accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

a new class of truly consistent splitting schemes for incompressible flows a new class of splitting schemes for incompressible flows is introduced the new schemes are based on a weak form of the pressure poisson equation and at each time step only require to solve a set of helmholtz type equations for the velocity and a poisson equation in the weak form for the pressure just as pressure correction and velocity correction schemes however unlike pressure correction and velocity correction schemes the new splitting schemes are free of splitting errors and deliver full accuracy on the vorticity and the pressure

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

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

an optimally accurate discrete regularization for second order timestepping methods for navier stokes equations abstract we propose a new optimally accurate numerical regularization stabilization for a family of second order timestepping methods for the navier stokes equations nse the method combines a linear treatment of the advection term together with stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure we rigorously prove that the entire new family of methods are unconditionally stable and o t 2 accurate the idea of curvature stabilization is new to cfd and is intended as an improvement over the commonly used speed stabilization which is only first order accurate in time and can have an adverse effect on important flow quantities such as drag coefficients numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows

the scalar auxiliary variable sav approach for gradient flows abstract we propose a new approach which we term as scalar auxiliary variable sav approach to construct efficient and accurate time discretization schemes for a large class of gradient flows the sav approach is built upon the recently introduced ieq approach it enjoys all advantages of the ieq approach but overcomes most of its shortcomings in particular the sav approach leads to numerical schemes that are unconditionally energy stable and extremely efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step the scheme is not restricted to specific forms of the nonlinear part of the free energy so it applies to a large class of gradient flows numerical results are presented to show that the accuracy and effectiveness of the sav approach over the existing methods

the single machine total tardiness scheduling problem review and extensions abstract we review the latest theoretical developments for the single machine total tardiness 1 t scheduling problem and propose extensions to some of them we also review and in some cases extend exact algorithms fully polynomial time approximation schemes heuristic algorithms special cases and generalizations of the 1 t problem our findings indicate that the 1 t problem continues to attract significant research interest from both a theoretical and a practical perspective even though the problem is ordinary np hard the current state of the art algorithms are capable of solving problems with up to 500 jobs

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

a triangular spectral element method applications to the incompressible navier stokes equations abstract encouraged by the success of spectral elements methods in computational fluid dynamics and p type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers we introduce here a complete formulation using a modal basis which has been implemented in a new code ne t r the new basis has the following properties jacobi polynomials of mixed weights semi orthogonality hierarchical structure generalized tensor warped product variable order and a new apex co ordinate system allowing automated integration with gaussian quadrature we have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations we use this notation to formulate the linear advection and helmholtz equations in an efficient manner and show that we recover the following properties well conditioned matrices asymptotic operation count of o n 3 scaling o n 2 of the spectral radius of the weak convective operator and exponential convergence using polynomials up to 40 having constructed and numerically analysed these equations we are then able to solve the incompressible navier stokes equations using a high order splitting scheme we demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the stokes and navier stokes problems

simultaneous influence of additive and multiplicative noise on stationary dissipative solitons we investigate the simultaneous influence of spatially homogeneous multiplicative noise as well as of spatially correlated additive noise on the formation of localized patterns in the framework of the cubic quintic complex ginzburg landau equation depending on the ratio between the strength of additive and multiplicative noise we find a number of distinctly different types of behavior including explosions collapse filling in and spatio temporal disorder as well as intermittent behavior of all types listed techniques used to analyze the results include snapshots x t plots and plots of the spatially and temporally averaged amplitude as a function of the strength of multiplicative noise while keeping the strength of additive noise fixed typically 50 realizations are used for averaging to obtain the corresponding data points in these diagrams for the widths of these distribution as a function of additive noise we obtain a linear decrease in the limit of fairly large but fixed values of the multiplicative noise to summarize our findings concisely we show three dimensional plots of the mean pattern amplitude and the generalized susceptibility as a function of the strengths of additive and multiplicative noise we critically compare the results of our investigations with those obtained in the two limiting cases of purely additive and of purely multiplicative noise

a faster fully polynomial approximation scheme for the single machine total tardiness problem abstract lawler e l lawler a fully polynomial approximation scheme for the total tardiness problem operations research letters 1 1982 207 208 proposed a fully polynomial approximation scheme for the single machine total tardiness problem which runs in o n 7 e time where n is the number of jobs and e is the desired level of approximation a faster fully polynomial approximation scheme running in o n 5 log n n 5 e time is presented in this note by applying an alternative rounding scheme in conjunction with implementing kovalyov s m y kovalyov improving the complexities of approximation algorithms for optimization problems operations research letters 17 1995 85 87 bound improvement procedure

an eigen based high order expansion basis for structured spectral elements we present an eigen based high order expansion basis for the spectral element approach with structured elements the new basis exhibits a numerical efficiency significantly superior in terms of the conditioning of coefficient matrices and the number of iterations to convergence for the conjugate gradient solver to the commonly used jacobi polynomial based expansion basis this basis results in extremely sparse mass matrices and it is very amenable to the diagonal preconditioning ample numerical experiments demonstrate that with the new basis and a simple diagonal preconditioner the number of conjugate gradient iterations to convergence has essentially no dependence or only a very weak dependence on the element order the expansion bases are constructed by a tensor product of a set of special one dimensional 1d basis functions the 1d interior modes are constructed such that the interior mass and stiffness matrices are simultaneously diagonal and have identical condition numbers the 1d vertex modes are constructed to be orthogonal to all the interior modes the performance of the new basis has been investigated and compared with other expansion bases

a time stepping scheme involving constant coefficient matrices for phase field simulations of two phase incompressible flows with large density ratios we present an efficient time stepping scheme for simulations of the coupled navier stokes cahn hilliard equations for the phase field approach the scheme has several attractive characteristics i it is suitable for large density ratios and numerical experiments with density ratios up to 1000 have been presented ii it involves only constant time independent coefficient matrices for all flow variables which can be pre computed during pre processing so it effectively overcomes the performance bottleneck induced by variable coefficient matrices associated with the variable density and variable viscosity iii it completely de couples the computations of the velocity pressure and the phase field function strategy for spectral element type spatial discretizations to overcome the difficulty associated with the large spatial order of the cahn hilliard equation is also discussed ample numerical simulations demonstrate that the current algorithm together with the navier stokes cahn hilliard phase field approach is an efficient and effective method for studying two phase flows involving large density ratios moving contact lines and interfacial topology changes

an unconditionally stable rotational velocity correction scheme for incompressible flows we present an unconditionally stable splitting scheme for incompressible navier stokes equations based on the rotational velocity correction formulation the main advantages of the scheme are i it allows the use of time step sizes considerably larger than the widely used semi implicit type schemes the time step size is only constrained by accuracy ii it does not require the velocity and pressure approximation spaces to satisfy the usual inf sup condition in particular the equal order finite element spectral element approximation spaces can be used iii it only requires solving a pressure poisson equation and a linear convection diffusion equation at each time step numerical tests indicate that the computational cost of the new scheme for each time step under identical time step sizes is even less expensive than the semi implicit scheme with low element orders therefore the total computational cost of the new scheme can be significantly less than the usual semi implicit scheme

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

numerical approximation of incompressible navier stokes equations based on an auxiliary energy variable abstract we present a numerical scheme for approximating the incompressible navier stokes equations based on an auxiliary variable associated with the total system energy by introducing a dynamic equation for the auxiliary variable and reformulating the navier stokes equations into an equivalent system the scheme satisfies a discrete energy stability property in terms of a modified energy and it allows for an efficient solution algorithm and implementation within each time step the algorithm involves the computations of two pressure fields and two velocity fields by solving several de coupled individual linear algebraic systems with constant coefficient matrices together with the solution of a nonlinear algebraic equation about a scalar number involving a negligible cost a number of numerical experiments are presented to demonstrate the accuracy and the performance of the presented algorithm

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

the stabilized extrapolated trapezoidal finite element method for the navier stokes equations abstract we consider a fully discrete stabilized finite element method for the navier stokes equations which is unconditionally stable and has second order temporal accuracy of o k 2 hk spatial error the method involves a simple artificial viscosity stabilization of the linear system for the approximation of the new time level connected to anti diffusion of its effects at the old time level lowering the cell reynolds number of the oseen problem to o 1 the method requires only the solution of one linear system arising from an oseen problem per time step the effectiveness of the method is illustrated in several numerical experiments

sub iteration leads to accuracy and stability enhancements of semi implicit schemes for the navier stokes equations we present an iterative semi implicit scheme for the incompressible navier stokes equations which is stable at cfl numbers well above the nominal limit we have implemented this scheme in conjunction with spectral discretizations which suffer from serious time step limitations at very high resolution however the approach we present is general and can be adopted with finite element and finite difference discretizations as well specifically at each time level the nonlinear convective term and the pressure boundary condition both of which are treated explicitly in time are updated using fixed point iteration and aitken relaxation eigenvalue analysis shows that this scheme is unconditionally stable for stokes flows while numerical results suggest that the same is true for steady navier stokes flows as well this finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters in unsteady flows second and third order temporal accuracy is obtained for the velocity field at cfl number 5 14 using analytical solutions systematic accuracy stability and cost comparisons are presented against the standard semi implicit method and a recently proposed fully implicit scheme that does not require newton s iterations in addition to its enhanced accuracy and stability the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully implicit schemes

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

some remarks on the decomposition properties of the single machine total tardiness problem abstract the note extends lawler s e l lawler a pseudopolynomial algorithm for sequencing jobs to minimize total tardiness annals of discrete mathematics 1 1977 331 342 fundamental decomposition theorem this extension serves as a basis for alternate criteria that recognize when the problem is non decomposable it also shows that even relaxing a single criterion to a weak inequality may contradict lawler s 1977 decomposition theorem

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

velocity correction schemes for the incompressible navier stokes equations in general coordinate systems this paper presents methods of including coordinate transformations into the solution of the incompressible navier stokes equations using the velocity correction scheme which is commonly used in the numerical solution of unsteady incompressible flows this is important when the transformation leads to symmetries that allow the use of more efficient numerical techniques like employing a fourier expansion to discretize a homogeneous direction two different approaches are presented in the first approach all the influence of the mapping is treated explicitly while in the second the mapping terms related to convection are treated explicitly with the pressure and viscous terms treated implicitly through numerical results we demonstrate how these methods maintain the accuracy of the underlying high order method and further apply the discretisation strategy to problems where mixed fourier spectral hp element discretisations can be applied thereby extending the usefulness of this discretisation technique