continued fractions and the potential models of confinement reply to a comment the analytic continued fractional formula for energies of the bound states in the harmonium potential v r ar 1 br cr2 as suggested by singh et al 1978 1979 and criticised by flessas 1982 is correct if and only if b 0 this is the special case of the more general result the author extends the analytic construction of the green function and the rigorous specification of its validity to the whole class of potentials with r2v r polynomial in ralpha alpha positive rational number

eigenvalue bounds for transformations of solvable potentials we study smooth transformations of potentials for which exact bound state solutions of schrodinger s equation are known eigenvalue approximation formulae are obtained which provide lower or upper energy bounds according to whether the transformation function g is convex or concave detailed results are presented for perturbed coulomb potentials of the form and

a coarse grid projection method for accelerating incompressible flow computations we present a coarse grid projection cgp method for accelerating incompressible flow computations which is applicable to methods involving poisson equations as incompressibility constraints the cgp methodology is a modular approach that facilitates data transfer with simple interpolations and uses black box solvers for the poisson and advection diffusion equations in the flow solver after solving the poisson equation on a coarsened grid an interpolation scheme is used to obtain the fine data for subsequent time stepping on the full grid a particular version of the method is applied here to the vorticity stream function primitive variable and vorticity velocity formulations of incompressible navier stokes equations we compute several benchmark flow problems on two dimensional cartesian and non cartesian grids as well as a three dimensional flow problem the method is found to accelerate these computations while retaining a level of accuracy close to that of the fine resolution field which is significantly better than the accuracy obtained for a similar computation performed solely using a coarse grid a linear acceleration rate is obtained for all the cases we consider due to the linear cost elliptic poisson solver used with reduction factors in computational time between 2 and 42 the computational savings are larger when a suboptimal poisson solver is used we also find that the computational savings increase with increasing distortion ratio on non cartesian grids making the cgp method a useful tool for accelerating generalized curvilinear incompressible flow solvers

an asymptotic preserving all speed scheme for the euler and navier stokes equations we present an asymptotic preserving all speed scheme for the simulation of compressible flows valid at all mach numbers ranging from very small to order unity the scheme is based on a semi implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly this discretization which is the key to the asymptotic preserving property provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime the divergence free condition on the velocity in the incompressible regime is respected and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations the implicit treatment of the acoustic part allows the time step to be independent of the mach number the scheme is conservative and applies to steady or unsteady flows and to general equations of state one and two dimensional numerical results provide a validation of the asymptotic preserving all speed properties

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

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

eigenvalues of the schrodinger equation with coulomb potentials plus linear and harmonic radial terms analytic approximations for the eigenvalues have been obtained using the two point quasifractional approximants method here we extend and generalize previous work where the method was applied to determine eigenenergies of two dimensional coulomb potentials in the presence of a magnetic field of arbitrary strength we have also shown the advantage of our results with respect to other calculations using the shifted 1 n method or pade approximants

the asymptotic iteration method for the eigenenergies of the schrodinger equation with the potential v r z r gr λr2 the eigenenergies of the schrodinger equation with coulomb potential plus linear and harmonic radial terms are investigated using the perturbation technique within the framework of the asymptotic iteration method the obtained eigenenergies are compared with those obtained from the shifted 1 n expansion method the moment method and also with the exact values obtained from the super symmetric method it is found that asymptotically this method gives accurate results for arbitrary g z and

the discrete continuity equation in primitive variable solutions of incompressible flow abstract the use of a non staggered computational grid for the numerical solutions of the incompressible flow equations has many advantages over the use of a staggered grid a penalty however is inherent in the finite difference approximations of the governing equations on non staggered grids in the primitive variable solutions the penalty is that the discrete continuity equation does not converge to machine accuracy rather it converges to a source term which is proportional to the fourth order derivative of the pressure the time increment and the square of the grid spacing an approach which minimizes the error in the discrete continuity equation is developed numerical results obtained for the driven cavity problem confirm the analytical developments

the hill determinant approach to the coulomb plus linear confinement an application of the so called hill determinant method to the bound state eigenvalue problem in the elementary quarkonium potential v r a r br is described proved and illustrated for a few examples an improvement of the method which is based on an extended continued fraction formulation of the eigenvalue condition is also proposed

analysis of artificial pressure equations in numerical simulations of a turbulent channel flow abstract recently several methods have been proposed to simulate incompressible fluid flows using an artificial pressure evolution equation avoiding the resolution of a poisson equation these methods can be seen as various levels of approximation of the compressible navier stokes equation in the low mach number limit we study the simulation of incompressible wall bounded flows using several artificial pressure equations in order to determine the most relevant approximations the simulations are stable using a finite difference method in a staggered grid system even without diffusive term and converge to the incompressible solution both in direct numerical simulations and for coarser meshes to be used in large eddy simulations a pressure equation with a convective and a diffusive term produces a more accurate solution than a compressible solver or methods involving more approximations this suggests that it is near to an optimal level of approximation the presence of a convective term in the pressure evolution equation is in particular crucial for the accuracy of the method the rate of convergence of the solution in terms of artificial mach number is studied numerically and validates the theoretical quadratic convergence rate we demonstrate that this property can be used to accelerate the rate of convergence using an extrapolation in terms of artificial mach number since the approach is based on an explicit and local system of equations the numerical procedure is massively parallelisable and has low memory requirements

finite volume methods for two dimensional incompressible flows with complex boundaries abstract the paper discusses possible ways of developing accurate and efficient finite volume methods for complex incompressible flows the issues of boundary fitted grid generation choice of coupled or uncoupled procedures cartesian or grid aligned velocity components staggered or non staggered variable arrangement discretization methods solution algorithms and convergence acceleration by multigrid methods are addressed an example of a finite volume method for general two dimensional flows is described in some detail and applications of this method to a variety of flows are presented

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

a three dimensional incompressible navier stokes flow solver using primitive variables an implicit finite difference computer code has been developed to solve the incompressible navier stokes equations in a three dimensional curvilinear coordinate system the pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation the solution procedure employs an implicit approximate factorization scheme the reynolds stresses which are uncoupled from the implicit scheme are lagged by one time step to facilitate implementing various levels of the turbulence model test problems for external and internal flows are computer and the results are compared with existing experimental data the application of this technique for general three dimensional problems is then demonstrated

a coupled implicit method for chemical non equilibrium flows at all speeds a time accurate coupled solution procedure is described for the chemical nonequilibrium navier stokes equations over a wide range of mach numbers this method employs the strong conservation form of the governing equations but uses primitive variables p g u v h y i as unknowns real gas properties and nonequilibrium chemistry are considered numerical tests include steady convergent divergent nozzle flows with air dissociation recombination chemistry dump combustor flows withn pentane air chemistry and nonreacting unsteady driven cavity flows numerical results for both the steady and unsteady flows demonstrate the efficiency and robustness of the present algorithm for mach numbers ranging from the incompressible limit to supersonic speeds c 1993 academic press inc

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

numerical solutions for the pressure poisson equation with neumann boundary conditions using a non staggered grid 1 abstract numerical solutions are obtained for the pressure poisson equation with neumann boundary conditions using a non staggered grid the existence of a solution for this equation requires the satisfaction of a compatibility condition which relates the source of the poisson equation and the neumann boundary conditions this compatibility condition is not automatically satisfied on non staggered grids failure to satisfy the compatibility condition leads to non convergent iterative solutions consistent finite difference approximations for the pressure equation with neumann boundary conditions are developed to satisfy the compatibility condition on non staggered grids the method is applied to calculate the pressure coefficient in a driven cavity when given the velocity field the velocity is computed from the stream function vorticity formulation of the navier stokes equations

on the classical theory of ordinary linear differential equations of the second order and the schrodinger equation for power law potentials the power law potentials in the schrodinger equation solved recently are shown to come from the classical treatment of the singularities of a linear second order differential equation this allows one to enlarge the class of solvable power law potentials

on the power series construction of bound states i the energies as zeros of the infinite hill determinants for a broad class of potentials we show that the nonvariational so called hill determinant intuitive identification of binding energies with zeros of certain infinite determinants may be given a rigorous mathematical foundation the essence of the construction lies in an appropriate restriction of the class of the admissible ansatze this eliminates the undesirable confluence of the physical and unphysical components of r that may take place in the general case

finite element method for incompressible viscous flows with adaptive hybrid grids a new adaptive finite element numerical method has been developed for the unsteady navier stokes equations of incompressible flow in two dimensions the momentum equations combined with a pressure correction equation are solved employing a nonstaggered grid the solution is advanced in time with an explicit implicit marching scheme an adaptive algorithm has been implemented which refines the grid locally to resolve detected flow features a combination of quadrilateral as well as triangular cells provides a stable and accurate numerical treatment of grid interfaces that are located within regions of high gradients applications of the developed adaptive algorithm include both steady and unsteady flows with low and high reynolds numbers comparisons with analytical as well as experimental data evaluate accuracy and robustness of the method i introduction i ncompressible flows are frequently encountered in engineering applications during the past two decades a significant number of numerical algorithms have been developed for solution of the incompressible navier stokes equations 1 liie lack of pressure term in the continuity equation makes solution of the momentum equations with the divergence free constraint more difficult in the case of incompressible flows the conservation of mass acts as a constraint condition that the velocity field must satisfy whereas in compressible flows the conservation of mass is given through a partial differential equation for the temporal variation of density the infinite speed of sound in the incompressible case requires an implicit treatment of the pressure furthermore spatial discretization for pressure and velocity may produce oscillatory solutions one approach followed is to formulate the equations in terms of a stream function and a vorticity extension of this method to three dimensions is not possible different formulations have been used in three dimensions such as the vorticity velocity approach another approach is to use the compressible flow equations and solve them for low mach numbers the required time step for such computations is very small because the speed of sound approaches infinity at the incompressible limit a method that uses compressible like governing equations is the artificial compressibility approach 2 a time derivative of the pressure is added to the continuity equation and the incompressible flowfield is treated as compressible during the transient stage time accuracy of the simulation is usually not preserved another class of algorithms uses a special poisson equation for the pressure field 5 8 the usual computational procedure is to assume an initial pressure field and then an iterative process is defined until the continuity equation is satisfied a major issue of the corresponding pressure and velocity spatial discretization is oscillations in the pressure field to reject these modes staggered grids have been employed by several of these algorithms 9 10 on the other hand employment of nonstaggered grids11 13 requires dissipation in the algorithms stability of both approaches with high reynolds number flows is an important issue a review of numerical methods for incompressible flows as well as references

coupled fully implicit solution procedure for the steady incompressible navier stokes equations abstract this paper presents a new fully implicit procedure for the solution of the steady incompressible navier stokes equations in primitive variables the momentum equations are coupled with a poisson type equation for the pressure and solved using the beam and warming approximate factorization method the present formulation does not require the iterative solution of the pressure equation at each time step thus the major drawback of the pressure poisson approach which made it prohibitively expensive for complex three dimensional applications is eliminated numerical solutions for the problem of the two dimensional driven cavity are obtained using a non staggered grid at re 100 400 and 1000 all the computed results are obtained without any artificial dissipation this feature of the present procedure demonstrates its excellent convergence and stability characteristics those characteristics result from the coupling of the pressure equation which is elliptic in space with the momentum equations

a high resolution pressure based algorithm for fluid flow at all speeds abstract a new collocated finite volume based solution procedure for predicting viscous compressible and incompressible flows is presented the technique is equally applicable in the subsonic transonic and supersonic regimes pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low mach numbers the newly developed algorithm has two new features i the use of the normalized variable and space formulation nvsf methodology to bound the convective fluxes and ii the use of a high resolution scheme in calculating interface density values to enhance the shock capturing property of the algorithm the virtues of the newly developed method are demonstrated by solving a wide range of flows spanning the subsonic transonic and supersonic spectrum results obtained indicate higher accuracy when calculating interface density values using a high resolution scheme

comments on a paper concerning the analytic continued fraction technique for bound states the author shows that the treatment of bound and confined states for a class of confinement potentials by application of the analytic continued fraction method as presented in a recent paper dalta et al ibid vol 13 p 3161 1980 is erroneous and may lead to wrong results for the eigenvalues of the relevant schrodinger equation

pressure based control volume finite element method for flow at all speeds we present a collocated pressure based method for the solution of viscouslinviscid fluid flow problems incorporating compressible and incompressible regimes the solution domain is discretized using a control volume based finite element method the fully implicit method does not experience any stability difficulties due to compressibility effects at low mach numbers the pressure velocity decoupling problem is fully resolved herein for flow regimes from incompressible through compressible flow transonic and supersonic flows are covered by including the pressure density coupling which arises in high speed flows in the continuity equation this is done in such a manner that the continuity equation remains a constraint equation for pressure in all flow regimes test problems with wide variations in geometry and fluid physics have been successfully solved demonstrating the generality and computational capability of the method the scheme is both efficient and robust the accuracy of the method has been checked by comparing the results with other numerical results available in the literature and with the exact solution

efficient solution of the euler and navier stokes equations with a vectorized multiple grid algorithm a multiple grid algorithm for use in efficiently obtaining steady solutions to the euler and navier stokes equations is presented the convergence of the explicit maccormack algorithm on a fine grid is accelerated by propagating transients from the domain using a sequence of successively coarser grids both the fine and coarsegrid schemes are readily vectorizable the combination of multiple grid ding and vectorization results in substantially reduced computational times for the numerical solution of a wide range of flow problems results are presented for subsonic transonic and supersonic inviscid flows and for subsonic attached and separated laminar viscous flows work reduction factors over a scalar single grid algorithm range as high as 76 8

preconditioned methods for solving the incompressible low speed compressible equations abstract acceleration methods are presented for solving the steady state incompressible equations these systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state we also consider the compressible equations in conservation form with slow flow two arbitrary functions and are introduced in the general preconditioning an analysis of this system is presented and an optimal value for is determined given a constant it is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed several generalizations to the compressible equations are presented which extend previous results

numerical solutions for the incompressible navier stokes equations in primitive variables using a non staggered grid 11 abstract in part i of this study a method was developed for the solution of the pressure poisson equation with neumann boundary conditions on a non staggered grid this method was used to determine the pressure when given the velocity field from a stream function vorticity solution in part ii the pressure equation is solved iteratively with the momentum navier stokes equations on a non staggered grid in this case the solution of the pressure equation not only provides the pressure but also serves to indirectly satisfy the continuity equation this primitive variables formulation has a major advantage over the stream function vorticity method in its applicability for three dimensional flow numerical results are obtained and compared with the stream function vorticity results for the driven cavity of part i

computation of flows with arbitrary equations of state the extension of time marching computations to fluids with arbitrary equations of state is demonstrated by means of stability analyses simplified problems and practical applications most of the examples use the properties of supercritical hydrogen for which the density varies by more than an order of magnitude for small changes in pressure and temperature but representative computations for incompressible fluids and perfect gases are also given to demonstrate the generality of the procedure because representative flow velocities in typical supercritical fluids applications are much lower than the speed of sound convergence enhancement through eigenvalue control is often necessary this is accomplished through a generalization of earlier preconditioning methods that enables efficient computation of arbitrary equation of state fluids perfect gases and incompressible fluids by a single procedure the present approach thus provides a single method that is uniformly applicable to all equations of state

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

the hill determinant method in application to the sextic oscillator limitations and improvement using the sextic anharmonic oscillator a chi 2 b chi 4 c chi 6 as a testing ground it is shown why the hill determinant method has a limited range of applicability a possible improvement is suggested and discussed

vorticity velocity formulation for three dimensional steady compressible flows the vorticity velocity formulation of the navier stokes equations is extended to the solution of three dimensional compressible fluid flow and heat transfer problems the basic governing equations are expressed in terms of three poisson like equations for the velocity components together with a vorticity transport equation and an energy equation the resulting seven coupled partial differential equations are solved by a finite difference method on a single grid and a discrete solution is obtained by combining a steady state and a time dependent newton s method once a converged solution is obtained one of the velocity equations can be removed from the system and replaced by the continuity equation and a conservative solution is obtained by using the previous solution as a starting estimate for newton s method with only a few additional iterations the numerical procedure is evaluated by applying it to natural and mixed convection problems the formulation is found to be stable at high rayleigh numbers and it may be applied to a wide variety of flow and heat transfer problems