black box multigrid for nonsymmetric problems one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in previous work this defect was overcome and a code written in which one needed only to specify the logically rectangular positive definite matrix problem the code did everything else necessary to set up the auxiliary coarser problems to achieve a multigrid solution in this paper the previous work is extended to a single nonsymmetric equation

accuracy analysis of explicit runge kutta methods applied to the incompressible navier stokes equations this paper investigates the temporal accuracy of the velocity and pressure when explicit runge kutta methods are applied to the incompressible navier stokes equations it is shown that at least up to and including fourth order the velocity attains the classical order of accuracy without further constraints however in case of a time dependent gradient operator which can appear in case of time varying meshes additional order conditions need to be satisfied to ensure the correct order of accuracy furthermore the pressure is only first order accurate unless additional order conditions are satisfied two new methods that lead to a second order accurate pressure are proposed which are applicable to a certain class of three and four stage methods a special case appears when the boundary conditions for the continuity equation are independent of time since in that case the pressure can be computed to the same accuracy as the velocity field without additional cost relevant computations of decaying vortices and of an actuator disk in a time dependent inflow support the analysis and the proposed methods

on the effect of numerical errors in large eddy simulations of turbulent flows aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods analytical and numerical studies show that aliasing errors are more destructive for spectral and high order finite difference calculations than for low order finite difference simulations numerical errors have different effects for different forms of the nonlinear terms in the navier stokes equations for divergence and convective forms spectral methods are energy conserving only if dealiasing is performed for skew symmetric and rotational forms both spectral and finite difference methods are energy conserving even in the presence of aliasing errors it is shown that discrepancies between the results of dealiased spectral and standard nondialiased finite difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences the relative importance of aliasing and truncation errors as compared to subgrid scale model terms in large eddy simulations is analyzed and discussed for low order finite difference simulations truncation errors can exceed the magnitude of the subgrid scale term

energy conserving runge kutta methods for the incompressible navier stokes equations energy conserving methods have recently gained popularity for the spatial discretization of the incompressible navier stokes equations in this paper implicit runge kutta methods are investigated which keep this property when integrating in time firstly a number of energy conserving runge kutta methods based on gauss radau and lobatto quadrature are constructed these methods are suitable for convection dominated problems such as turbulent flows because they do not introduce artificial diffusion and are stable for any time step secondly to obtain robust time integration methods that work also for stiff problems the energy conserving methods are extended to a new class of additive runge kutta methods which combine energy conservation with l stability in this class the radau iia b method has the best properties results for a number of test cases on two stage methods indicate that for pure convection problems the additive radau iia b method is competitive with the gauss methods however for stiff problems such as convection dominated flows with thin boundary layers both the higher order gauss and radau iia b method suffer from order reduction overall the gauss methods are the preferred method for energy conserving time integration of the incompressible navier stokes equations

energy preserving turbulent simulations at a reduced computational cost energy conserving discretizations are widely regarded as a fundamental requirement for high fidelity simulations of turbulent flows the skew symmetric splitting of the nonlinear term is a well known approach to obtain semi discrete conservation of energy in the inviscid limit however its computation is roughly twice as expensive as that of the divergence or advective forms alone a novel time advancement strategy that retains the conservation properties of skew symmetric based schemes at a reduced computational cost has been developed this method is based on properly constructed runge kutta schemes in which a different form advective or divergence for the convective term is adopted at each stage a general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew symmetric based techniques

some random approximation theorems with applications fan 8 proved a very interesting theorem which provides a tool to study xed point theory in connection with best approximation various aspects of that theorem have been studied by fan 9 ha 10 singh and watson 22 lin 14 lin and yen 16 and many others fan s theorem has been of great importance in nonlinear analysis approximation theory game theory and minimax theorems the study of random approximations and random xed points have been a very active area of research in probabilistic functional analysis in the last decade see sehgal and waters 21 sehgal and singh 20 papageorgiou 18 lin 15 xu 25 tan and yuan 23 24 and beg and shahzad 1 4 most random xed point theorems in banach spaces deal with condensing or nonexpansive random operators what about the random operators which are neither of the above cases the interesting case would be a 1 set contractive random operator the class of 1 set contractive random operators includes condensing and nonexpansive random operators besides it also includes other important random operators such as semicontractive random operators and lane locally almost nonexpansive random operators recently beg and shahzad 3 studied these random operators in details and gave many results regarding random approximations and random

random fixed point equations and inverse problems using collage method for contraction mappings abstract in this paper we are interested in the direct and inverse problems for the following class of random fixed point equations t w x w x w where t x x is a given operator is a probability space and x is a polish metric space the inverse problem is solved by recourse to the collage theorem for contractive maps we then consider two applications i random integral equations and ii random iterated function systems with greyscale maps rifsm for which noise is added to the classical ifsm

coincidence points generalized i nonexpansive multimaps and applications we establish coincidence point results for multimaps satisfying generalized i nonexpansive type conditions several invariant approximation results are obtained as applications a random coincidence point result is also proved and as applications random invariant approximation results are obtained our results unify extend and complement several well known results

a fixed point theorem for condensing operators and applications to hammerstein integral equations in banach spaces abstract we prove a fixed point theorem of furi pera type for condensing operators this will then be used to establish the existence of solutions to certain integral equations of hammerstein type in banach spaces

basic smoothing procedures for the multigrid treatment of elliptic 3d operators from 2d multigrid it is a fundamental insight that pointwise relaxation combined with standard coarsening gives no reasonable smoothing effect if the given elliptic operator shows appreciably anisotropic behavior corresponding questions and practical consequences are discussed in this paper for the 3d case in the general anisotropic 3d case even line relaxation is not sufficient if standard coarsening is maintained instead plane relaxation is necessary in certain cases if plane relaxation is applied correctly and performed by use of appropriate 2d multigrid the resulting 3d multigrid method has an asymptotic complexity of o n where n number of 3d grid points and is highly efficient even for moderate values of n in contrast to the common opinion plane relaxation turns out to be a simple and general smoothing concept for standard elliptic 3d problems

cone compression and expansion fixed point theorems in frechet spaces with applications abstract new fixed point theory for multivalued maps between frechet spaces is presented applications are given to illustrate how the theory can be applied in practice

a parametric level set based collage method for an inverse problem in elliptic partial differential equations in this work based on the collage theorem we develop a new numerical approach to reconstruct the locations of discontinuity of the conduction coefficient in elliptic partial differential equation

on some versions of incomplete block matrix factorization iterative methods abstract various forms of preconditioning matrices for iterative acceleration methods are discussed the preconditioning is based on two versions of incomplete block matrix factorization

fixed point theory for admissible multimaps defined on closed subsets of frechet spaces abstract this paper presents new fixed point results for a general class of maps defined on frechet spaces our results in particular apply to kakutani acyclic o neill approximable admissible with respect to gorniewicz and u c maps

explicit runge kutta schemes for incompressible flow with improved energy conservation properties the application of pseudo symplectic runge kutta methods to the incompressible navier stokes equations is discussed in this work in contrast to fully energy conserving implicit methods these are explicit schemes of order p that preserve kinetic energy to order q with q p use of explicit methods with improved energy conservation properties is appealing for convection dominated problems especially in case of direct and large eddy simulation of turbulent flows a number of pseudo symplectic methods are constructed for application to the incompressible navier stokes equations and compared in terms of accuracy and efficiency by means of numerical simulations

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

random fixed points of k set and pseudo contractive random maps abstract let be a measurable space x and y separable banach spaces and c a weakly compact subset of x let f c y and t c y be continuous random operators then the deterministic solvability of the equation f x t x 0 x c implies the stochastic solvability of it provided that f t is demiclosed at zero and t c is bounded for each as applications random fixed points of various types of pseudo contractive and k set contractive random operators are obtained

kinetic energy conservation issues associated with the collocated mesh scheme for incompressible flow the collocated mesh scheme is often favored over the staggered mesh scheme for turbulence simulation in complex geometries due to its simpler form in curvilinear coordinates the collocated mesh scheme does not conserve kinetic energy however and few careful checks of the impact of these errors have been made in this work analysis is used to identify two sources of kinetic energy conservation error in the collocated mesh scheme 1 errors arising from the interpolations used to estimate the velocity on the cell faces and 2 errors associated with the slightly inconsistent pressure field used to ensure mass conservation for the cell face volume fluxes it is shown that the interpolation error can be eliminated through the use of first order accurate centered interpolation operators with mesh independent weights the pressure error appears to be intrinsic to the scheme and it is shown to scale as o t2 x2 the effects of the conservation errors are investigated numerically through simulations of inviscid flow over an airfoil and in large eddy simulations of turbulent channel flow neither the interpolation error nor the pressure error appear to lead to significant problems in the channel flow simulations where viscous dissipation is present and where the cartesian mesh is stretched in only one direction the inviscid airfoil simulations performed on a curvilinear mesh show a much greater sensitivity to the interpolation error the standard second order centered interpolation is shown to lead to severe numerical oscillations while the kinetic energy conserving first order centered interpolation produces solutions that are almost as smooth as those obtained with a third order upwind interpolation when compared in channel flow simulations however the first order centered interpolation is shown to be far superior to the third order upwind interpolation the latter being adversely affected by numerical dissipation only slight differences were noted when comparing channel flow simulations using first order and second order centered interpolations these results suggest that numerical oscillations can be controlled in curvilinear coordinates through the use of properly constructed non dissipative centered interpolations

aspects of multigrid methods for problems in three dimensions smoothing methods are analyzed for use in multigrid methods applied to flow problems in three dimensions a three dimensional version of incomplete block factorization is described the anisotropic diffusion and convection diffusion equations are used as model problems unlike the two dimensional case none of the smoothing methods considered is suitable for all relevant values of the physical parameters

random fixed points of set valued maps some random fixed point theorems for set valued random operators under very mild conditions are established some recent results of o regan proc amer math soc 126 1998 3045 3053 and computers math applic 35 1998 27 34 are improved significantly the discussion in this paper underlines in addition to generality the unifying aspects of our result

using the collage method to solve odes inverse problems with multiple data sets abstract we consider inverse problems for ordinary differential equations odes where multiple noisy observational data sets are available the inverse problem for a single data set asks us to find an ode or system thereof that admits the interpolated data set as an approximate solution such an inverse problem can be cast within the framework of fixed point equations so that we seek to approximate the data values by a fixed point of an operator related to the odes the collage coding method replaces the minimization of this approximation error by the minimization of the so called collage distance the purpose of this paper is to formulate and explore various methods for dealing with the collection of collage distances corresponding to multiple noisy data sets we present three schemes averaging the parameter estimates obtained by minimizing each of the separate collage distances and two approaches arising from goal programming numerous examples are presented

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

multigrid methods development of fast solvers numerical and programming aspects are discussed of multigrid algorithms for the solution of discretized linear elliptic equations the aim is to obtain software that is perceived and can be used just like any standard subroutine for solving systems of linear equations the user has to specify only the matrix and the right hand side and remains unaware of the underlying multigrid method we find that a large class of equations can be solved efficiently in this way the equation may be non self adjoint and its coefficients are arbitrary special attention is given to the treatment of the convection diffusion equation at high peclet number details are given of an available portable fortran code which vectorizes satisfactorily on vector machines cp time statistics are given for a cyber 205

random fixed points of 1 set contractive random maps in frechet spaces a random fixed point theorem for more general 1 set contractive random operators in frechet spaces is proved as applications some random fixed point theorems for nonexpansive semicontractive and lane random operators are also derived thus the results of itoh j math anal appl 67 1979 261 273 lin proc amer math soc 123 1995 1167 1176 proc amer math soc 103 1988 1129 1135 shahzad j math anal appl 203 1996 712 718 tan and yuan stochastic anal appl 15 1997 103 123 and xu proc amer math soc 110 1990 495 500 are extended

an efficient time advancing strategy for energy preserving simulations energy conserving numerical methods are widely employed within the broad area of convection dominated systems semi discrete conservation of energy is usually obtained by adopting the so called skew symmetric splitting of the non linear convective term defined as a suitable average of the divergence and advective forms although generally allowing global conservation of kinetic energy it has the drawback of being roughly twice as expensive as standard divergence or advective forms alone in this paper a general theoretical framework has been developed to derive an efficient time advancement strategy in the context of explicit runge kutta schemes the novel technique retains the conservation properties of skew symmetric based discretizations at a reduced computational cost it is found that optimal energy conservation can be achieved by properly constructed runge kutta methods in which only divergence and advective forms for the convective term are used as a consequence a considerable improvement in computational efficiency over existing practices is achieved the overall procedure has proved to be able to produce new schemes with a specified order of accuracy on both solution and energy the effectiveness of the method as well as the asymptotic behavior of the schemes is demonstrated by numerical simulation of burgers equation

multi grid solution of three dimensional problems with discontinuous coefficients a three dimensional multi grid algorithm is developed and compared to an efficient three dimensional incomplete cholesky conjugate gradient iccg algorithm on a suite of test problems with discontinouous and or anisotropic coefficients the multi grid algorithm used plane relaxation and a definition of coarse grid operators which is an extension of that used for two dimensional problems with discontinuous coefficients numerical results show that the three dimensional results are not nearly as good as the two dimensional results on similar problems

numerically stable formulations of convective terms for turbulent compressible flows abstract a systematic analysis of the discrete conservation properties of non dissipative central difference approximations of the compressible navier stokes equations is reported a generalized splitting of the nonlinear convective terms is considered and energy preserving formulations are fully characterized by deriving a two parameter family of split forms previously developed formulations reported in literature are shown to be particular members of this family novel splittings are introduced and discussed as well furthermore the conservation properties yielded by different choices for the energy equation i e total and internal energy entropy are analyzed thoroughly it is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family local conservation of primary invariants which is a fundamental property to build high fidelity shock capturing methods is also discussed in the paper numerical tests performed for the taylor green vortex at zero viscosity fully confirm the theoretical findings and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results

random fixed point theorems and approximation in cones a very general random fixed point theorem is first proved as applications some random approximation theorems and random fixed point theorems are obtained recent results due to itoh j math anal appl 67 1979 261 273 lin proc amer math soc 103 1988 502 506 and xu proc amer math soc 110 1990 495 500 either are improved or are direct consequences of our results the stochastic versions of the recent results of lin 1988 and li proc amer math soc 97 1986 277 280 are also given

solution of an inverse problem for fractals and other sets abstract the problem of providing succinct approximate descriptions of given bounded subsets of rn can be solved by application of the contraction mapping principle

inverse problems for random differential equations using the collage method for random contraction mappings in this paper we are concerned with differential equations with random coefficients which will be considered as random fixed point equations of the form t w x w x w w w here t wxx x is a random integral operator w f p is a probability space and x is a complete metric space we consider the following inverse problem for such equations given a set of realizations of the fixed point of t possibly the interpolations of different observational data sets determine the operator t or the mean value of its random components as appropriate we solve the inverse problem for this class of equations by using the collage theorem for contraction mappings

a simple cell centered multigrid method for 3d interface problems abstract a multigrid algorithm is presented for cell centered discretizations of interface problems instead of constructing the coarse grid operators by means of the galerkin approximation the coarse grid operators are obtained by discretization on the coarse grids the advantage of this approach is that we obtain m matrices on all grids and that the sparsity pattern of the fine grid matrix is retained on all grids moreover the coarse grid operators are very easy to construct numerical results of several test problems are presented

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications