on extremal pure nash equilibria for mixed extensions of normal form games abstract in this paper we derive conditions under which mixed extensions of normal form games have least and greatest nash equilibria in pure strategies and either of them gives best utilities among all mixed nash equilibria when strategy spaces are complete separable metric spaces equipped with closed partial orderings and the values of utility functions are in separable ordered banach spaces the obtained results are applied to supermodular normal form games whose strategy spaces are multidimensional

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

numerical investigation of the laminar boundary layer flow around an impulsively moved circular cylinder abstract an implicit difference scheme of splitting type is developed for solving unsteady boundary layer equations with an unfavorable adverse pressure gradient the results obtained compare quantitatively well with the results of lagrangian numerical schemes and indicate that a singularity evolves after a finite time approximately t 3 in dimensionless units

alternating direction adaptive grid generation an alternating direction method has been developed to adaptively resolve numerical solutions to physical problems by moving the points of a coordinate grid with the solution or with the salient quantities derived from it a surface grid is defined over the grid on the physical region grid movement is computed on this abstractly defined solution surface with the use of pointwise weights and is then projected back to determine new locations in the physical region the weights cause clustering relative to the uniform conditions of surface arc length and are formed with magnitudes of quantities that are used for point attraction when the salient quantities have been used to form the surface the gradients are implicitly revolved and surface curvature becomes the primary remaining quantity in each coordinate direction the magnitude of normal curvature is used with arbitrary weights a movement cycle generally consists of a sweep through the coordinate curves in a given direction on a curve by curve basis and then in alternating directions to do the same thing until all directions have been covered

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

conformal mappings onto multiply connected regions with specified boundary shapes abstract the author describes a method of calculating conformal mappings of any given finitely connected region onto a region with arbitrarily specified boundary shapes if the specified shapes are rectangles than this method can be used to generate conformal grids which should be useful for numerical solution of many partial differential equations for example in calculating the airflow past an airflow with flaps or the flow of cooling water past fuel pins in a nuclear reactor he has proved that there exists a conformal mapping of any given finitely connected region onto a region with arbitrarily specified boundary shapes the construction in the proof has been adapted for computer implementation some examples have been worked to determine the region bounded by circles which is the image of a given region in the extended complex plane under a conformal mapping taking infinity to infinity a region whose outer boundary is a rectangle is mapped conformally to a region with all rectangular boundaries and the vertices of the outer boundaries correspond mappings onto regions bounded by rectangles should be of considerable use in grid generation for numerical solution of partial differential equations grids have been calculated for one simple example author

fast generation of three dimensional computational boundary conforming periodic grids of c type a fast computer program grid3c has been developed to generate multilevel three dimensional c type periodic boundary conforming grids for the calculation of realistic turbomachinery and propeller flow fields the technique is based on two analytic functions that conformally map a cascade of semi infinite slits to a cascade of doubly infinite strips on different riemann sheets up to four consecutively refined three dimensional grids can be automatically generated and permanently stored on four different computer tapes grid nonorthogonality is introduced by a separate coordinate shearing and stretching performed in each of three coordinate directions the grids can be easily clustered closer to the blade surface the trailing and leading edges and the hub or shroud regions by changing appropriate input parameters hub and duct or outer free boundary can have different axisymmetric shapes a vortex sheet of arbitrary thickness emanating smoothly from the blade trailing edge is generated automatically by grid3c blade cross sectional shape chord length twist angle sweep angle and dihedral angle can vary in an arbitrary smooth fashion in the spanwise direction input coordinates must be cartesian while the output grid coordinates can be cartesian or cylindrical

the determination of the poles of the mapping function and their use in numerical conformal mapping abstract let be the function which maps conformally a simply connected domain onto the unit disc this paper is concerned with the problem of determining the dominant poles of in compl and of using this information in order to obtain accurate numerical approximations to by means of the bergman kernel method

a noniterative method for the generation of orthogonal coordinates in doubly connected regions in this paper a noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves has been developed the basic generating equation is the gaussian equation for a euclidean space under the constraint of orthogonality the resulting equation has been solved analytically for the case of isothermic coordinates a coordinate transformation then yields nonisothermic coordinates in which the outgoing coordinate can be distributed in any desired manner the method has been applied in many cases and these test results demonstrate that the proposed method can be readily applied in a variety of problems

coordinate generation with precise controls over mesh properties abstract coordinate generation techniques with precise controls over mesh properties are mathematically developed the controls are precise because mesh properties can be explicitly specified within a local region independent of the mesh elsewhere the local regions can adjoin boundaries where a particular mesh form can greatly simplify a problem or can be used for a smooth juncture between distinct coordinate systems where in effect a branch cut with a prescribed geometry and mesh distribution can be obtained away from boundaries a local region can be given a particular mesh form to model internal objects or to simplify a problem

analysis and application of finite volume element methods to a class of partial differential equations abstract the finite volume element fve methods for a class of partial differential equations are discussed and analyzed in this paper the new initial values are introduced in the finite volume element schemes and we obtain optimal error estimates in l p and w 1 p 2 p as well as some superconvergence estimates in w 1 p 2 p the main results in this paper perfect the theory of the finite volume element methods

use of streamline coordinates in the numerical solution of compressible flow problems abstract iterative correction of streamline geometry is used as the basis of a numerical method for subsonic isentropic steady flow problems direct solution of coupled ordinary differential equations is applicable to the corresponding supersonic problem the method is formulated in three dimensions but for illustrative purposes is applied to a two dimensional jet involving a free surface condition the method may be particularly economical where intuition or experience provides a good starting guess for streamline geometry

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

general curvilinear coordinate systems the basic ideas of the construction and use of numerically generated boundary fitted coordinate systems for the numerical solution of partial differential equations are discussed with such coordinate systems all computation can be done on a fixed square grid in the rectangular transformed region regardless of the shape or movement of the physical boundaries a number of different types of configurations for the transformed region and the basic transformation relations from a cartesian system to a general curvilinear system are given the material of this paper is applicable to all types of coordinate system generation

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

generation of boundary conforming grids around wing body configurations using transfinite interpolation a computational procedure for generating three dimensional nonorthogonal surface fitted mesh systems around wing fuselage configurations is presented the method is based on the concept of transfinite interpolation which has been extended to handle very general mapping function specifications at the boundaries thereby making it possible to generate single block mappings with geometry data specified only at the outer boundaries of the computational domain since it is a direct algebraic mapping technique the method is very inexpensive in terms of computer cost different types of possible mappings are compared with respect to resolution and economy of nodal points a procedure for a novel type of mapping designated type o o is described and several plots of generated grids demonstrate the capabilities of the method the singular lines inherent in every three dimensional mesh for this type of surface geometry are also discussed

adaptive gridding for finite difference solutions to heat and mass transfer problems abstract the authors purpose in writing this paper is to review some of their recent work in the calculation of optimal meshes for the solution of parabolic and elliptic partial differential equations pde they first explain our strategies for the adaptive placement of mesh points in addition they make some speculation as to promising avenues for future research in mesh adaptation finally they discuss examples of the application of adaptive gridding to problems of heat and mass transfer they draw these examples from our work in combustion modeling in obtaining numerical solutions of pdes the spatial derivatives are often approximated by discrete representations on a mesh network the accuracy of any numerical solution depends in an important way on the relationship of the location of the mesh points to changes in the dependent variables the authors objective is to investigate finite difference methods in which the mesh networks adapt themselves dynamically to obtain accurate solutions such methods represent an important advance in overcoming a major shortcoming of traditional fixed mesh methods which are often unable to resolve accurately steep fronts or sharp peaks

high order finite difference methods for a second order dual phase lagging models of microscale heat transfer two adi solvers are given for 2d and 3d heat conduct equations at microscale they are uniquely solvable it is shown that they are of order 2 in time and order 4 in space numerical results exhibit the efficiency of the solvers in this paper a compact alternating direction implicit adi method which combines the fourth order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators is firstly presented for solving two dimensional 2d second order dual phase lagging models of microscale heat transfer by the discrete energy method it is shown that it is second order accurate in time and fourth order accurate in space with respect to l2 norms additionally the compact adi method is successfully generalized to solve corresponding three dimensional 3d problem also the convergence result of the solver for 3d case is given rigorously finally numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results

high level continuity for coordinate generation with precise controls abstract the development of precisely controlled coordinate generation techniques is continued from a first study to include the higher order smoothness which is necessary for three dimensional applications in the first study the controls came from the use of local piecewise linear interpolants in the general multisurface transformation the consequent integration therein resulted in coordinates with continuity up to first derivatives and with the capability to prescribe uniformity either locally or globally for the family of transverse coordinate curves the admission of uniformity placed a constraint upon the general interpolants which was trivially satisfied in the piecewise linear case with smoother piecewise constructions the constraint is used herein along with a requirement for coordinate curves to have the most general possible curvature properties the result is a class of coordinate transformations with continuity extended up to higher derivatives that retains the precise local controls displayed in the first study and that can be used in three or more dimensions

a vectorized finite volume adaptive grid algorithm for navier stokes calculations an adaptive grid finite volume method has been used to solve the navier stokes equations for complete forebody and afterbody flowfields around blunt bodies the code which is applicable for axisymmetric or two dimensional flows allows the mesh to adjust during the computation to provide a closer spacing of mesh points in regions of high gradients thus minimizing the number of required computational points the solution technique is explicit utilizing a maximum time step advancement at each grid point to accelerate convergence to the steady state the code has been fully vectorized for efficient solution on the cyber 203 computer a very flexible rezoning routine is used to concentrate mesh points anywhere in the field either by a user defined weighting function or by allowing high gradient regions to adjust the grid the grid adjustment routine is implicit in nature and represents a very small portion of the total computational cost currently the code runs in approximately 1 6 x 10 5 seconds per grid point per iteration

a mixed collocation finite difference method for 3d microscopic heat transport problems three dimensional time dependent initial boundary value problems of a novel microscopic heat equation are solved by the mixed collocation finite difference method in and on the boundaries of a particle when the thickness is much smaller than both the length and width the collocation method on fixed grid size is used to approximate the space operator whereas the finite difference scheme is used for time discretization this new mixed method is applied to a novel heat problem in a particle in order to compute the temperature distribution in and on the particle s surface the second derivatives of the basis functions for the spectral approximation are derived direct substitution of derivatives in the model transforms the differential equation into a linear system of equations that is solved by the specific preconditioned conjugate gradient method the high order accuracy and resolution achieved by the proposed method allows one to obtain engineering accuracy solution on coarse meshes the consistency stability and convergence analysis are provided and numerical results are presented

the high order multistep adi solver for two dimensional nonlinear delayed reaction diffusion equations with variable coefficients abstract in this paper a high order compact multistep alternating direction implicit adi difference scheme is constructed and analyzed for two dimensional 2d nonlinear delayed reaction diffusion equations ndres with variable coefficients by the discrete energy method it is presented that it achieves the convergence rate of o 2 h x 4 h y 4 with respect to h 1 and l 2 norms in non constrained temporal grids and is unconditionally stable finally numerical results demonstrate the effectiveness of our adi solver and exhibit the correctness of theoretical results

two dimensional unsteady euler equation solver for arbitrarily shaped flow regions a new technique is described for solving supersonic fluid dynamics problems containing multiple regions of continuous flow each bounded by a permeable or impermeable surface region boundaries are in general arbitrarily shaped and time dependent discretization of such a region for solution by conventional finite difference procedures is accomplished using an elliptic solver which alleviates the dependence on a particular base coordinate system multiple regions are coupled together through the boundary conditions the technique has been applied to a variety of problems including a shock diffraction problem and supersonic flow over a pointed ogive

on extremal solutions of inclusion problems with applications to game theory abstract a recursion principle generalized iteration methods and the axiom of choice are applied to prove the existence of extremal fixed points of set valued mappings in posets extremal solutions of an inclusion problem and extremal nash equilibria for a normal form game

unconditionally stable finite difference scheme and iterative solution of 2d microscale heat transport equation abstract a two dimensional time dependent heat transport equation at the microscale is derived a second order finite difference scheme in both time and space is introduced and the unconditional stability of the finite difference scheme is proved a computational procedure is designed to solve the discretized linear system at each time step by using a preconditioned conjugate gradient method numerical results are presented to validate the accuracy of the finite difference scheme and the efficiency of the proposed computational procedure

existence and comparison results for fixed points of multifunctions with applications to normal form games in this paper we apply generalized iteration methods to prove comparison results which show how fixed points of a multifunction can be bounded by least and greatest fixed points of single valued functions as an application we prove existence and comparison results for fixed points of multifunctions these results are applied to normal form games by proving existence and comparison results for pure and mixed nash equilibria and their utilities

application of adaptive grids to fluid flow problems with asymptotic solutions proper grid point location is important in the asymptotic finite difference solution of any fluid flow or heat transfer problem this paper uses a new technique that provides a simple way of moving the mesh points in physical space in order to reduce the error in the computed asymptotic solution relative to that obtained using a fixed mesh applications to fluid flow problems are presented including boundary layer flow and inviscid supersonic flow over cylinders and wedges with associated detached shocks the treatment of curved bound aries stationary and nonstationary boundaries and systems of partial differential equations is discussed significant error reductions are demonstrated he accuracy of the finite difference solution of a partial differential equation is dependent upon the suitability of the grid used in computing the solution it is clear that two solutions to the same problem will not have equal accuracy for the same number of mesh points if the points are located differently in each case the typical grid point allocation problem in computational fluid dynamics may be stated in the following way if a solution to a set of partial differential equations is required on a physical domain subject to prescribed boundary conditions how should a fixed number of grid points be distributed in order to assure that the solution obtained is the best the term best as used here implies that some measure of error has been used in the point allocation since the size and complexity of the typical problem attempted using computational methods have in creased dramatically it is imperative that grid points be used as efficiently as possible to minimize storage requirements the construction of finite difference grid systems has received a great deal of attention over the past five years several techniques have been proposed for producing ac ceptable body fitted mesh systems probably the most well known of these methods is that of thompson et al 1 2 in which a solution of laplace s equation produces the grid in physical space this technique is ideally suited to elliptic problems where a grid point allocation can be completed at the beginning of the calculation hindman et al 3 produced an adaptive grid scheme for time dependent flows the grid speeds in physical space were obtained by using the idea proposed by thompson thames and mastin but in ad dition differentiating the equations for physical grid point location with respect to time at each time step a matrix inversion led to expressions for the components of the grid point velocity unfortunately hindman s work included only the effect of boundary motion on the grid speeds the technique used to determine grid point motion in the present study is that proposed by rai and anderson 4 the influence of both boundary motion and the details of the internal flowfield are included in the calculation of the grid point distribution numerical solutions for a number of fluid flow problems are presented showing the applicability of the technique to both viscous and inviscid flows these examples

transfinite mappings and their application to grid generation the two essential ingredients of any boundary value problem are the field equations which describe the physics of the problem and a set of relations which specify the geometry of the problem domain mesh generators or grid generators are preprocessors whicd decompose the problem domain into a large number of interconnected finite elements or curvilinear finite difference stencils a number of such techiques have been developed over the past decade to alleviate the frustation and reduce the time involved in the tedious manual subdividing of a complex shaped region or 3 d structure into finite elements our purpose here is to describe how the techniques of bivariate and trivariate blending function interpolation which were originally developed for and applied to geometric problems of computer aided design of sculptured surface and 3 d solids can be adapted and applied to the geometric problems of grid generation in contrast to other techniques which require the numerical solution of complex partial differential equations and hence a great deal of computing the transfinite methods proposed herein are computationally inexpensive

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

conformal grid generation this paper treats conformal mapping as it relates to the generation of grids to be used for flow simulation classical and contemporary mapping methods are discussed and compared in some detail numerous suggestions are included to help the reader avoid common pitfalls and the mapping methods believed most promising are identified

fixed point theorems and their applications to theory of nash equilibria abstract in this paper we prove fixed point theorems for set valued mappings in products of posets applications to the theory of nash equilibria are presented

h1 galerkin mixed finite element methods for pseudo hyperbolic equations h 1 galerkin mixed finite element methods are discussed for a class of second order pseudo hyperbolic equations depending on the physical quantities of interest two methods are discussed optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension an extension to problems in two and three space variables is also discussed the existence and uniqueness are derived and it is showed that the h 1 galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the lbb consistency condition

a compact difference scheme for numerical solutions of second order dual phase lagging models of microscale heat transfer dual phase lagging dpl models constitute a family of non fourier models of heat conduction that allow for the presence of time lags in the heat flux and the temperature gradient these lags may need to be considered when modeling microscale heat transfer and thus dpl models have found application in the last years in a wide range of theoretical and technical heat transfer problems consequently analytical solutions and methods for computing numerical approximations have been proposed for particular dpl models in different settings in this work a compact difference scheme for second order dpl models is developed providing higher order precision than a previously proposed method the scheme is shown to be unconditionally stable and convergent and its accuracy is illustrated with numerical examples

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

assessing the quality of curvilinear coordinate meshes by decomposing the jacobian matrix an algebraic decomposition of the jacobian matrix j x i email protected j which relates physical and computational variables is presented this invertible decomposition parameterizes the mesh by the physically intuitive qualities of cell orientation cell orthogonality cell volume and cell aspect ratio this decompostion can be used to analyze numerically generated curvilinear coordinate meshes and to assess the contribution of the mesh to the truncation error for any specific differential operator and algorithm this is worked out in detail for laplace s equation in nonconservative and conservative forms an analysis by g h k 1 of the mesh contribution to truncation error for the full potential code tair is given in abbreviated form the variables introduced here and their derivatives are also natural lagrange multipliers for adaptive mesh algorithms based on a variational principle

on the numerical solution of initial boundary value problems in one space dimension the numerical solution of initial boundary value problems of the form a u x t u t b u x t u x c u x t is considered particular emphasis is placed on the solution of problems with large gradients e g shocks and boundary layers a mesh selection technique is described that accurately places points in regions where the solution is rapidly changing this is accomplished by a transformation of the original equations to a new coordinate system finite difference solutions for two sample problems are calculated

three dimensional grid generation using poisson equations a method for constructing a boundary fitted grid system for arbitrary three dimensional 3 d configurations is presented the grid generation procedure is formulated through the poison equations a feature of the scheme is that grid lines in the grid system are smothly interconnected between the boundary co ordinates the control of the grid distributions in the interior of the system has implemented by the forcing functions appearing in the poisson equations and by the boundary coordinates reasonable cell sizes and shapes are obtained even for extreme boundary shapes which normally tend to cause poorly controlled grid distributions

algebraic grid generation three methods are described for transforming grids in bounded two and three dimensional physical domains into a uniform grid in a rectangular computational domain the methods are based on mathematical interpolation functions and do not require the solution of differential equations or the use of complex variables they are simply referred to as algebraic methods and are called transfinite interpolation the multisurface method and the two boundary technique the primary advantage of the methods is that they provide explicit control of physical grid shape and physical grid spacing secondly they require relatively few computations consequently the application of inter active computer graphics in conjunction with the methods is advocated for rapid generation of grids the basic mathematical structure of each method is described and particular attention is given to surface representation parameter variable generation and control function generation physical boundary topology and grid derivative requirements are presented

an efficient full potential implicit method based on characteristics for supersonic flows la methode utilise la theorie de la propagation de signal caracteristique pour simuler avec precision la structure de l ecoulement qui inclut des ondes de choc et un ecoulement transversal mixte elliptique hyperbolique

solution of viscous internal flows on curvilinear grids generated by the schwarz christoffel transformation abstract a method is presented for combining an accurate orthogonal curvilinear coordinate generation procedure with a fast accurate and stable forward marching viscous flow solution technique to solve for the flow field in arbitrary axisymmetric ducts in this method the coordinates are generated from the plane potential flow streamlines and potential lines using the schwarz christoffel transformation with a composite finite difference formula which is valid everywhere and which treats the poles exactly by analytic integration since the coordinate streamlines approximate the actual streamlines the equations of motion for viscous compressible flow can be parabolized so as to solve for both the boundary layers and core flow in a single streamwise pass the versatility of the method is demonstrated by two examples of viscous compressible swirling flow through complex radial gas turbine passages author

numerical generation of three dimensional coordinates between bodies of arbitrary shapes abstract this paper is devoted to the numerical solution of a set of second order elliptic partial differential equations for the generation of three dimensional curvilinear coordinates between two arbitrary shaped bodies the central idea of the method is to generate a series of surfaces between the given inner and the outer boundary surfaces and then to connect these surfaces in such a manner so as to have a sufficiently differentiable three dimensional coordinate net in the enclosed region it is important to state here that the proposed equations for the numerical solution form a consistent set of second order elliptic equations which are a consequence of the equations of gauss for a surface additional constraints are then imposed which besides yielding the simplest form of equations for numerical purposes also preserve the essential geometric properties of the generated surfaces author

the use of solution adaptive grids in solving partial differential equations the grid point distribution used in solving a partial differential equation using a numerical method has a substantial influence on the quality of the solution an adaptive grid which adjusts as the solution changes provides the best results when the number of grid points available for use during the calculation is fixed basic concepts used in generating and applying adaptive grids are reviewed in this paper and examples illustrating applications of these concepts are presented

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

an alternative discretization and solution procedure for the dual phase lag equation we describe an alternative numerical treatment of the dual phase lag equation often used to account for microscale short time heat transport the approach consists of an undecomposed formulation of the partial differential equation resulting from taylor expansion with respect to lag times of the original delay partial differential equation trapezoidal integration in time and centered differencing in space provide an accurate discretization as demonstrated by comparisons with analytical and experimental results in one dimension and via grid function convergence tests in three dimensions for relatively fine 3 d grids the approach is approximately six times faster than a standard explicit scheme and nearly three times faster than an implicit method employing conjugate gradient iteration at each time step

applications and generalizations of variational methods for generating adaptive meshes abstract generating computation meshes for irregular regions have been of interest to a lot of people in many areas of research for a long time one technique that has met with sucess over the long run has been to generate methods using an elliptic equation or a system of elliptic equations the technique in its simplest form uses a system of laplace equations which are solved by direct or iterative methods as people gained more experience with this method source terms were added to the laplace equations to gain additional control of the mesh in variable coefficients of the derivatives were added for further flexibility in this paper the authors work with a method that systematically generates a set of elliptic equations without having to explicitly perturb a set of laplace equations with source terms and variable coefficients this technique uses the variational methods often associated with elliptic equations following this introduction they briefly discuss the variational formulation in two dimensional cartesian geometry then the formulation will be generalized to three dimensions next several three dimensional test problems will be shown after displaying these three dimensional test results the author wil the exhibit an application of the mesh generation technique in two dimensions this application involves generating an adaptive mesh for a supersonic flow past a step in a wind tunnel author

conformal mappings for internal viscous flow problems abstract isolated occlusions in a straight channel or pipe can be described analytically in a fashion that allows efficient accurate computation of nonlinear flow fields through conformal transformations

error induced by coordinate systems it is pointed out that the choice of a curvilinear coordinate system can have a substantial effect on the error in the numerical solution of a partial differential equation the truncation error is dependent not only on the higher order derivatives of the solution and the local grid spacing but also on the rate of change of the grid spacing and on the departure of the grid from orthogonality in connection with the present investigation an analysis is conducted of the local truncation error in the approximation of first and second order derivatives on a curvilinear grid attention is given to a number of examples which illustrate the two fundamental sources of truncation error in the numerical solution of partial differential equations on curvilinear coordinate systems the first is the grid spacing and changes in grid spacing which is measured by the first and second order derivatives of the functions defining the coordinate system the second source is the higher order derivatives of the solution itself

patched coordinate systems abstract numerical grid generation has been an excellent tool for producing curvilinear body fitted coordinate systems curvilinear coordinate systems or grids are commonly used in the solution of partial differential equations in domains surrounding arbitrary geometrical boundary shapes body fitted grids are particularly advantageous in the treatment of surface boundary conditions and usually yield a degree of simplicity in the logic required to solve the hosted partial differential equations in practice numerical grid generation usually involves transformation of the physical domain of interest into a geometrically simple domain such as a rectangular block or assembly of blocks the solution of grid generation equations in the simple domain products the coordinates of a corresponding grid in the physical domain subject to a variety of grid control procedures aimed at producing favorable grid characteristics this process is usually straightforward when the topology of the physical domain is simple enough to allow transformation to a single rectangular domain but when dealing with geometrically and topologically complex domains such as surround an aircraft configuration the total issue of grid generation becomes more complex the domain in general cannot be mapped into a single block the configuration surface geometry itself may be nonanalytic and these features will be manifest in any grid surrounding such complex boundary shapes

upwind schemes and boundary conditions with applications to euler equations in general geometries abstract a unified treatment of several upwind shock capturing algorithms is presented each algorithm has a riemann initial value problem as its basis the treatment of boundaries involves solving the associated riemann initial boundary value problem the first author s algorithm applied to multidimensional euler equations in general geometries is then presented its worth is verified by various calculations which include mach 8 supersonic flow past a circular cylinder

an unconditionally stable finite difference scheme for solving a 3d heat transport equation in a sub microscale thin film heat transport at the microscale is of vital importance in microtechnology applications the heat transport equation is different from the traditional heat diffusion equation since a second order derivative of temperature with respect to time and a third order mixed derivative of temperature with respect to space and time are introduced in this study we develop a finite difference scheme with two levels in time for the 3d heat transport equation in a sub microscale thin film it is shown by the discrete energy method that the scheme is unconditionally stable the 3d implicit scheme is then solved by using a preconditioned richardson iteration so that only a tridiagonal linear system is solved for each iteration the numerical procedure is employed to obtain the temperature rise in a gold sub microscale thin film

numerical solution of transonic wing flowfields a fast fully implicit approximate factorization algorithm designed to solve the conservative full potential equation is used to compute lifting wing flowfields with embedded transonic flow the computer program twing transonic wing analysis uses an elliptic solver numerical grid generation routine and is capable of analyzing arbitrary wing geometries with sweep twist taper and section variation computed flowfield results for several wing geometries are presented they demonstrate substantial improvement in convergence speed relative to the flo28 computer code

elliptic grid generation various types of generating systems for boundary conforming coordinate systems based on the numerical solution of systems of elliptic partial differential equations are discussed particular emphasis is given to the determination of functions in these equations which control the distribution of the curvilinear coordinate lines in the field

the finite element numerical analysis for a class nonlinear evolution equations a complete stability and convergence analysis is given for two level scheme for the approximate solution of a class generalized nerve conduction equation defining a kind of truncation error and using a kind of technique optimal order estimates in l 2 is constructively obtained

marching grid generation using parabolic partial differential equations abstract the unique aspect of grid generation in computational fluid dynamics is that the grid generation equations have no physical meanings so any equation may be used for this purpose if the grids generated are useful this aspect provides a vast freedom in developing new methods of grid generation this paper explores feasibility of using parabolic partial differential equations for grid generation the advantages of using parabolic partial differential equations are as follows 1 parabolic equations are initial value problems so grids are generated by a marching algorithm like the hyperbolic grid generation method 2 the parabolic partial differential equations have most properties of the elliptic equations particularly the diffusion effect which smooths out any singularity of the inner boundary condition if any and 3 the prescribed outer boundary conditions may be satisfied the importance of the marching algorithm stated above is twofold first computational time required is only a very small fraction of that for the elliptic grid generation equations second the fast memory space required during grid generation can be substantially reduced from that required by the elliptic grid generation method the authors consider the grid generation for a two dimensional airfoil flow calculations for simplicity of discussions although the method described in this paper is not restricted to two dimensions or airfoil problems

grid evolution in time asymptotic problems the selection of the proper coordinate system in solving any fluid flow or heat transfer problem is a very important consideration a new technique of moving mesh points in physical space is introduced so as to reduce the error in a computed asymptotic solution relative to that obtained using a fixed mesh the technique has been used to solve the simple viscous burgers equation in one and two dimensions substantial error reductions are demonstrated the treatment of boundary points and the effect of using different error criteria in generating grids are discussed

a compact finite difference scheme for solving a one dimensional heat transport equation at the microscale 431 heat transport at the microscale is of vital importance in microtechnology applications the heat transport equation is different from the traditional heat diffusion equation since a second order derivative of temperature with respect to time and a third order mixed derivative of temperature with respect to space and time are introduced in this study we develop a high order compact finite difference scheme for the heat transport equation at the microscale it is shown by the discrete fourier analysis method that the scheme is unconditionally stable numerical results show that the solution is accurate

coordinate system control adaptive meshes the discussion of the formulation and properties of the adaptive mesh algorithm is as complete as space permits as should be clear from the discussion adaptive gridding is most useful for singular perturbation problems where localized regions with large gradients develop spontaneously often choosing the correct weight function is not trivial and sometimes it is appropriate to cause the mesh to respond to structure in several dependent variables simultaneously

the numerical differentiation of discrete functions using polynomial interpolation methods a fortran subroutine package called dermod has been written for calculating numerical approximations for the spatial derivatives of a function defined only on a discrete set of data points the routines are designed to complement many existing codes for solving ordinary and partial differential equations and for the interpolation of tabular data we describe some new numerical differentiation algorithms discuss a mapping procedure for nonuniform grids and explain the program methodology used in designing the software

automatic algebraic coordinate generation a computer software system has been developed to automatically generate two dimensional coordinates from algebraic transformations for topologically complex regions a smooth assembly of the transformations can be used to automatically produce a composite mesh where a general gridded format is retained the algebraic mesh generation system consists of a collection of operator subroutines which are applied to an established data structure and which automatically perform the necessary parts of mesh construction from a sequence of multisurface transformations the system operators are discussed taking into account the data base the order of application direct surface generators geometric surface operators surface generators from existing surfaces transverse operators mesh operators assembly operators and data visualization operators attention is given to applications related to airfoils

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

generalized coordinate forms of governing fluid equations and associated geometrically induced errors the governing equations of fluid flow may be cast into various forms upon application of a generalized coordinate mapping these forms are the nonconservation law form nclf and strong weak and chain rule conservation law forms sclf wclf crclf respectively this paper describes the geometrically induced errors resulting from the failure to satisfy a certain consistency condition for each of these four forms and also demonstrates the ability of the crclf to produce exactly the same numerical solution as the wclf provided a condition is met on where to evaluate the transformation metrics it is also demonstrated that considerably fewer arithmetic operations are required to advance the solution from n to n 1 when the crclf is used in com parison to both the sclf and wclf he effort devoted to the field of computational fluid dynamics is progressing at an ever increasing rate at certain points in time during this natural evolution of numerical methods schemes and ideas it is sometimes in structive to pause to reflect upon past work in an attempt to gain the proper perspective it is important that this reflection reaches all of the way to the fundamental rules and practices used to develop numerical techniques some of these practices are often taken for granted much too soon after their in troduction into the literature only through a global view of this past effort can subtle mutually experienced problem areas and possible causes be identified the present work deals with the numerical solution to the transformed fluid flow equations i e euler equations etc using the finite difference approach the transformed equations are obtained through application of a generalized mapping from physical coordinates to computational coordinates at the time of application of the mapping to the original equations and prior to choosing the numerical in tegration scheme the analyst must decide in which form the equations should be written the choices are the non conservation law form nclf strong conservation law form sclf weak conservation law form wclf and chain rule conservation law form crclf this decision is one of the fundamental practices alluded to in the previous paragraph the present work illustrates how such a fundamental decision can strongly influence both the amount of analysis required to develop a consistent algorithm and the number of arithmetic operations required to execute the algorithm in addition it is shown that large solution errors and in some cases instabilities can result from failure to implement the results of such an analysis

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

an integral equation method for the numerical conformal mapping of interior exterior and doubly connected domains a numerical method based on the integral equation formulation of symm is described for computing approximations to the mapping functions which accomplish the following conformal maps a the mapping of a domain interior to a closed jordan curve onto the interior of the unit disc b the mapping of a domain exterior to a closed jordan curve onto the exterior of the unit disc c the mapping of a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on approximating the unknown source density by cubic splines and singular functions and is particularly suited for the mapping of difficult domains having sharp corners

a general mapping procedure for variable area duct acoustics a general mapping procedure is described and applied to the study of noise propagation in variable area ducts the mapping provides a boundary fitted co ordinate system which is ideal for the finite difference solution of acoustic fields with irregular boundaries without the burden of large matrices required by finite element methods the procedure is first described in general and then applied to a particular two dimensional geometry under current experimental investigation this method should be ideally suited to the study of high frequency noise propagation in variable area ducts and in cases where the far field is included in the calculation procedure moreover the current approach can be directly extended to three dimensions resulting in numerical calculation over a rectangular parallelepiped in the transformed plane

steady viscous flows by compact differences in boundary fitted coordinates fourth order compact differencing is applied to the steady solution of two dimensional viscous incompressible flows at moderate reynolds number the physical region where the fluid flow occurs is mapped onto a rectangle by means of the boundary fitted coordinates transformation method the design of the general numerical algorithm rests upon discretization of velocity components and pressure field at the same grid points the validity of this procedure is assessed by the investigation of a stokes test problem fourth order numerical results are compared to the analytical solution second order results and chebyshev tau approximation it is shown that fourth order differences provide good precision particularly when the ability of generating irregular meshes is fully exploited standard problems as poiseuille and couette flows and the square cavity problem are solved the fourth order results on coarse mesh compare favourably with other techniques such as finite element method and second order differences a global convergence analysis was performed on a two dimensional problem with smooth boundary conditions one observes a rate of convergence of order four for the velocity components and of order three for the pressure for the square cavity problem with the two corner singularities the rates of convergence are decreased by almost an order of magnitude the solution of a plane constricted channel flow enhances the overall improvement in accuracy gained by the treatment of the geometrically complex region through the mapping technique

runge kutta methods and viscous wave equations we study the numerical time integration of a class of viscous wave equations by means of runge kutta methods the viscous wave equation is an extension of the standard second order wave equation including advection diffusion terms differentiated in time the viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient a stable runge kutta methods are then very good candidates for time integration in particular diagonally implicit ones special attention is paid to the question how the a stability property can be translated to this non standard class of viscous wave equations