simulations of singularity dynamics in liquid crystal flows a c0 finite element approach in this paper we present a c 0 finite element method for a 2d hydrodynamic liquid crystal model which is simpler than existing c 1 element methods and mixed element formulation the energy law is formally justified and the energy decay is used as a validation tool for our numerical computation a splitting method combined with only a few fixed point iteration for the penalty term of the director field is applied to reduce the size of the stiffness matrix and to keep the stiffness matrix time independent the latter avoids solving a linear system at every time step and largely reduces the computational time especially when direct linear system solvers are used our approach is verified by comparing its computational results with those obtained by c 1 elements and by mixed formulation through numerical experiments of a few other splittings and explicit implicit strategies we recommend a fast and reliable algorithm for this model a number of examples are computed to demonstrate the algorithm

accurate three dimensional lid driven cavity flow a chebyshev collocation method in space is introduced which allows an accurate calculation of three dimensional lid driven cavity flows the time integration is carried out by an adams bashforth backward euler scheme the accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flow field this way the most severe discontinuities in the boundary conditions which arise along the lines where moving and stationary walls meet are taken care of analytically and thus do not spoil the numerical part of the solution calculations are carried out for no slip boundary conditions at the cavity end walls as well as for periodic end wall conditions in general the spatial accuracy is better than fifth order for rigid end wall conditions the accuracy is reduced near the end walls to o n 3 2 but recovers in the bulk tabulated data are provided for the most interesting flow properties

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

long time numerical solution of the navier stokes equations based on a sequential regularization formulation the sequential regularization method is a reformulation of the unsteady navier stokes equations from the viewpoint of constrained dynamical systems or the approximate helmholtz hodge projection in this paper we study the long time behavior of the sequential regularization formulation we give a uniform in time estimate between the solution of the reformulated system and that of the navier stokes equations we also conduct an error analysis for the temporal discrete system and show that the error bound is independent of time a couple of long time flow examples are computed to demonstrate this method

upwind differencing scheme for the time accurate incompressible navier stokes equations the two dimensional incompressible navier stokes equations are solved in a time accurate manner using the method of pseudocompres sibility using this method subiterations in pseudotime are required to satisfy the continuity equation at each time step an upwind differencing scheme based on flux difference splitting is used to compute the convective terms the upwind differencing is biased based on the sign of the local eigenvalue of the jacobian matrix of the convective fluxes both third order and fifth order differencing schemes are used on the convective fluxes throughout the grid s interior the equations are solved using an implicit line relaxation scheme this solution scheme is stable and is capable of running at large time steps in pseudo time leading to fast convergence for each physical time step a variety of computed results are presented to validate the present scheme results for the flow over an oscillating plate are compared with the exact analytic solution and good agreement is seen excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure flow solutions over a circular cylinder with vortex shedding are also presented finally the flow past an airfoil at 90 angle of attack is computed

impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in banach spaces abstract in this paper we deal with the existence of impulsive mild solutions for semilinear differential inclusions with nonlocal conditions where the linear part generates an evolution system and the nonlinearity satisfies the lower scorza dragoni property our theorems extend the existence propositions proved by fan in 2010 an example is presented

application of dispersion relation preserving theory to develop a two dimensional convection diffusion scheme in this paper a finite difference scheme is developed within the nine point semi discretization framework for the convection diffusion equation the employed pade approximation renders a fourth order temporal accuracy and the spatial approximation of convection terms accommodates the dispersion relation the artificial viscosity introduced in the two dimensional convection diffusion reaction cdr equation for stability reasons is analytically derived constraints on the mesh size and time interval for rendering a monotonic matrix are also rigorously derived to validate the proposed method we investigate several problems that are amenable to the exact solutions the results with good rates of convergence are obtained for the investigated scalar and navier stokes problems

mathematical models for the control of a pest population by infected pest in this paper we propose two mathematical models concerning continuous and impulsive pest control strategies respectively therefore our models are the ordinary differential equations and the impulsive differential equations as a result the global asymptotic stability of the equilibria of the ordinary differential equations is studied in the case when an impulsive control is used it is observed that there exists a globally asymptotically stable susceptible pest eradication periodic solution when the amount of infective pests released periodically is larger than some critical value when the amount of infective pests released is less than this critical value the system is shown to be permanent which implies that the trivial susceptible pest eradication solution loses its stability finally by means of numerical simulation we obtain the critical values of the control variable under different methods of release of infected pests

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

sequential regularization methods for nonlinear higher index daes sequential regularization methods relate to a combination of stabilization methods and the usual penalty method for differential equations with algebraic equality constraints this paper extends an earlier work siam j numer anal 33 1996 pp 1921 1940 to nonlinear problems and to differential algebraic equations daes with an index higher than 2 rather than having one winning method this is a class of methods from which a number of variants are singled out as being particularly effective methods in certain circumstances we propose sequential regularization methods for index 2 and index 3 daes both with and without constraint singularities in the case of no constraint singularity we prove convergence results numerical experiments confirm our theoretical predictions and demonstrate the viability of the proposed methods the examples include constrained multibody systems

a note on the solvability of singular boundary value problems theorem suppose that 4 at u 1 e cw 11 x r pw e cp 11 and q t e w 11 withd0 q t 0 on 0 11 p o 0 and p t q t bounded on 0 11 b s l at df 00 s q t df ml m2 i iiwmo

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

a fully explicit method for incompressible flow computation abstract a new formulation of the navier stokes equations is introduced to solve incompressible flow problems it keeps the benefits of the penalty method that is velocity and pressure can be obtained separately and no pressure poisson equation is involved unlike the penalty method the formulation is more stable or less stiff and then explicit time integration can be applied for easy implementation no linear or nonlinear system need be solved in the method in the case that a large number of time steps are needed a parallelization based on domain decomposition is applied to reduce the computational time with the explicit time integration the parallel implementation and its message passing are very simple as well

an energy law preserving c0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics in this paper we use finite element methods to simulate the hydrodynamical systems governing the motions of nematic liquid crystals in a bounded domain w we reformulate the original model in the weak form which is consistent with the continuous dissipative energy law for the flow and director fields in w 1 2 s w s 0 is an arbitrarily small number this enables us to use convenient conformal c 0 finite elements in solving the problem moreover a discrete energy law is derived for a modified midpoint time discretization scheme a fixed iterative method is used to solve the resulted nonlinear system so that a matrix free time evolution may be achieved and velocity and director variables may be solved separately a number of hydrodynamical liquid crystal examples are computed to demonstrate the effects of the parameters and the performance of the method

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

a divergence free condition compensated method for incompressible navier stokes equations the present study aims to develop a new formulation to effectively calculate the incompressible navier stokes solutions in non staggered grids the distinguished feature of the proposed method which avoids directly solving the divergence free equation is to add a rigorously derived source term to the momentum equation to ensure satisfaction of the fluid incompressibility for the sake of numerical accuracy dispersion relation preserving upwind scheme developed within the two dimensional context was employed to approximate the convection terms the validity of the proposed mass preserving navier stokes method is justified by solving two benchmark problems at high reynolds and rayleigh numbers based on the simulated navier stokes solutions the proposed formulation is shown to outperform the conventional segregated method in terms of the reduction of cpu time 2007 elsevier b v all rights reserved

parallel auxiliary space amg solver for h div problems in this paper we present a family of scalable preconditioners for matrices arising in the discretization of h div problems using the lowest order raviart thomas finite elements our approach belongs to the class of auxiliary space based methods and requires only the finite element stiffness matrix plus some minimal additional discretization information about the topology and orientation of mesh entities we provide a detailed algebraic description of the theory parallel implementation and different variants of this parallel auxiliary space divergence solver ads and discuss its relations to the hiptmair xu hx auxiliary space decomposition of h div siam j numer anal 45 2007 pp 2483 2509 and to the auxiliary space maxwell solver ams j comput math 27 2009 pp 604 623 an extensive set of numerical experiments demonstrates the robustness and scalability of our implementation on large scale h div problems with large jumps in the material coefficients

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

analysis of a sequential regularization method for the unsteady navier stokes equations the incompressibility constraint makes navier stokes equations difficult a reformulation to a better posed problem is needed before solving it numerically the sequential regularization method srm is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods in the paper we study the existence and uniqueness for the solution of the srm and provide a simple proof of the convergence of the solution of the srm to the solution of the navier stokes equations we also give error estimates for the time discretized srm formulation

an efficient numerical method for the equations of steady and unsteady flows of homogeneous incompressible newtonian fluid in the present paper we present some numerical methods to solve the equations of steady and unsteady flows such as those in the microcirculatory bed and large blood vessels arteries and veins respectively in the case of steady flows the method does not need neither any boundary conditions on pressure nor any small parameter and the main computation consists of solving some poisson equations in the case of unsteady flows the scheme uses a consistent neumann boundary condition for the pressure poisson equation at each time step a poisson and heat equation are solved for the pressure and each velocity component respectively the accuracy and efficiency of scheme are checked by a set of numerical tests

some current cfd issues relevant to the incompressible navier stokes equations abstract the goals of this paper are to carefully define a particular class of well set incompressible navier stokes problems in the continuum partial differential equation pde setting and to discuss some relevant and sometimes poorly understood issues related to these well posed pde problems both in the continuum world and in its computer counterpart emphasized are three common formulations used to generate computer codes the primitive variable velocity pressure formulation and two derived variable formulations pressure poisson equation and vorticity transport equation

stability and convergence of efficient navier stokes solvers via a commutator estimate for strong solutions of the incompressible navier stokes equations in bounded domains with velocity specified at the boundary we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure these schemes require no solution of stationary stokes systems nor any compatibility between velocity and pressure spaces to ensure an inf sup condition and are representative of a class of highly efficient computational methods that have recently emerged the proofs are simple based upon a new sharp estimate for the commutator of the laplacian and helmholtz projection operators this allows us to treat an unconstrained formulation of the navier stokes equations as a perturbed diffusion equation c 2006 wiley periodicals inc

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

a predicted sequential regularization method for index 2 hessenberg daes the sequential regularization method srm is a dynamic iterative method for the numerical solution of higher index differential algebraic equations daes the srm has the advantage of being based on a regularized problem that is less stiff than those produced by standard regularization methods consequently nonstiff integrators may be used making the srm a competitive alternative to popular integrators in past work the number of srm iterations was taken to be roughly equal to the order of the numerical method used in each dynamic iteration in this paper we propose a predicted srm psrm that reduces the number of iterations in each dynamic iteration to one we give a new error analysis for explicit runge kutta methods applied to linear index 2 hessenberg daes with or without singularities we also give numerical examples to confirm the predicted convergence rates for the psrm extrapolation formulas and methods based on the differential part of the daes serve as a predictor and the srm iteration serves as a corrector implementation of higher order schemes for the psrm makes use of continuous extensions of runge kutta methods in particular we give a prediction scheme for the algebraic variable at intermediate stage points that suppresses order reduction in the differential variable near a singularity moreover the srm psrm provides new insight into operator splitting and fast convergence rates for waveform relaxation

an iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media the miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure velocity equation and the concentration equation an iterative perturbation procedure is proposed and analyzed for the pressure velocity equation which is capable of producing as accurate a velocity approximation as the mixed finite element method and which requires the solution of symmetric positive definite linear systems only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution trivially applying perturbation methods can only give an error o epsilon while our iterative scheme can improve the error to o epsilon m at the m th iteration level where epsilon is a small positive number thus the convergence rate of our iterative procedure is o epsilon and consequently a small number of iterations is required theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method

an element by element bicgstab iterative method for three dimensional steady navier stokes equations construction of a stabilized galerkin upwind finite element model for steady and incompressible navier stokes equations in three dimensions is the main theme of this study in the time independent context the weighted residuals statement is kept biased in favor of the upstream flow direction by adding an artificial damping term of physical plausibility to the galerkin framework this upwind approach has significant advantage of seeking solutions free from cross stream diffusion error finite element solutions have been found by mixed formulation implemented in quadratic cubic elements which are characterized as possessing the so called lbb ladyzhenskaya babuska brezzi condition an element by element bicgstab solution solver is intended to alleviate difficulties regarding the asymmetry and indefiniteness arising from the use of a mixed formulation for incompressible fluid flows the developed three dimensional finite element code is first rectified by solving a problem amenable to analytic solution a well known lid driven cavity flow problem in a cubical cavity is also studied

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

an effective explicit pressure gradient scheme implemented in the two level non staggered grids for incompressible navier stokes equations in this paper an improved two level method is presented for effectively solving the incompressible navier stokes equations this proposed method solves a smaller system of nonlinear navier stokes equations on the coarse mesh and needs to solve the oseen type linearized equations of motion only once on the fine mesh level within the proposed two level framework a prolongation operator which is required to linearize the convective terms at the fine mesh level using the convergent navier stokes solutions computed at the coarse mesh level is rigorously derived to increase the prediction accuracy this indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic solution convergence can therefore be accelerated for the sake of numerical accuracy momentum equations are discretized by employing the general solution for the two dimensional convection diffusion reaction model equation the convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms the converged solution is thus very high in accuracy as well as in yielding a quadratic spatial rate of convergence for the sake of programming simplicity and computational efficiency pressure gradient terms are rigorously discretized within the explicit framework in the non staggered grid system the proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme which accommodates the dispersion relation preserving property have been both rigorously justified from the predicted navier stokes solutions

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

element by element parallel computation of incompressible navier stokes equations in three dimensions development of a stable finite element model for solving steady incompressible viscous fluid flows in three dimensions is the main theme of the present study for stability reasons weighting functions are designed in favor of field variables on the upstream side for accuracy reasons it is required that weighting functions be equipped with the streamline operator so that false diffusion errors can be largely suppressed in the steady state analysis of navier stokes equations we adopt the mixed formulation to preserve mass conservation on quadratic elements which accommodate the ladyzhenskaya babuska brezzi lbb stability condition to resolve difficulties arising from asymmetry and indefiniteness in the resulting large size matrix equations we abandon the elimination like solution solver because the storage demand exceeds the ability of our hardware to solve for three dimensional problems a modern iteration solver known as the biconjugate gradiant stabilized bicgstab solution solver is thus implemented in an element by element fashion to effectively alleviate the problem for performance reasons the finite element code developed here should be implemented in a hardware environment which is suited to the use of an iterative solver to this end our analysis is implemented in shared memory parallel architectures cray c 90 and j 90 we benchmark the parallel computing performance through a lid driven cavity flow problem and a problem amenable to analytic solution

trivariate spline approximations of 3d navier stokes equations we present numerical approximations of the 3d steady state navier stokes equations in velocity pressure formulation using trivariate splines of arbitrary degree d and arbitrary smoothness r with r d using functional arguments we derive the discrete navier stokes equations in terms of b coefficients of trivariate splines over a tetrahedral partition of any given polygonal domain smoothness conditions boundary conditions and the divergence free condition are enforced through lagrange multipliers the pressure is computed by solving a poisson equation with neumann boundary conditions we have implemented this approach in matlab and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem

numerical exploration of flow topology and vortex stability in a curved duct we performed incompressible flow simulation in a square duct with 90 bend and a curvature radius of 2 3 to extend our understanding of the vortical flow development in the bend the solutions for the flow investigated at the reynolds number of re 790 are obtained in a tri quadratic element system where velocities stagger the pressure working variable using the streamline upwind finite element model and the bicgstab iterative solver the simulated results reveal that centrifugal force convects the quickly moving fluid particles towards the outer wall the axial velocity as a result shows twin peaks in the curved channel at about 66 the secondary flow shows three complex pairs of vortices also noteworthy is the formation of a downstream spiralling flow motion to better elucidate the dominating three dimensional flow nature the topological study of limiting streamlines was undertaken insight into the longitudinal flow instability is gained by tracking the formation and diminishing of limiting cycles copyright 2006 john wiley sons ltd

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

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

on error estimates of projection methods for navier stokes equations first order schemes in this paper projection methods or fractional step methods are studied in the semi discretized form for the navier stokes equations in a two or three dimensional bounded domain error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method a modified projection scheme which leads to improved error estimates is also proposed

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

on the existence of mild solutions of semilinear evolution differential inclusions abstract in this paper we deal with a cauchy problem governed by the following semilinear evolution differential inclusion x t a t x t f t x t and with initial data x 0 x 0 e where a t t 0 d is a family of linear operators in the banach space e generating an evolution operator and f is a caratheodory type multifunction we prove the existence of local and global mild solutions of the problem moreover we obtain the compactness of the set of all global mild solutions in order to obtain these results we define a generalized cauchy operator our existence theorems respectively contain the analogous results provided by kamenskii obukhovskii and zecca condensing multivalued maps and semilinear differential inclusions in banach spaces de gruyter ser nonlinear anal appl vol 7 de gruyter berlin 2001 for inclusions with constant operator

a hybrid multigrid method for the unsteady incompressible navier stokes equations an approach is presented for the high reynolds incompressible flow based on multigrid method and sequential regular method in this paper the velocity pressure increment and sequential regular equations are derived from the navier stokes equation the pressure increment optimum sweep algorithm is discussed the numerical results of the close square flow show that our algorithm improves the convergence rate for high reynolds unsteady flow

a weighted meshfree collocation method for incompressible flows using radial basis functions abstract a weighted strong form collocation method using radial basis functions and explicit time integration is proposed to solve the incompressible navier stokes equations the velocities and pressure are solved directly at the same time step and the continuity equation is satisfied at each time step which improve the solution accuracy and stability no artificial compressibility coefficient needs to be introduced for modeling the incompressible flows and no pressure oscillation arises in the numerical solutions optimal convergence can be achieved by imposing the derived proper weights on the boundaries and the continuity equation radial basis collocation method in a lagrangian form is quite easy to capture the moving boundary or free surface in flow problems moreover solid boundary conditions can be enforced directly and no special treatments are required further critical time step for the explicit time integration is predicted in the stability analysis and the influences on the stability are evaluated numerical studies validate the high accuracy as well as good stability of the presented method

the controllability of an impulsive integro differential process with nonlocal feedback controls abstract the controllability of an impulsive process governed by a parametric integro differential equation involving a volterra operator is shown the model is studied via the existence of impulsive mild solutions for a semilinear integro differential inclusion a discussion on the impulse functions is presented

immersed boundary methods for general finite difference and finite volume navier stokes solvers we present an immersed boundary algorithm for incompressible flows with complex boundaries suitable for cartesian or curvilinear grid system the key stages of any immersed boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary a line in 2d a surface in 3d and the spreading of a force field from the immersed boundary to the neighboring mesh points to enforce the desired boundary conditions on the immersed boundary points we propose a technique that uses the reproducing kernel particle method w k liu s jun y f zhang reproducing kernel particle methods int j numer methods fluids 20 8 1995 1081 1106 for the interpolation and spreading unlike other methods presented in the literature the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved independent of the grid topology uniform or non uniform cartesian or curvilinear the technique is easy to implement and is able to maintain the order of the original underlying spatial discretization applications to two and three dimensional flows in cartesian and non cartesian grid system with uniform and non uniform meshes are presented

sequential regularization methods for higher index daes with constraint singularities the linear index 2 case standard stabilization techniques for higher index differential algebraic equations daes often involve elimination of the algebraic solution components this may not work well if there are singularity points where the constraint jacobian matrix becomes rank deficient this paper proposes instead a sequential regularization method srm a functional iteration procedure for solving problems with isolated singularities which have smooth differential solution components for linear index 2 daes we consider both initial value problems ivps and boundary value problems bvps the convergence of the srm is described and proved in detail we believe that this is the first convergence proof for any method for daes with this type of constraint singularities moreover the regularization parameter in our method is not necessarily very small so the srm is an important improvement over usual regularization methods various aspects of the subsequent numerical discretization of the regularized problems are discussed as well and some numerical verifications are carried out

existence uniqueness results for a class of singular boundary value problems ii abstract in this paper we establish existence uniqueness of solution of a class of singular boundary value problem p x y x q x f x y for 0 x b and y 0 a 1 y b 1 y b 1 where p x satisfies i p x 0 in 0 b ii p x c 1 0 r and for some r b iii x p x p x is analytic in z z r and q x satisfies i q x 0 in 0 b ii q x l 1 0 b and for some r b iii x 2 q x p x is analytic in z z r with quite general conditions on f x y region for multiple solutions have also been determined

dynamics of the diffusive nicholson s blowflies equation with distributed delay in this paper we study the diffusive nicholson s blowflies equation generalizing previous works we model the generation delay by using an integral convolution over all past times and results are obtained for general delay kernels as far as possible the linearized stability of the non zero uniform steady state is studied in detail mainly by using the principle of the argument global stability both of this state and of the zero state are studied by using energy methods and by employing a comparison principle for delay equations finally we consider what bifurcations are possible from the non zero uniform state in the case when it is unstable

a sequential regularization method for time dependent incompressible navier stokes equations the objective of the paper is to present a method called the sequential regularization method srm for the nonstationary incompressible navier stokes equations from the viewpoint of regularization of differential algebraic equations daes and to provide a way to apply a dae method to partial differential algebraic equations pdaes the srm is a functional iterative procedure it is proved that its convergence rate is o epsilon m where m is the number of the srm iterations and epsilon is the regularization parameter the discretization and implementation issues of the method are considered in particular a simple explicit difference scheme is analyzed and its stability is proved under the usual step size condition of explicit schemes it appears that the srm formulation is new in the navier stokes context unlike other regularizations or pseudocompressibility methods in the navier stokes context the regularization parameter epsilon in the srm need not be very small and the regularized problem in the sequence may be essentially nonstiff in time direction for any epsilon hence the stability condition is independent of epsilon even for explicit time discretization numerical experiments are given to verify our theoretical results

error estimate of the p 1 nonconforming finite element method for the penalized unsteady navier stokes equations we consider a finite element method for the penalty formulation of the time dependent navier stokes equations usually the improper choice of the finite element space will lead that the error estimate inversely depends on the penalty parameter epsilon we use the classical p 1 nonconforming finite element space for the spatial discretization optimal epsilon uniform error estimations for both velocity and pressure are obtained

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

integral conditions for the pressure in the computation of incompressible viscous flows abstract the problem of finding the correct conditions for the pressure in the time discretized navier stokes equations when the incompressibility constraint is replaced by a poisson equation for the pressure is critically examined it is shown that the pressure conditions required in a nonfractional step scheme to formulate the problem as a system of split second order equations are of an integral character and similar to the previously discovered integral conditions for the vorticity the novel integral conditions for the pressure are used to derive a finite element method which is very similar to that developed by glowinski and pironneau and is the finite element counterpart of the influence matrix method of kleiser and schumann

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

three dimensional flow in cavity at yaw abstract this study introduces a building block prototype model of three dimensional 3 d flows about protrusions and cavities with an arbitrary angle between the external flow and rigid elements the considered flow in a 3 d cavity has the lid of cavity moving along the lid diagonal case a the proposed case is taken as a typical representative of essentially 3 d highly separated vortical flows having simple single block rectangular geometry of computational domain this 3 d lid driven cavity flow is studied by numerical solution of the 3 d viscous navier stocks equations using fluent software with second order upwind schemes for convective terms computational results are compared to the classical benchmark case where the lid moves in the direction parallel to the cavity side walls case b the volume and plane integrals of momentum in the direction of lid motion and perpendicular to it are introduced to compare cases in a quantitative way the integral momentum of the secondary flow transverse to the lid motion appears to be about an order of magnitude larger than that in case b in the proposed flow the number and strength of secondary vortices depend substantially on the re number of the flow whereas in case b only one pair of secondary vortices appears for arbitrary re number for a rectangular 3 d 3 1 1 cavity the integral momentum of the secondary flow in case a is an order of magnitude larger than that in the case b the flowfield in case a includes a curvilinear separation line and non symmetrical vortices

a novel concept for the design of immersed interface methods the objective of this paper is to present a novel robust high order accurate immersed interface method iim for advection diffusion type equations in contrast to other immersed methods that were designed for consistency and accuracy with a posteriori check of the numerical stability we combine local taylor series expansion at irregular grid points with a local stability constraint as part of the design process stability investigations of the iim are employed to demonstrate that the local stability constraint is sufficient for obtaining a globally stable method as long as the neumann number is less than its limiting value one of the key aspects of this iim is that the irregular finite difference stencils can be isolated from the rest of the computational domain to validate our novel immersed interface approach two dimensional and three dimensional test cases for model equations are presented in addition this method is applied to the incompressible navier stokes equations to conduct stability investigations of a boundary layer flow over a rough surface and for investigations of pulsatile stenotic flows stability investigations of wall bounded flows are challenging for immersed methods because the near wall accuracy is important for correctly capturing the characteristics of the hydrodynamic instability mechanisms in particular regarding the wave relation between the wave velocity components close to the wall

impulsive semilinear differential inclusions topological structure of the solution set and solutions on non compact domains abstract this paper deals with an impulsive cauchy problem governed by the semilinear evolution differential inclusion x t a t x t f t x t where a t t 0 b is a family of linear operators not necessarily bounded in a banach space e generating an evolution operator and f is a caratheodory type multifunction first a theorem on the compactness of the set of all mild solutions for the problem is given then this result is applied to obtain the existence of mild solutions for the impulsive cauchy problem defined on non compact domains

development of a dispersively accurate conservative level set scheme for capturing interface in two phase flows a two step interface capturing scheme implemented within the framework of conservative level set method is developed in this study to simulate the gas water two phase fluid flow in addition to solving the pure advection equation which is used to advect the level set function for tracking interface both nonlinear and stabilized features are taken into account for the level set function so that a sharply varying interface can be stably predicted to preserve the conservative property the mapping given by x x u t is performed between two coordinates x and x for the advection diffusion equation in a flow field with velocity u to capture the interface the flux term capable of compressing the level set contours is also adopted in the construction of linear inviscid burgers equation which is indispensable in the level set method to resolve the physically sharp interface without incurring contact discontinuity oscillations a damping term which is nonlinear in terms of the level set function and can render an adequate artificial diffusion to stabilize the contact surface is added into the reinitialization step in the modified level set method for accurately predicting the level set function the advection scheme for solving the linear inviscid burgers equation in the advection step of the modified level set method is developed to accommodate the true dispersion relation the solution computed from the resulting two dimensional dispersion relation preserving advection scheme can minimize the phase error less artificial damping is needed to damp the oscillations in the vicinity of contact surface and the interface can be less numerically smeared for the sake of programming simplicity the incompressible two phase flow will be discretized in non staggered grids without incurring checkerboard oscillations by the developed explicit compact scheme for the approximation of pressure gradient terms for the verification of the proposed two dimensional dispersion relation preserving scheme and the non staggered incompressible flow solver three benchmark problems have been chosen in this study the proposed conservative level set method for capturing the interface in incompressible fluid flows is also verified by solving the dam break bubble rising in water droplet falling in water and rayleigh taylor instability problems good agreements with the referenced solutions are demonstrated for all the four investigated problems

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

fully compact higher order computation of steady state natural convection in a square cavity the flow in a thermally driven square cavity with adiabatic top and bottom walls and differentially heated vertical walls for a wide range of rayleigh numbers 10 3 or ra or 10 7 has been computed with a fourth order accurate higher order compact scheme which was used earlier only for the stream function vorticity psi omega form of the two dimensional steady state navier stokes equations the boundary conditions used are also compact and of identical accuracy in particular a compact fourth order accurate neumann boundary condition has been developed for temperature at the adiabatic walls the treatment of the derivative source term is also compact and has been done in such a way as to give fourth order accuracy and easy assimilation with the solution procedure as the discretization for the psi omega formulation boundary conditions and source term treatment are all fourth order accurate highly accurate solutions are obtained on relatively coarser grids unlike other compact solution procedure in literature for this physical configuration the present method is fully compact and fully higher order accurate also use of conjugate gradient and hybrid biconjugate gradient stabilized algorithms to solve the symmetric and nonsymmetric algebraic systems at every outer iteration ensures good convergence behavior of the method even at higher rayleigh numbers

multigrid in h div on axisymmetric domains abstract in this paper we will construct and analyze a multigrid algorithm that can be applied to weighted h div problems on a two dimensional domain these problems arise after performing a dimension reduction to a three dimensional axisymmetric h div problem we will use recently developed fourier finite element spaces that can be applied to axisymmetric h div problems with general data we prove that if the axisymmetric domain is convex then the multigrid v cycle with modern smoothers will converge uniformly with respect to the meshsize

on a class of weakly regular singular two point boundary value problems ii abstract existence of a unique solution of a class of weakly regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established where p x satisfies i p x 0 on 0 b ii p x c 1 0 r and for some r b iii xp x p x is analytic in z z r with taylor expansion xp x p x b 0 b 1 x b 0 0 1 and with quite general conditions on f x y these conditions on f x y are sharp which is seen through one example regions for multiple solutions have also been determined

energy law preserving c0 finite element schemes for phase field models in two phase flow computations we use the idea in 33 to develop the energy law preserving method and compute the diffusive interface phase field models of allen cahn and cahn hilliard type respectively governing the motion of two phase incompressible flows we discretize these two models using a c 0 finite element in space and a modified midpoint scheme in time to increase the stability in the pressure variable we treat the divergence free condition by a penalty formulation under which the discrete energy law can still be derived for these diffusive interface models through an example we demonstrate that the energy law preserving method is beneficial for computing these multi phase flow models we also demonstrate that when applying the energy law preserving method to the model of cahn hilliard type un physical interfacial oscillations may occur we examine the source of such oscillations and a remedy is presented to eliminate the oscillations a few two phase incompressible flow examples are computed to show the good performance of our method