soliton solutions for quasilinear schrodinger equations the critical exponential case abstract quasilinear elliptic equations in r 2 of second order with critical exponential growth are considered by using a change of variable the quasilinear equations are reduced to semilinear equations whose respective associated functionals are well defined in h 1 r 2 and satisfy the geometric hypotheses of the mountain pass theorem using this fact we obtain a cerami sequence converging weakly to a solution v in the proof that v is nontrivial the main tool is the concentration compactness principle p l lions the concentration compactness principle in the calculus of variations the locally compact case part i and ii ann inst h poincare anal non lineaire 1 1984 109 145 223 283 combined with test functions connected with optimal trudinger moser inequality

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

nonlinear schrodinger equation with unbounded or decaying radial potentials involving exponential critical growth in r2 abstract we study the existence and multiplicity of solutions for the following class of nonlinear schrodinger equations u v x u q x f u in r 2 where v and q are unbounded or decaying radial potentials and the nonlinearity f s has exponential critical growth the approaches used here are based on a version of the trudinger moser inequality and a minimax theorem

multiple solutions for quasilinear schrodinger equations involving critical exponent by using lions second concentration compactness principle and concentration compactness principle at infinity to prove that the ps condition holds locally and by minimax methods and the krasnoselski genus theory we establish the multiplicity of solutions for a class of quasilinear schrodinger equations arising from physics

essential singularity in the xy spin 1 chain with uniaxial anisotropy we have performed finite cell calculations on the one dimensional spin 1 xy model with transverse anisotropy infinite size extrapolation of the results shows strong evidence for a transition at t 0 by increasing the strength of the anisotropy with peculiar scaling properties characteristic of an essential singularity thus this quantum chain which is a simple truncated version of the quantum formulation of the classical 2d xy model behaves very similarly to its classical analogue

on a nonhomogeneous and singular quasilinear equation involving critical growth in r2 abstract this paper establishes sufficient conditions for the existence and multiplicity of solutions for nonhomogeneous and singular quasilinear equations of the form u v x u u 2 u g x u x a h x in r 2 where a 0 2 v x is a continuous positive potential bounded away from zero and which can be large at infinity the nonlinearity g x s is allowed to enjoy the critical exponential growth with respect to the trudinger moser inequality and the nonhomogeneous term h belongs to l q r 2 for some q 1 2 by combining variational arguments in a nonstandard orlicz space context with a singular version of the trudinger moser inequality we obtain the existence of two distinct solutions when h q is sufficiently small schrodinger equations of this type have been studied as models of several physical phenomena

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

unsteady solution of incompressible navier stokes equations abstract a new time accurate finite difference numerical method for solving incompressible navier stokes equations is presented with primitive variables as the unknowns the numerical scheme is a crank nicolson implicit treatment of all terms of the equations with central differencing for space derivatives the pressure and the nonlinearities in convection terms are not lagged to obtain the incompressible solution at the advanced time level a continuous auxiliary system is introduced in pseudo time with artificial compressibility time accurate solutions are presented for two dimensional fluid flows in a square cavity with an impulsively starting lid and with an oscillating lid

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

stability analysis of uniform and non uniform annular passages conducting incompressible laminar flows for small and large amplitude oscillatory motions of the outer cylinder abstract a computational method is developed involving the simultaneous integration of the navier stokes and structural equations for the purpose of studying the stability of concentric annular passages conducting incompressible laminar flows it is assumed that one side of the annulus i e the centre body is fixed and the outer cylindrical duct is flexibly supported the outer cylinder is displaced from its equilibrium position and is then released in this situation the fluid part of the problem is solved by an accurate method using a three point backward implicit scheme followed by a pseudo time iteration using an artificial compressibility factor the fluid equations are discretized in space based on a finite difference formulation and primitive variables for which stretched staggered grids are used the resulting equations are cast in delta form and are solved using an adi scheme the fluid forces acting on the vibrating cylinder are calculated from the integration of the unsteady pressure and shear stresses resulting from the unsteady primitive variables calculated the equations of motion of the structure subjected to the calculated fluid forces are solved using the runge kutta scheme to obtain the displacement of the moving cylinder the problem is solved a for small amplitude motions by means of the so called mean position mp analysis and b for large amplitude oscillations of the outer cylinder for which a time dependent coordinate transformation tdct is used to fix the computational domain both these approaches mp and tdct are applied to uniform and non uniform backstep shaped annuli for translational motion of the cylinder the problem is also solved for i rocking motion and uniform annuli and ii translational motion for diffuser shaped annuli with only mp analysis when the mp analysis is considered it is shown that for translational motion of the outer cylinder the most stable configuration is that of a uniform geometry and the least stable one is the backward step geometry for rocking motion in uniform annular geometry for some system parameters and high enough flow velocity the outer cylinder can develop flutter limit cycle oscillation the comparison between the results of mp and tdct analyses for uniform and backstep geometries for translational motion of the cylinder indicates that the outer cylinder is less stable when tdct is used and the coupled frequency of oscillation is also changed considerably

finite lattice methods in quantum hamiltonian field theory i o 2 and o 3 heisenberg models for pt i see ibid vol 14 p 241 57 1981 two efficient methods for finding the low lying states of hamiltonians on finite lattices are described the first involves constructing a finite representation of the hamiltonian using strong coupling eigenstates while the second is based on the lanczos recursion method the methods are used to determine the mass gap of the o 2 and o 3 heisenberg hamiltonians in 1 1 dimensions for a sequence of finite chains the critical behaviour of the infinite chain is then analysed by extrapolating the finite lattice estimates using finite size scaling a remarkably sensitive test is developed for the presence of a phase transition for the o 2 model data this test yields strong evidence for a phase transition with the weak coupling phase massless while in the o 3 case the test supports although more weakly the absence of any transition

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

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

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

on a class of periodic quasilinear schrodinger equations involving critical growth in r2 abstract we consider the equation u v x u k 2 u 2 u g x u u 0 x r 2 where v r 2 r and g r 2 r r are two continuous 1 periodic functions and k is a positive constant also we assume g behaves like exp u 4 as u we prove the existence of at least one weak solution u h 1 r 2 with u 2 h 1 r 2 the mountain pass in a suitable orlicz space together with the trudinger moser inequality are employed to establish this result such equations arise when one seeks for standing wave solutions for the corresponding quasilinear schrodinger equations schrodinger equations of this type have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

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

conservation properties of vectorial operator splitting this work is concerned with the conservation properties of a new vectorial operator splitting scheme for solving the incompressible navier stokes equations it is proven that the difference approximation of the advection operator conserves square of velocity components and the kinetic energy as the differential operator does while pressure term conserves only the kinetic energy some analytical requirements necessary to be satisfied of difference schemes for incompressible navier stokes equations are formulated and discussed the properties of the methods are illustrated with results from numerical computations for lid driven cavity flow

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

solutions for a quasilinear schrodinger equation a dual approach abstract we consider quasilinear stationary schrodinger equations of the form 1 u u 2 u g x u x r n introducing a change of unknown we transform the search of solutions u x of 1 into the search of solutions v x of the semilinear equation 2 v 1 1 2f 2 v g x f v x r n where f is suitably chosen if v is a classical solution of 2 then u f v is a classical solution of 1 variational methods are then used to obtain various existence results

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

quasilinear schrodinger equations with singular and vanishing potentials involving exponential critical growth in r2 abstract we study the existence and nonexistence of solution for the following class of quasilinear schrodinger equations u v x u u 2 u q x h u x r 2 u x 0 as x where v and q are potentials that can be singular at the origin unbounded or vanishing at infinity and the nonlinearity h s is allowed to satisfy the exponential critical growth with respect to the trudinger moser inequality by combining variational methods in a suitable weighted orlicz space with a version of the trudinger moser inequality for this space we obtain the existence of a nonnegative ground state solution for this we have used some regularity results and we need to explore a symmetric criticality type argument moreover under some conditions on v q and h s we prove that this problem does not have positive radial solution schrodinger equations of this type have been studied as models of several physical phenomena

a nonhomogeneous elliptic problem involving critical growth in dimension two abstract in this paper we study a class of nonhomogeneous schrodinger equations u v x u f u h x in the whole two dimension space where v x is a continuous positive potential bounded away from zero and which can be large at the infinity the main difficulty in this paper is the lack of compactness due to the unboundedness of the domain besides the fact that the nonlinear term f s is allowed to enjoy the critical exponential growth by means of the trudinger moser inequality by combining variational arguments and a version of the trudinger moser inequality we establish the existence of two distinct solutions when h is suitably small

time marching solution of incompressible navier stokes equations for internal flow abstract a factored adi finite difference scheme has been developed for solution of the two dimensional incompressible navier stokes equations by the artificial compressibility method the scheme employs primitive variables with central differencing on a staggered grid the resulting close coupling between pressure and velocity produces enhanced stability and eliminates the need for artificial damping a spatially variable time step based on a fixed courant number is used to improve computational efficiency numerical results have been obtained for flows in a straight channel a curved rotating channel and a driven cavity in the latter case at reynolds number of 10 000 with local cell reynolds numbers as high as 100 a solution on a 40 40 stretched grid shows no spatial oscillations in the flow variables

finite lattice calculation of the quantum mechanical n vector model the truncated quantum mechanical n vector model in one dimension is studied by means of the phenomenological renormalisation group prg and the block renormalisation group brg for general values of n particular emphasis is put on the extrapolation of the finite size results using the finite size scaling hypothesis accurate estimates of the critical properties can be made for 0 or approximately n or approximately 2 the two methods used are critically compared