect while the line search filter inexact secant methods have superlinear local convergence rate finally the results of numerical experiments indicate that the proposed methods are efficient for the given test problems

boundary conditions for the solution of compressible navier stokes equations by an implicit factored method a method is presented for formulating the boundary conditions in implicit finite difference form needed for obtaining solutions to the compressible navier stokes equations by the beam and warming implicit factored method the usefulness of the method was demonstrated a by establishing the boundary conditions applicable to the analysis of the flow inside an axisymmetric piston cylinder configuration and b by calculating velocities and mass fractions inside the cylinder for different geometries and different operating conditions stability selection of time step and grid sizes and computer time requirements are discussed in reference to the piston cylinder problem analyzed

upwind compact finite difference schemes it was shown by ciment leventhal and weinberg j comput phys 28 1978 135 that the standard compact finite difference scheme may break down in convection dominated problems an upwinding of the method which maintains the fourth order accuracy is suggested and favorable numerical results are found for a number of test problems

the effect of strong heat addition on the convergence of implicit schemes the effect of substantial amounts of heat addition on the convergence characteristics of implicit algorithms for flow in a converging diverging nozzle is investigated heat addition has an adverse effect on convergence because it decreases the mach number at the inlet introduces a strong energy source term and produces steep temperature gradients the nozzle convergence rate has analogous detrimental effects and is also studied comparisons of both variations show that when equivalent procedures are used beam and warming s algorithm converges for somewhat wider ranges of parameters than does maccormack s the combination of local time steps which provide uniform values of cfl throughout the grid boundary conditions based on the method of characteristics and euler implicit differencing in time provides convergence that is nearly independent of heat addition and is in close agreement with the predictions of stability theory the incorporation of real gas effects was tested and shown to have a minor effect

effect of grid aspect ratio on convergence the effects of high aspect ratio grids on algorithm convergence are considered by means of vector stability theory and computational experiments the results indicate that approximately factored implicit schemes experience convergence deterioration because of nonoptimum local time stepping procedures and increased need for viscous preconditioning based on this insight an enhanced algorithm is devised using improved selection of the local time step appropriate definition of the viscous preconditioning matrix and proper implementation of the boundary conditions the new algorithm provides uniformly efficient convergence at all aspect ratios for both euler and navier stokes computations for a variety of test problems

the influence of differencing and cfl number on implicit time dependent non linear calculations abstract this paper is concerned with the stability and precision of transient implicit finite difference algorithms the test problem is the burgers equation the influence of space discretisation and initial values is studied linear analysis provides incomplete stability conditions for both continuous and weak solutions and we derive a simple criterion of augmented stability taking into account the non linearity of the problem the real stability limits appear to be a combination of the linear stability analysis criterion and of this augmented stability criterion

local convergence of a trust region algorithm with line search filter technique for nonlinear constrained optimization a trust region algorithm in association with line search filter technique for solving nonlinear equality constrained programming is proposed in this paper in the current iteration the trial step providing sufficient descent is generated by solving a corresponding trust region subproblem then the step size is decided by backtracking line search together with filter technique to obtain the next iteration point the advantage of this method is that resolving trust region subproblem many times to determine a new iteration point in traditional trust region method can be avoided and hence the expensive computation can be lessened and the difficult decisions in regard to the choice of penalty parameters in the merit functions can be avoided by using filter technique second order correction steps are introduced in the proposed algorithm to overcome maratos effect convergence analysis shows that fast local convergence can be achieved under some mild assumptions the preliminary numerical results are reported

viscous shock layer flowfield analysis by an explicit implicit method the present work extends the recently reported implicit analogue of maccormack s earlier widely used explicit method to external axisymmetric laminar flows with strong entropy gradients the details of the numerics of the implicit part are provided in a body oriented coordinate system with a moving outer shock boundary during the transient part of the solutions the limiting values of the courant number are obtained when the shock boundary is treated explicitly the solution algorithm outlined includes the treatment of the source term associated with the equations in weak conservation form from the results obtained for two sample problems it becomes clear that accuracy of predictions is indeed very good at higher values of the courant number there is a significant saving in overall computing time depending on the courant number used and the flow reynolds number these properties combined with the simplicity of programming the implicit analogue may appeal to researchers for using it in the analysis of three dimensional flow problems

model order reduction with preservation of passivity non expansivity and markov moments a new model order reduction technique is presented which preserves passivity and non expansivity it is a projection based method which exploits the solution of linear matrix inequalities to generate a descriptor state space format which preserves positive realness and bounded realness in the case of both non singular and singular systems solving the linear matrix inequality can be replaced by equivalently solving an algebraic riccati equation which is known to be a more efficient approach a new algebraic riccati equation and a frequency inversion technique are also presented to specifically deal with the important singular case the preservation of markov moments is also guaranteed by the judicious choice of a projection matrix three pertinent examples comparing the present approach with positive real balanced truncation show the strength and accuracy of the present approach

a finite difference method for solving unsteady viscous flow problems a method has been developed to solve the unsteady compressible navier stokes equations with the property of consistency and the ability of minimizing the equation stiffness it relies on innovative extensions of the stateof the art finite difference techniques and is composed of 1 the upwind scheme for split flux and the central scheme for conventional flux terms in the inviscid and viscous regions respectively 2 the characteristic treatment of both inviscid and viscous boundaries 3 an adi procedure compatible with interior and boundary points 4 a scalar matrix solver including viscous terms the performance of this method is assessed with three sample problems a shock reflected from the wall the shock induced boundary layer separation and a transient internal nozzle flow the results from the present method an existing hybrid block method and a well known two step explicit method are compared and discussed it is concluded that this method has an optimal tradeoff between the solution accuracy and computational economy and other desirable properties for analyzing transient viscous flow problems

dynamical approach study of spurious steady state numerical solutions of nonlinear differential equations i the dynamics of time discretization and its implications for algorithm development in computational fluid dynamics abstract the goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady state numerical solutions when using the time dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms this interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as the dynamics of numerics and the numerics of dynamics at the present time this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications this is the first of a series of research papers under the same topic our hope is to reach researchers in the fields of computational fluid dynamics cfd and in particular hypersonic and combustion related cfd by simple examples in which the exact solutions of the governing equations are known the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results one striking phenomenon is that with the same initial data the continuum and its discretized counterpart can asymptotically approach different stable solutions this behavior is especially important for employing a time dependent approach to the steady state since the initial data are usually not known and a freestream condition or an intelligent guess for the initial conditions is often used with the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart it is not easy to delineate the true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum part i concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in cfd and to serve as an introduction for parts 11 and iii of the same series of research papers

computation of low speed flow with heat addition a perturbation expansion is used to obtain a system of conservation laws for compressible flows that is valid at arbitrarily low mach numbers these equations are rendered hyperbolic by adding an artificial time derivative to the energy equation thus introducing pseudoacoustic waves with a speed the same order as the particle velocity traditional time iterative schemes are shown to be effective in solving this system numerically stability calculations of the complete vector system indicate unconditional stability at all mach numbers in the absence of gravity but the source term introduced by buoyancy becomes destabilizing at froude numbers below about 1 this instability is amplified by approximate factorization thus precluding solutions with gravity below this mach number level computations of strong heat addition in low mach number flow both with and without gravity confirm the stability predictions convergent solutions are maintained and are used to show the effect of gravity on such flowfields

near wall k e computation of transonic turbomachinery flows with tip clearance acomputational methodforthenumericalintegration ofthefavre reynolds averaged three dimensionalcompressible navier stokes equations in axial turbomachinery using the launder sharma near wall k turbulence closure is developed the mean e ow and turbulence transport equations are discretized using a e nite volume method based on muscl van leere ux vector splitting with van albada limiters and are integrated in time using a fully coupled approximately factored implicit backward euler method the resulting scheme is robust and was found stable for local time steps of courant friedrichs lewy number cfl 20 the computational domain is discretized using a basic h o h grid the tip clearance gap is discretized using a e ne o type grid the radial distribution ofnodeswithin thetip clearancegap is independent of theblade rowo grid and a bufferoverlap grid is used to convey information boundary conditions at periodicity boundaries and at domain interfaces are treated using e ve phantom nodes this procedure ensures stability at high cfl results are presented for the nasa 37 rotor at an operating point near surge computations are compared with measurements both for blade to blade mach number contours and pitchwise distributions and for radial distributions downstream of the blades results are obtained using three grids of 10 6 2 10 6 and 3 10 6 points with 21 31 and 41 radial stations within the tip clearancegap respectively demonstratingthatresultsaregridindependent comparisonwithmeasurementsis satisfactory with theexception of pressureratio overestimation due to unsatisfactory prediction ofe ow separation by the turbulence model

second order accurate boundary conditions for compressible flows the accuracy and the consistency of the methods commonly used for the numerical application of the boundary conditions on the surface of solid bodies immerged in an inviscid compressible flow is examined a new method is proposed for the application of second order accurate boundary conditions in the case of isentropic irrotational compressible flows the method is applied to a circular cylinder in uniform subsonic and low transonic flows the conservation law equations are solved using the second order accurate difference scheme of maccormack the results indicate a definite improvement over lower accuracy methods the method can be easily generalized to apply on three dimensional solid surfaces

synthesis of positive real multivariable feedback systems this is a tutorial paper describing the application of infinity norm optimal control theory to the synthesis of sector bounded and positive real multivariable feedback systems the paper gives a concise treatment of the practical aspects of the theory describing how it has been implemented via state space oriented computer algorithms included is a concise summary of the state space approach to infinity norm control developed by doyle and by safonov and verma based on the hankel model reduction theory and incorporating the multivariable hankel results of glover

a new penalty free type algorithm based on trust region techniques abstract in this paper we propose a new penalty free type method for nonlinear equality constrained problems the new algorithm uses trust region framework and feasibility safeguarding technique moreover it has no choice of penalty parameter and penalty function as a merit function and it does not use the filter technique to avoid the penalty function either we analyze the global convergence of the main algorithm under the standard assumptions the preliminary numerical tests are reported

implicit conservative schemes for the euler equations a new approach to a characteristic modeling scheme is presented that does not require the governing equations to be written in characteristi c variables or the flux terms to be split into positive and negative parts the method is based on the observation that for certain finite volume schemes the upwind influence can be accounted for through the conditions applied at the boundaries of individual cells instead of flux differencing the method is developed and applied to the one dimensional nozzle flow equations for subsonic transonic and supersonic flows and is then extended to two dimensions using an approximate factorization for the problem of a shock wave reflecting from a flat plate the present solution algorithm reduces the computational work per time step by 40 of that required by standard central difference implicit calculations i introduction n umerical methods that attempt to model the physics of the differential equations being approximated by properly accounting for the directions in which signals are propagated can be called characteristic modeling schemes the standard approach to characteristic modeling schemes is through the implementation of upwind schemes see refs 1 7 to devise an upwind scheme the primative fluxes are split into contributions whose jacobians have eigenvalues that are either 0 or 0 respectively at every point on the numerical mesh these fluxes are then differenced either upwind or downwind of that point

the convergence rate for difference approximations to general mixed initial boundary value problems the convergence rate is investigated for the solutions to difference schemes approximating the mixed initial boundary value problem for general systems of differential equations it is shown that if an energy estimate holds then the extra boundary conditions can be of one order lower accuracy without destroying the convergence rate expected from the approximation at inner points if the maximal order of the derivatives occurring in the boundary conditions is low enough then even lower accuracy can be permitted for the extra boundary conditions

nonoscillatory solution of the steady state inviscid burgers equation by mathematical programming abstract numerical solution of the steady state inviscid burgers equation u2 0 on 0 1 u 0 g0 u 1 g1 g0 g1 is considered to obtain the physically relevant discontinuous solution of this problem the equation is singularly perturbed by adding a small amount of viscosity u u2 0 a cell entered finite difference scheme that uses two points for the inviscid part and four points for the viscous part is proposed the overdetermined nonlinear system that results is solved by a homotopy based on the newton linearizer and a linear programming algorithm used to solve l problems difficulties in capturing interior layers centered at node points are observed however the computational results for interior layers centered between node points and for boundary layers show accurate nonoscillatory solutions with discontinuities captured in one cell on both coarse and fine grids the amount of artificial viscosity required to obtain these solutions is small 10 9 compared to the amount of artificial viscosity required by standard methods a partial proof of pointwise and l1 0 1 convergence of the linear spline interpolant of the numerical solution to the inviscid solution h 0 and h 0 is given for one case

upwind iteration methods for the cell vertex scheme in one dimension this paper describes and analyses a series of methods for solving the algebraic equations obtained from the cell vertex finite volume discretisation in one dimension the objective is to explore the possibilities for improved iteration methods that may be applied to cell vertex discretisations of the navier stokes equations in higher dimensions in general there is no natural one to one correspondence between cellbased residuals and nodal unknowns for this system in order to devise iteration schemes it is therefore necessary to provide a mapping between cells and nodes the family of methods introduced here is based on the application of standard iterative techniques to a nodal residual formed of a combination of neighbouring cell based residuals it includes the familiar lax wendroff iteration upwind iteration schemes and marching schemes capable of attaining convergence rates independent of the number of algebraic equations the aim in each case is to set to zero the residual for each cell apart from exceptional cells such as those containing shocks the final results show that matrix based upwind iteration methods using cell residuals modified to take account of critical points and applying several local iterations converge in around 15 iterations

an implicit form for the osher upwind scheme conservative upwind schemes for the euler equations such as the osher scheme accurately resolve flow discontinuities and correctly model the physics of the problem however these schemes require many more arithmetic operations per integration step than simple central difference schemes and hence result in large computing times an implicit version of the first order and second order accurate osher schemes in two spatial dimensions and generalized coordinates is developed in this study because implicit schemes permit the use of large integration steps in many cases they require fewer integration steps to reach steady state especially in calculations on grids with widely varying mesh cell sizes the implicit scheme developed in this study accelerated convergence speeds by almost an order of magnitude in the problems considered test cases include quasi one dimensional nozzle flow and supersonic flow past a cylinder

implicit methods of second order accuracy for the euler equations this paper presents a study of noniterative implicit finite difference methods for hyperbolic systems new space centered methods involving two time levels are considered an analysis is made of various properties such as solvability stability dissipation dispersion and efficient solution of the algebraic systems the computation of shock waves with a cfl number larger than the unity is discussed unconditional stability results are proved for a case having several space variables accurate solutions of the euler equations are obtained that offer major reductions in computing costs over explicit methods

line search filter methods for nonlinear programming line search methods are proposed for nonlinear programming using fletcher and leyffer s filter method math program 91 2002 pp 239 269 which replaces the traditional merit function their

matrix nearness based guaranteed passive system approximation abstract in this paper we present a new approach towards global passive approximation in order to find a passive real rational transfer function g s that is an arbitrarily close approximation of the passive transfer function nearest to a non passive square transfer function h s it is based on existing solutions to pertinent matrix nearness problems it is shown that the key point in constructing the passive real rational transfer function g s is to find a good rational approximation of the well known ramp function over an interval defined by the minimum and maximum dissipation of h s the proposed algorithms rely on the stable anti stable decomposition of a given transfer function pertinent examples are given to show the scope and accuracy of the proposed algorithms

adaptive least squares finite element methods for linear transport equations based on an h div flux reformulation abstract in this paper we study the least squares finite element methods lsfem for the linear hyperbolic transport equations the linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions while the normal component of the flux across the mesh faces needs to be continuous traditional lsfems using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions in order to separate the continuity requirements a new flux variable is introduced with this reformulation the continuities of the flux and the solution can be handled separately in natural h div l 2 conforming finite element spaces several variants of the methods are developed to handle the inflow boundary condition strongly or weakly with the reformulation the new lsfems can handle discontinuous solutions and boundary conditions much better than the traditional lsfems with continuous polynomial approximations with least squares functionals as a posteriori error estimators the adaptive methods can naturally identify error sources including singularity and non matching discontinuity for discontinuity aligned mesh no extra error is introduced if an r t 0 p 0 pair is used to approximate the flux and the solution the new adaptive lsfems can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods extensive numerical tests are performed to show the effectiveness of the methods developed in the paper

boundary approximations for implicit schemes for one dimensional inviscid equations of gasdynamics the applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary value problems of the hyperbolic type for the numerical experiments the one dimensional inviscid gasdynamic equations in conservation law form are selected a class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space time extrapolations as implicit or explicit boundary schemes is emphasized some numerical examples with various inflow outflow conditions highlight the commonly discussed issues explicit vs implicit boundary schemes and unconditionally stable schemes hen finite difference schemes are used to solve initial boundary value problems for the equations of fluid dynamics it is well known that most methods require more boundary conditions than those required by the governing partial differential equations these additional boundary conditions for the finite difference equations are often called numerical boundary conditions the numerical boundary conditions cannot be imposed arbitrarily but are determined in general using interior information for example by extrapolation or uncentered approximations in this paper any numerical procedure used to provide a numerical boundary condition will be called a boundary scheme whatever schemes are used for the numerical boundary conditions it is a common practice to assume that the boundary scheme has a local effect and will not affect the solution globally during the early 1970s kreiss 1 2 osher 3 gustafsson et al 4 varah 5 and gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with boundary schemes included since then further progress has been made in the theory of linear difference approximations for initial boundary value problems of the hyperbolic and parabolic type 7 12 because improper treatment of the boundary conditions can lead to instability and inaccuracy even though we start with a stable interior scheme i e scheme for the interior points it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and boundary schemes surveys of recent developments and extensive bibliographies are included in papers by coughran13 and yee 14 the purpose of this paper is to examine the applicability to practical calculations for nonlinear gasdynamic problems of recent theoretical stability analyses of implicit difference approximations for initial boundary value problems of the hyperbolic type as numerical computations have progressed the use of the conservative form of the gasdynamic equations has gained popularity for physical reasons it is sometimes desirable to specify boundary conditions in the nonconservative variables and to compute with conservative variables in the interior we will consider the additional complications introduced by this procedure

analysis of difference schemes for a stationary shock problem we consider the limit function u of the solutions u varepsilon of a singularly perturbed boundary value problem in general u will be discontinuous two difference approximations of the boundary value problem are studied and the discrete values for varepsilon 0 h to 0 are compared with u error estimates of order o h and o h 2 are proved and the sharpness of numerical layers is established

near wall reynolds stress three dimensional transonic flow computation a computational method for the favre reynolds averaged three dimensional compressible navier stokes equations using near wall reynolds stress closure is described the near wall reynolds stress closure uses the launder shima formulation for the reynolds stresses and the jones launder sharma modified dissipation e equation the mean flow and turbulence transport equations are discretized using a finite volume method based on muscl van leer flux vector splitting with van albada limiters the mean flow and turbulence equations are integrated in time using a fully coupled approximately factored implicit backward euler method the resulting scheme is robust and achieves optimal convergence with local time step courant friedrichs lewy 50 the turbulence closure is validated by comparison with classic flat plate boundary layer data results are presented for the three dimensional delery transonic channel test case and compared with k e computations an analysis of the limitations of the closure is attempted and possible improvements are suggested

on the numerical solution of conservation laws by minimizing residuals the numerical solution of conservation laws by minimizing the residuals of an overdetermined set of discrete equations is studied previous research has shown that for certain formulations minimizing the residuals in the l1 norm will yield solutions that resolve discontinuities that are very sharp and correctly placed in this study we analyze a previously proposed method that numerically solves the 2d advection equation with discontinuous data the method is able to resolve the discontinuity over one mesh cell without generating spurious oscillations however we have found that incorrect solutions are generated for some data this had led us to formulate and prove two theorems concerning these results we also provide an analysis of the solution procedure along with suggestions for developing future schemes that are more applicable to a wide range of problems

a note on strict passivity we show that there exists an explicit descriptor state space format which actually describes all strictly passive transfer functions a key advantage of this explicitly strictly passive descriptor state space format resides in its relation with congruence projection based reduced order modeling where the resulting reduced order model is also cast in this same format another advantage of the format is that it allows for a simple construction of strictly passive random systems generators

steady shock tracking and newton s method applied to one dimensional duct flow abstract a new computational approach combining shock tracking and newton s method is applied to steady one dimensional flow through a variable area duct on a transformed computational grid where the shock is fixed the physical shock location appears explicitly as an unknown in a set of finite difference equations and is coupled to the other unknowns the space time characteristics for the associated time dependent problem are used in formulating the boundary conditions the resulting system is solved by newton s method the computed results agree very well with an exact solution and the newton iterates converge rapidly in comparison to some explicit shock capturing time asymptotic methods

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

boundary conditions for euler equations the mathematical theory of boundary conditions is applied to euler equations of gasdynamics to select appropriate boundary conditions for numerical experiments we begin with the presentation of the mathematical analysis namely the so called normal modes analysis the nice feature of the theory is that appropriate boundary conditions are characterized by an algebraic condition known as the uniform kreiss s condition we illustrate this by presenting a formal program that determines whether boundary conditions are appropriate finally examples of good and bad boundary conditions handled by the program are given

curvature based wall boundary condition for the euler equations on unstructured grids a curvature corrected symmetry technique developed by dadone and grossman aiaa j vol 32 1994 for a structured grid euler solver has been extended to unstructured grids and to arbitrary curved boundaries in this paper the local curvature is estimated numerically with a robust procedure and is used successfully in the boundary treatment numerical results for 2d inviscid flows around circular and elliptic cylinders and the naca0012 airfoil with the new boundary condition showed dramatic improvements over those with a conventional non curvature corrected approach in all cases spurious entropy productions with the new boundary treatment are significantly reduced and sometimes by several orders of magnitude

euler equations implicit schemes and boundary conditions implicit boundary condition procedures are presented for use with implicit finite difference schemes for the unsteady euler equations this new boundary point treatment is based on the mathematical theory of characteristics for hyperbolic systems of equations along with the theoretical background the practical application of the method to several types of boundaries is also explained using several examples the specific boundary conditions covered include subsonic inflow and outflow surface tangency and shock waves the example problems include one dimensional laval nozzle flow dual throat rocket engine nozzle flow and supersonic flow past a sphere the implicit boundary treatment permits the use of large time steps allowing the finite difference algorithm to converge to the asymptotic steady state much faster than schemes that use explicitly applied boundary conditions at least an order of magnitude increase in computational speed is demonstrated in the examples shown background t he growing popularity of solutions to the euler equations in transonics and their continued application in supersonics have increased the need for quicker solutions the potential of implicit schemes in this direction has not been fully exploited for want of correct implicit application of boundary conditions the predominant use of implicit algorithms for the navier stokes equations has partly been responsible for the neglect of implicit boundary point treatment for the euler equations thus there is a need for correct and stable procedures for the easy implicit application of boundary conditions such methods will serve the two purposes of 1 reaching time asymptotic steady state faster and 2 permitting a time step for truly unsteady flow that is not necessarily restricted by the cfl stability criterion but is based upon the magnitude of the transients for clues and information on how to construct such boundary condition procedures one must turn to the mathematical theory of characteristi cs for hyperbolic systems of equations the unsteady euler equations belong to this category the theory for hyperbolic systems is rich with information on signal propagation directions the characteristics theory clearly points to the number of boundary conditions that may and need be prescribed without overdetermining the solution boundary condition procedures based on this theory have been known and applied for several years by kentzer 1 porter and coakley 2 de neef 3 and others in earlier work by this author 4 5 easily understood and implementable methods for boundary point treatment were presented however all of the above techniques were developed for explicit finite difference schemes it seems that it must be easy to extend such methodologies based on mathematical theory for hyperbolic systems to implicit finite difference schemes and indeed it is simple enough the rest of this paper describes such implicit boundary condition procedures the given examples illustrate in detail the application of the proposed methodology to specific types of boundaries and demonstrate the merits of the new scheme

on a parabolic quadrilateral finite element for compressible flows abstract we present a 9 node finite element for compressible flow solutions a high order derivative upwind term and a shock capturing term are employed for stability and accuracy of the formulation we give the solutions of various example problems to illustrate our experiences with the element

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

reply by authors to g degrez references 1 casier f deconinck h and hirsch ch a class of bidiagonal schemes for solving the euler equations aiaa journal vol 22 nov 1984 pp 1556 1563 2 maccormack r w the effect of viscosity in hypervelocity impact cratering aiaa paper 69 354 april 1969 3 maccormack r w a numerical method for solving the equations of compressible viscous flow aiaa journal vol 20 sept 1982 pp 1275 1281

calculation of shocked one dimensional flows on abruptly changing grids by mathematical programming abstract the steady state inviscid and nearly inviscid burgers equations and the euler equations for steady state shocked flow in a quasi 1 dimensional nozzle are discretized by cell centered finite differences on the cells of an arbitrarily spaced grid inviscid and viscous terms are approximated to second order by 2 point schemes and 4 point schemes respectively there results an overdetermined system of nonlinear algebraic equations this system is solved by a mathematical programming procedure that minimizes a weighted sum of the absolute values of the residuals the l 1 norm of the vector of residuals in this algorithm which is nonconservative no upwinding switches arbitrary constants or heuristic quantities are used the artificial viscosity used to solve the inviscid burgers equation is small 10 15 u no artificial viscosity of any kind is used for the euler equations the numerical solutions of both the viscous and the inviscid problems are accurate and nonoscillatory on grids with abrupt refinements in mesh length by factors as high as 10 4 shocks are invariably captured in one cell and this cell is rarely more than three cells away from the cell in which the physical shock occurs

a new result on passivity preserving model reduction abstract the problem of model reduction with preservation of passivity is investigated the approach is based on positive real interpolation and is inspired by the similarity between lowner and pick matrices the former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions it follows that interpolation of the original set of data together with an appropriately defined mirror image set of data yields automatically positive real interpolants subsequently we show how this result can be implemented using a krylov projection procedure the ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large scale systems