algebraic splitting for incompressible navier stokes equations fully discretized incompressible navier stokes equations are solved by splitting the algebraic system with an approximate factorization this splitting affects the temporal convergence order of velocity and pressure the splitting error is proportional to the pressure variable and a simple analysis shows that the original convergence order of the time integration scheme can be retained by solving for incremental pressure the combination of splitting and incremental pressure is shown to be equivalent to an error correcting method using the full pressure in numerical experiments employing a third order time integration scheme and various orders for the pressure increment the splitting error is shown to control the convergence order and the full order of the scheme is recaptured for both velocity and pressure the difference between perturbing the momentum or the continuity equation is also explored

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

projection method ii godunov ryabenki analysis this is the second of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of the present paper is to explain why the accuracy of the velocity approximation is not affected by 1 the numerical boundary layers in the approximation of pressure and the intermediate velocity field and 2 the noncommutativity of the projection operator and the laplacian this is done by using a godunov ryabenki type of analysis in a rigorous fashion by doing so we hope to be able to convey the message that normal mode analysis is basically sufficient for understanding the stability and accuracy of a finite difference method for the navier stokes equation even in the presence of boundaries as an example we analyze the second order projection method based on pressure increment formulations used by van kan and bell colella and glaz the leading order error term in this case is of o dt and behaves as high frequency oscillations over the whole domain compared with the o dt 1 2 numerical boundary layers found in the second order kim moin method

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

spurious transients of projection methods in microflow simulations the temporal behavior of projection methods for viscous incompressible lowreynolds number ows is adressed the methods considered result from algebraically splitting the linear system corresponding to each time step in such a way that the computation of velocity is segregated from that of pressure each method is characterized by two possibly equal approximate inverses b1 and b2 of the momentum equation velocity matrix plus a parameter which renders the method non incremental if 0 or incremental if 1 the classical rst order projection method together with more sophisticated methods perot s second order method yosida method pseudoexact factorization method and their incremental variants are put into the same algebraic form and their accuracy numerically tested splitting errors of rst

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

the accuracy of the fractional step method we analyze the accuracy of the fractional step method of kim and moin j comput phys 59 1985 pp 308 323 for the incompressible navier stokes equations we show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method we also show that the pressure in any projection method can be at best first order accurate our analysis is simpler and more direct than the previous analyses of this method we also show that there is no numerical boundary layer for velocity or pressure but there is one for the auxiliary pressure variable

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

a class of fully second order accurate projection methods for solving the incompressible navier stokes equations in this paper a continuous projection method is designed and analyzed the continuous projection method consists of a set of partial differential equations which can be regarded as an approximation of the navier stokes n s equations in each time interval of a given time discretization the local truncation error lte analysis is applied to the continuous projection methods which yields a sufficient condition for the continuous projection methods to be temporally second order accurate based on this sufficient condition a fully second order accurate discrete projection method is proposed a heuristic stability analysis is performed to this projection method showing that the present projection method can be stable the stability of the present scheme is further verified through numerical experiments the second order accuracy of the present projection method is confirmed by several numerical test cases

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

a hybrid reconstructed discontinuous galerkin and continuous galerkin finite element method for incompressible flows on unstructured grids a hybrid reconstructed discontinuous galerkin and continuous galerkin method based on an incremental pressure projection formulation termed rdg p n p m cg p n in this paper is developed for solving the unsteady incompressible navier stokes equations on unstructured grids in this method a reconstructed discontinuous galerkin method rdg p n p m is used to discretize the velocity and a standard continuous galerkin method cg p n is used to approximate the pressure the rdg p n p m cg p n method is designed to increase the accuracy of the hybrid dg p n cg p n method and yet still satisfy lady enskaja babuska brezzi lbb condition thus avoiding the pressure checkerboard instability an upwind method is used to discretize the nonlinear convective fluxes in the momentum equations in order to suppress spurious oscillations in the velocity field a number of incompressible flow problems for a variety of flow conditions are computed to numerically assess the spatial order of convergence of the rdg p n p m cg p n method the numerical experiments indicate that both rdg p 0 p 1 cg p 1 and rdg p 1 p 2 cg p 1 methods can attain the designed 2nd order and 3rd order accuracy in space for the velocity respectively moreover the 3rd order rdg p 1 p 2 cg p 1 method significantly outperforms its 2nd order rdg p 0 p 1 cg p 1 and rdg p 1 p 1 cg p 1 counterparts being able to not only increase the accuracy of the velocity by one order but also improve the accuracy of the pressure

approximate factorization as a high order splitting for the implicit incompressible flow equations abstract we apply the method of approximate factorization to derive a second order accurate splitting of the incompressible flow equations stokes or navier stokes equations this is novel because the method of approximate factorization was believed inapplicable to this type of equation system we demonstrate the resulting splitting on a second order crank nicolson discretization and point out its intimate relationship to some existing second order accurate splitting projection methods further we use the approximate factorization method to derive entirely new splittings we first develop a new generalized second order accurate splitting which may be specialized to a variety of applications we indicate its application to finite elements checker board free cell centered discretizations and ocean modeling we then generalized the original splitting to an arbitrarily high order scheme

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

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

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

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

factorization methods for the numerical approximation of navier stokes equations we investigate a general approach for the numerical approximation of incompressible navier stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve the splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations several schemes based on approximate factorization are investigated for some of these methods a formal analogy with well known time advancing schemes such as the projection chorin temam s can be pointed out features and limits of this analogy that was earlier introduced in b perot j comp phys 108 1993 51 8 are addressed other new methods can also be formulated starting from this approach in particular we introduce the so called yosida method which can be investigated in the framework of quasi compressibility schemes numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

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

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

the energy levels of the sextic double well potential an infinite set of exact solutions of the sextic double well potential is derived these solutions that are obtained for special configurations of the well are shown to provide an excellent description of the entire negative energy spectrum for arbitrary geometry of the well a universal feature of this class of deep double well potentials is deduced

a generalization of the lindeberg principle we generalize lindeberg s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables the result is applied to get a similar theorem for smooth functions of exchangeable random variables this theorem allows us to identify for the first time the limiting spectral distributions of wigner matrices with exchangeable entries

a finite difference scheme for three dimensional incompressible flows in cylindrical coordinates a finite difference scheme for the direct simulation of the incompressible time dependent three dimensional navier stokes equations in cylindrical coordinates is presented the equations in primitive variables vr v vzandp are solved by a fractional step method together with an approximate factorization technique cylindrical coordinates are singular at the axis the introduction of the radial fluxqr r vron a staggered grid simplifies the treatment of the region atr 0 the method is tested by comparing the evolution of a free vortex ring and its collision with a wall with the theory experiments and other numerical results the formation of a tripolar vortex where the highest vorticity is atr 0 is also considered finally to emphasize the accurate treatment near the axis the motion of a lamb dipole crossing the origin is simulated

convergence of gauge method for incompressible flow a new formulation a gauge formulation of the incompressible navier stokes equations in terms of an auxiliary field a and a gauge variable u a was proposed recently by e and liu this paper provides a theoretical analysis of their formulation and verifies the computational advantages we discuss the implicit gauge method which uses backward euler or crank nicolson in time discretization however the boundary conditions for the auxiliary field a are implemented explicitly vertical extrapolation the resulting momentum equation is decoupled from the kinematic equation and the computational cost is reduced to solving a standard heat and poisson equation moreover such explicit boundary conditions for the auxiliary field a will be shown to be unconditionally stable for stokes equations for the full nonlinear navier stokes equations the time stepping constraint is reduced to the standard cfl constraint t x c we also prove first order convergence of the gauge method when we use mac grids as our spatial discretization the optimal error estimate for the velocity field is also obtained

a second order boundary fitted projection method for free surface flow computations this paper describes a new approach to the high fidelity simulation of axisymmetric free surface flows a boundary fitted grid is coupled with a new projection method for the solution of the navier stokes equations with second order accuracy in space and time two variants of this new method are developed by adapting two existing algorithms suitable for prescribed velocity boundary conditions to the case of prescribed normal and tangential stresses at the free boundary a normal mode analysis for a fixed boundary problem confirms the second order accuracy of the algorithms the approach is validated by comparison with a rayleigh plesset solution for an oscillating spherical bubble with an analysis of shape oscillations and with existing results for the buoyant rise of a deforming bubble for reynolds numbers up to 200 and weber numbers up to 12 in addition to the simulation of axisymmetric free surface flows of intrinsic interest the present approach is suitable for the validation of genuinely three dimensional calculations

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

runge kutta chebyshev projection method in this paper a fully explicit stabilized projection method called the runge kutta chebyshev rkc projection method is presented for the solution of incompressible navier stokes systems this method preserves the extended stability property of the rkc method for solving odes and it requires only one projection per step an additional projection on the time derivative of the velocity is performed whenever a second order approximation for the pressure is desired we demonstrate both by numerical experiments and by order analysis that the method is second order accurate in time for both the velocity and the pressure being explicit the rkc projection method is easy to implement and to parallelize hence it is an attractive candidate for the solution of large scale moderately stiff diffusion like problems