on error estimates of the penalty method for unsteady navier stokes equations the penalty method has been widely used for numerical computations of the unsteady navier stokes equations however the best error estimates available to the author s knowledge were not optimal an

adaptive unstructured volume remeshing ii application to two and three dimensional level set simulations of multiphase flow in part i adaptive unstructured volume remeshing i the method j comput phys in press we presented an adaptive remeshing algorithm that automatically adjusts the size of the elements of meshes of unstructured triangles 2d and unstructured tetrahedra 3d with time and position in the computational domain in order to efficiently resolve the relevant physical scales here we illustrate the performance of an implementation of the algorithm in finite element level set simulations of deformable droplet and fluid fluid interface interactions breakup and coalescence in multiphase flows the wide range of length scales characterizing the dynamics are accurately resolved as demonstrated by comparison to experiments and to theoretical and sharp interface boundary integral numerical results the computational cost is found to be competitive even with respect to boundary integral methods for the first time using an interface capturing level set method we successfully simulate the inertia driven impact and rebound of a liquid droplet from a liquid interface and find agreement with recent experimental results

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

an efficient time splitting approximation of the navier stokes equations with lps modeling abstract in this work the solution of the navier stokes equations nse is addressed by a finite element fe local projection stabilization lps method combined with an efficient time splitting approximation strategy the focus is on the high order term by term stabilization method that has one level in the sense that it is defined on a single mesh and in which the projection stabilized structure of standard lps methods is replaced by an alternative interpolation stabilized structure the main contribution is on numerically analyzing for the cited lps method the proposed time approximation via stable velocity pressure segregation using semi implicit backward differentiation formulas bdf an overview about theoretical results from the numerical analysis of the proposed method is given by highlighting the used mathematical tools numerical studies support the analytical results and illustrate the potential of the method for the efficient and accurate simulation of turbulent flows on relatively coarse grids smooth unsteady flows are simulated with optimal order of accuracy

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

non negative solutions for a class of non positone problems in the recent past many results have been established on non negative solutions to boundary value problems of the form where 0 f 0 0 positone problems in this paper we consider the impact on the non negative solutions when f 0 f u to be convex to guarantee uniqueness of positive solutions and f u to be appropriately concave for multiple positive solutions this is in contrast to the case of positone problems where the roles of convexity and concavity were interchanged to obtain similar results we further establish the existence of non negative solutions with interior zeros which did not exist in positone problems

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 efficient rotational pressure correction schemes for the coupling stokes darcy problem abstract in this paper the rotational pressure correction methods for the stokes darcy system are developed and analyzed the central advantage of these methods is a time dependent version of domain decomposition these methods have first order second order accuracy without the incompressibility constraint of the stokes darcy system their main feature is the implementation efficiency in that we only solve one vector valued elliptic equation and one scalar valued poisson equation for the stokes equations per time step the unconditional stability and long time stability are established and numerical experiments are also presented to show their performance

error estimates for an operator splitting method for navier stokes equations second order schemes we present in this paper an error analysis of a fractional step method for the approximation of the unsteady incompressible navier stokes equations under mild regularity assumptions on the continuous solution we obtain second order error estimates in the time step size both for velocity and pressure numerical results in agreement with the error analysis are also presented

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

open and traction boundary conditions for the incompressible navier stokes equations we present numerical schemes for the incompressible navier stokes equations nse with open and traction boundary conditions we use pressure poisson equation ppe formulation and propose new boundary conditions for the pressure on the open or traction boundaries after replacing the divergence free constraint by this pressure poisson equation we obtain an unconstrained nse for stokes equation with open boundary condition on a simple domain we prove unconditional stability of a first order semi implicit scheme where the pressure is treated explicitly and hence is decoupled from the computation of velocity using either boundary condition the schemes for the full nse that treat both convection and pressure terms explicitly work well with various spatial discretizations including spectral collocation and c 0 finite elements moreover when reynolds number is of o 1 and when the first order semi implicit time stepping is used time step size of o 1 is allowed in benchmark computations for the full nse besides standard stability and accuracy check various numerical results including flow over a backward facing step flow past a cylinder and flow in a bifurcated tube are reported numerically we have observed that using ppe formulation enables us to use the velocity pressure pairs that do not satisfy the standard inf sup compatibility condition our results extend that of johnston and liu h johnston j g liu accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term j comp phys 199 1 2004 221 259 which deals with no slip boundary conditions only

on error estimates of some higher order projection and penalty projection methods for navier stokes equations this paper is a continuation of our previous work 10 on projection methods we study first existing higher order projection schemes in the semidiscretized form for the navier stokes equations one error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent neumann boundary condition satisfied by the pressure approximations we then propose a penalty projection scheme for which we obtain improved error estimates

improvements on open and traction boundary conditions for navier stokes time splitting methods we present in this paper a numerical scheme for incompressible navier stokes equations with open and traction boundary conditions in the framework of pressure correction methods a new way to enforce this type of boundary condition is proposed and provides higher pressure and velocity convergence rates in space and time than found in the present state of the art we illustrate this result by computing some numerical and physical tests in particular we establish reference solutions of a laminar flow in a geometry where a bifurcation takes place and of the unsteady flow around a square cylinder

a pressure correction scheme for generalized form of energy stable open boundary conditions for incompressible flows we present a generalized form of open boundary conditions and an associated numerical algorithm for simulating incompressible flows involving open or outflow boundaries the generalized form represents a family of open boundary conditions which all ensure the energy stability of the system even in situations where strong vortices or backflows occur at the open outflow boundaries our numerical algorithm for treating these open boundary conditions is based on a rotational pressure correction type strategy with a formulation suitable for c 0 spectral element spatial discretizations we have introduced a discrete equation and associated boundary conditions for an auxiliary variable the algorithm contains constructions that prevent a numerical locking at the open outflow boundary in addition we have developed a scheme with a provable unconditional stability for a sub class of the open boundary conditions extensive numerical experiments have been presented to demonstrate the performance of our method for several flow problems involving open outflow boundaries we compare simulation results with the experimental data to demonstrate the accuracy of our algorithm long time simulations have been performed for a range of reynolds numbers at which strong vortices or backflows occur at the open outflow boundaries we show that the open boundary conditions and the numerical algorithm developed herein produce stable simulations in such situations

a semicoarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients the purpose of this paper is to present a semicoarsening multigrid algorithm for solving the finite difference discretization of symmetric and nonsymmetric two and three dimensional elliptic partial differential equations with highly discontinuous and anisotropic coefficients the discrete equations are assumed to be defined on a logically rectangular grid obtained possibly through grid generation for a problem defined on an irregular domain the basic algorithm is described along with some modifications which are designed to improve its efficiency and robustness for certain types of problem cases fortran codes that implement the two and three dimensional semicoarsening multigrid algorithms are described briefly and numerical results are presented

a finite element penalty projection method for incompressible flows the penalty projection method for the solution of navier stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage namely the solution of the momentum balance equation to constrain the divergence of the predicted velocity field after a presentation of the scheme in the time semi discrete formulation then in fully discrete form for a finite element discretization we assess its behaviour against a set of benchmark tests including in particular prescribed velocity and open boundary conditions the results demonstrate that the augmentation always produces beneficial effects as soon as the augmentation parameter takes a significant value the projection method splitting error is reduced pressure boundary layers are suppressed and the loss of spatial convergence of the incremental projection scheme in case of open boundary conditions does not occur anymore for high values of the augmentation parameter the results of coupled solvers are recovered consequently in contrast with standard penalty methods there is no need for a dependence of the augmentation parameter with the time step and this latter can be kept to reasonable values to avoid to degrade too severely the conditioning of the linear operator associated to the velocity prediction step

semicoarsening multigrid on distributed memory machines this paper presents the results of a scalability study for a three dimensional semicoarsening multigrid solver on a distributed memory computer in particular we are interested in the scalability of the solver how the solution time varies as both problem size and number of processors are increased for an iterative linear solver scalability involves both algorithmic issues and implementation issues we examine the scalability of the solver theoretically by constructing a simple parallel model and experimentally by results obtained on an ibm sp the results are compared with those obtained for other solvers on the same computer

a direction splitting algorithm for incompressible flow in complex geometries the paper proposes and examines the stability and convergence properties of an extension of the direction splitting methods towards parabolic and incompressible flow problems in complex possibly time dependent geometries two possible spatial discretization are considered one of them is based on a fictitious domain procedure and the other one is based on a grid adaptation around the domain boundary the stability analysis of the discretization of the parabolic or stokes problems in complex geometries is seriously complicated by the fact that the discrete second order spatial operators no longer commute if the geometry is different from a rectangle or parallelepiped in this case some of the classical direction splitting schemes loose stability in the three dimensional case we propose a modification of the douglas scheme to avoid this problem the numerical results suggest that the new technique is still unconditionally stable and retains the same convergence rate in both time and space as the classical crank nicolson time discretization with a central difference discrete spatial operators the stability in case of the unsteady stokes problem is analyzed only if the parabolic part is not split the analysis of the fully split scheme in this case remains an open problem

uniquely solvable and energy stable decoupled numerical schemes for the cahn hilliard stokes darcy system for two phase flows in karstic geometry we propose and analyze two novel decoupled numerical schemes for solving the cahn hilliard stokes darcy chsd model for two phase flows in karstic geometry in the first numerical scheme we explore a fractional step method operator splitting to decouple the phase field cahn hilliard equation from the velocity field stokes darcy fluid equations to further decouple the stokes darcy system we introduce a first order pressure stabilization term in the darcy solver in the second numerical scheme so that the stokes system is decoupled from the darcy system and hence the chsd system can be solved in a fully decoupled manner we show that both decoupled numerical schemes are uniquely solvable energy stable and mass conservative ample numerical results are presented to demonstrate the accuracy and efficiency of our schemes

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

least squares finite element formulations for viscous incompressible and compressible fluid flows abstract we present least squares based finite element formulations for the numerical solution of viscous fluid flows governed by the navier stokes equations as an alternate approach to the well known weak form galerkin finite element formulations the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise and the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix we address the issue of norm equivalence of the least squares functional and its implications on the resulting finite element model in particular we develop an understanding of the compromise that must exist between the optimality and practicality of the finite element model the latter measured in terms of c k regularity across inter element boundaries we show through numerical examples that for the traditional c 0 basis such a compromise is possible when high p levels are used to span the finite element spaces when a low p level solution is desired guidelines are presented to obtain a reliable least squares collocation solution numerical examples are presented to demonstrate the high and low p level approaches these include incompressible flow past two circular cylinders in a side by side arrangement for gap sizes s d 2 0 and 0 85 incompressible flow past a square cylinder and subsonic transonic and supersonic compressible flow past a circular cylinder in addition we present a discontinuous least squares formulation where c k regularity across inter element boundaries is enforced in a weak sense through the least squares functional allowing for h and p type non conformities in the computational domain

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

stabilized finite element method based on the crank nicolson extrapolation scheme for the time dependent navier stokes equations this paper provides an error analysis for the crank nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two dimensional time dependent navier stokes problem where the finite element space pair x h m h for the approximation u n h p n h of the velocity u and the pressure p is constructed by the low order finite element the q 1 p 0 quadrilateral element or the p 1 p 0 triangle element with mesh size h error estimates of the numerical solution u n h p n h to the exact solution u t n p t n with t n 0 t are derived

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

a pressure correction scheme for convection dominated incompressible flows with discontinuous velocity and continuous pressure in this work we present a pressure correction scheme for the incompressible navier stokes equations combining a discontinuous galerkin approximation for the velocity and a standard continuous galerkin approximation for the pressure the main interest of pressure correction algorithms is the reduced computational cost compared to monolithic strategies in this work we show how a proper discretization of the decoupled momentum equation can render this method suitable to simulate high reynolds regimes the proposed spatial velocity pressure approximation is lbb stable for equal polynomial orders and it allows adaptive p refinement for velocity and global p refinement for pressure the method is validated against a large set of classical two and three dimensional test cases covering a wide range of reynolds numbers in which it proves effective both in terms of accuracy and computational cost

a nonreflecting outlet boundary condition for incompressible unsteady navier stokes calculations abstract the goal of this work is to adapt a nonreflecting outlet boundary condition derived from a wave equation to the numerical solution of the full incompressible navier stokes equations for an elliptic unsteady free shear flow the numerical results show that a significant improvement is achieved with this nonreflecting boundary condition in comparison with the results obtained by using free boundary layer type conditions the physical phenomena studied concern the onset of the kelvin helmholtz instability in the free non forced shear layer and certain 2d characteristics of transition towards turbulence these phenomena are simulated naturally without imposing perturbations the frequency of the organized vortices and the spread of the mixing layer are correctly predicted the performances of the method are shown through comparison with the physical experiments owing to the nonreflecting boundary conditions the feedback noises are inhibited effectively so that the computation domain can be reduced and the dynamic characteristics of the flow are maintained up clearly to the outlet boundary

a robust and accurate outflow boundary condition for incompressible flow simulations on severely truncated unbounded domains we present a robust and accurate outflow boundary condition and an associated numerical algorithm for incompressible flow simulations on unbounded physical domains aiming at maximizing the domain truncation without adversely affecting the flow physics the proposed outflow boundary condition allows for the influx of kinetic energy into the domain through the outflow boundaries and prevents un controlled growth in the energy of the domain in such situations the numerical algorithm for the outflow boundary condition is developed on top of a rotational velocity correction type strategy to de couple the pressure and velocity computations and a special construction in the algorithmic formulation prevents the numerical locking at the outflow boundaries for time dependent problems extensive numerical tests for flow problems with bounded and semi bounded physical domains demonstrate that this outflow boundary condition and the numerical algorithm produce stable and accurate simulations on even severely truncated computational domains where strong vortices may be present at or exit the outflow boundaries the method developed herein has the potential to significantly expedite simulations of incompressible flows involving outflow or open boundaries and to enable such simulations at reynolds numbers significantly higher than the state of the art

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

on the outflow boundary condition for external incompressible flows a new approach this work presents a new procedure for extrapolating velocities at the outflow boundary in the computations of incompressible flows around rigid bodies the extrapolation procedure is based on the radial variation of the velocity field at large distances from the rigid body which can be inferred from mass conservation and vorticity considerations since the extrapolation is based on these physical considerations the proposed boundary condition is considered to be physically consistent it has been demonstrated via numerical simulations of 2d laminar incompressible viscous flow past a circular and a square cylinder at re 100 that the application of the proposed boundary condition allows one to limit the unbounded domain to a small size 6 8 times the characteristic size of the body without any significant change in the flow characteristics like the lift coefficient c l and the strouhal number st thus the proposed boundary conditions can enhance the computational efficiency of this class of flows

analysis of velocity components decoupled projection method for the incompressible navier stokes equations we study the temporal accuracy and stability of the velocity components decoupled projection method vdpm for fully discrete incompressible navier stokes equations in particular we investigate the effect of three formulations of the nonlinear convection term which include the advective skew symmetric and divergence forms on the temporal accuracy and stability second order temporal accuracy of the vdpm for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l 2 norm considering the energy evolution we demonstrate that the vdpm is stable when the time step is less than or equal to a constant stability diagrams which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices are obtained using von neumann analysis these diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability and the effects of the formulations of the nonlinear convection term are analyzed overall our results indicate that the vdpm along with an advective discrete convection operator is almost unconditionally stable second order accurate in time and computationally efficient because of the non iterative solution procedure in solving the decoupled momentum equations

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

gap estimates of the spectrum of hill s equation and action variables for kdv consider the schrodinger equation y v y y for a potential v of period 1 in the weighted sobolev space n z 0 r 0 h s c f x k f k e f n where f k k z denote the fourier coefficients of f when considered as a function of period 1

preconditioned spectral multi domain discretization of the incompressible navier stokes equations in this paper we present a preconditioned multi domain algorithm applied to the elliptic kernels arising from the spectral collocation of the incompressible navier stokes equations in three space dimensions with one homogeneous direction the technique based on the iterative solution of the schur complement matrix allows for efficient numerical solution of the operators in complex geometries consisting of a collection of non overlapping rectangular subdomains the method is shown to be nearly optimal in terms of condition number behavior in a double path of refinement strategy i e whenever both the number of legendre modes and the number of subdomains are significantly increased it is thus well suited for engineering applications in the fields of direct numerical simulation and large eddy simulation of turbulence

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

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

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

convergence analysis of a class of massively parallel direction splitting algorithms for the navier stokes equations in simple domains we provide a convergence analysis for a new fractional timestepping technique for the incompressible navier stokes equations based on direction splitting this new technique is of linear complexity unconditionally stable and convergent and suitable for massive parallelization

exact multiplicity of positive solutions of a semipositone problem with concave convex nonlinearity abstract we study the exact multiplicity of positive solutions and bifurcation diagrams of the dirichlet boundary value problem u x f u 0 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter f c 0 c 2 0 satisfies f 0 0 semipositone and f is concave convex on 0 and is asymptotic superlinear assuming additional suitable conditions on f on the u plane we give a classification of totally three qualitatively different bifurcation diagrams a reversed s shaped curve a broken reversed s shaped curve and a monotone decreasing curve our results improve those in j shi r shivaji exact multiplicity of solutions for classes of semipositone problems with concave convex nonlinearity discrete contin dyn syst 7 2002 559 571 we also give an application to determine completely the exact multiplicity of positive solutions and bifurcation diagrams of the problem with cubic nonlinearity u x u a u b u c 0 1 x 1 u 1 u 1 0 0 a b c

error analysis of a fractional time stepping technique for incompressible flows with variable density in this paper we analyze the convergence properties of a new fractional time stepping technique for the solution of the variable density incompressible navier stokes equations the main feature of this method is that contrary to other existing algorithms the pressure is determined by just solving one poisson equation per time step first order error estimates are proved and stability of a formally second order variant of the method is established

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

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

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

on the penalty projection method for the navier stokes equations with the mac mesh we deal with the time dependent navier stokes equations nse with dirichlet boundary conditions on the whole domain or on a part of the domain and open boundary conditions on the other part it is shown numerically that combining the penalty projection method with spatial discretization by the marker and cell scheme mac yields reasonably good results for solving the above mentioned problem the scheme which has been introduced combines the backward difference formula of second order bdf2 namely gear s scheme for the temporal approximation the second order richardson extrapolation for the nonlinear term and the penalty projection to split the velocity and pressure unknowns similarly to the results obtained for other projection methods we estimate the errors for the velocity and pressure in adequate norms via the energy method

error analysis of fully discrete velocity correction methods for incompressible flows a fully discrete version of the velocity correction method proposed by guermond and shen 2003 for the time dependent navier stokes equations is introduced and analyzed it is shown that when accounting for space discretization additional consistency terms which vanish when space is not discretized have to be added to establish stability and optimal convergence error estimates are derived for both the standard version and the rotational version of the method these error estimates are consistent with those by guermond and shen 2003 as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf sup condition

geometry of kdv 4 abel sums jacobi variety and theta function in the scattering case the study of the general projective curve x defined by the vanishing of an irreducible polynomial p x y originated in the discovery of addition formulas for integrals v r x y dx of rational functions on such a curve

evolution of bifurcation curves for semipositone problems when nonlinearities develop multiple zeroes we discuss existence uniqueness and multiplicity results on positive solutions to u x email protected u x 0 for email protected 0 1 u 0 0 u 1 where l 0 is a constant and u is a smooth function satisfying 0 0 semipositone in particular we follow a class of nonlinearities u as they evolve from having one positive zero to three positive zeros and compare the corresponding bifurcation curves we establish that the bifurcation curve splits into two connected components in the critical case when u develops the second zero

a finite element pressure correction scheme for the navier stokes equations with traction boundary condition abstract we consider the popular pressure correction scheme for the solution of the time dependent navier stokes equations with traction boundary condition a finite element based method to improve the performance of the classical approach is proposed the improvement is achieved by modifying the traction boundary condition for the provisional velocity u n 1 in each time step the corresponding term consists of a simple boundary functional involving the normal derivative of the pressure correction that can be evaluated in a natural and easy way in the context of finite elements computational results show a significant improvement of the solution in particular for the pressure in the case of smooth domains

efficient and accurate numerical schemes for a hydro dynamically coupled phase field diblock copolymer model in this paper we consider numerical approximations of a hydro dynamically coupled phase field diblock copolymer model in which the free energy contains a kinetic potential a gradient entropy a ginzburglandau double well potential and a long range nonlocal type potential we develop a set of second order time marching schemes for this system using the invariant energy quadratization approach for the double well potential the projection method for the navierstokes equation and a subtle implicit explicit treatment for the stress and convective term the resulting schemes are linear and lead to symmetric positive definite systems at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

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

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

optimal first order error estimates of a fully segregated scheme for the navier stokes equations abstract a first order linear fully discrete scheme is studied for the incompressible time dependent navier stokes equations in three dimensional domains this scheme is based on an incremental pressure projection method and decouples each component of the velocity and the pressure solving in each time step a linear convection diffusion problem for each component of the velocity and a poisson neumann problem for the pressure using an inf sup stable and continuous finite elements approach of order o h in space unconditional optimal error estimates of order o k h are deduced for velocity and pressure without imposing constraints on the mesh size h and the time step k finally some numerical results are performed to validate the theoretical analysis and also to compare the studied scheme with other current first order segregated schemes

a semi implicit augmented iim for navier stokes equations with open traction or free boundary conditions in this paper a new navier stokes solver based on a finite difference approximation is proposed to solve incompressible flows on irregular domains with open traction and free boundary conditions which can be applied to simulations of fluid structure interaction implicit solvent model for biomolecular applications and other free boundary or interface problems for some problems of this type the projection method and the augmented immersed interface method iim do not work well or does not work at all the proposed new navier stokes solver is based on the local pressure boundary method and a semi implicit augmented iim a fast poisson solver can be used in our algorithm which gives us the potential for developing fast overall solvers in the future the time discretization is based on a second order multi step method numerical tests with exact solutions are presented to validate the accuracy of the method application to fluid structure interaction between an incompressible fluid and a compressible gas bubble is also presented

error analysis of pressure correction schemes for the time dependent stokes equations with open boundary conditions the incompressible stokes equations with prescribed normal stress open boundary conditions on part of the boundary are considered it is shown that the standard pressure correction method is not suitable for approximating the stokes equations with open boundary conditions whereas the rotational pressure correction method yields reasonably good error estimates these results appear to be the first ever published for splitting schemes with open boundary conditions numerical results in agreement with the error estimates are presented

role of the lbb condition in weak spectral projection methods 405 this paper investigates the relevance of the ladyshenskaya babuska brezzi con dition in spectral projection methods we consider the stability and convergence properties for a first order nonincremental projection method and a second order incremental projection method both based on a spectral galerkin legendre spatial discretization we show that the convergence of both projection methods is controlled by the ability of the spectral framework to approximate correctly the steady stokes problem c 2001 elsevier science

numerical radiation boundary conditions for unsteady transonic flow a family of numerical boundary conditions for far field computational boundaries in calculations involving unsteady transonic flow is devised these boundary conditions are developed in a systematic fashion from general principles both numerical and analytic comparisons with other currently used methods are given

a gpav based unconditionally energy stable scheme for incompressible flows with outflow open boundaries abstract we present an unconditionally energy stable scheme for approximating the incompressible navier stokes equations on domains with outflow open boundaries the scheme combines the generalized positive auxiliary variable gpav approach and a rotational velocity correction type strategy and the adoption of the auxiliary variable simplifies the numerical treatment for the open boundary conditions the discrete energy stability of the proposed scheme has been proven irrespective of the time step sizes within each time step the scheme entails the computation of two velocity fields and two pressure fields by solving an individual de coupled helmholtz including poisson type equation with a constant pre computable coefficient matrix for each of these field variables the auxiliary variable being a scalar number is given by a well defined explicit formula within a time step which ensures the positivity of its computed values extensive numerical experiments with several flows involving outflow open boundaries in regimes where the backflow instability becomes severe have been presented to test the performance of the proposed method and to demonstrate its stability at large time step sizes

the interaction between quasilinear elastodynamics and the navier stokes equations the interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations coupled to one another along the moving material interface through the continuity of the velocity and traction vectors we prove the existence and uniqueness locally in time of strong solutions in sobolev spaces for quasilinear elastodynamics coupled to the incompressible navier stokes equations unlike our approach in 5 for the case of linear elastodynamics we cannot employ a fixed point argument on the nonlinear system itself and are instead forced to regularize it by a particular parabolic artificial viscosity term we proceed to show that with this specific regularization we obtain a time interval of existence which is independent of the artificial viscosity together with a priori estimates we identify the global solution in both phases as well as the interface motion as a weak limit in strong norms of our sequence of regularized problems

on the korteweg de vries equation convergent birkhoff normal form abstract the korteweg de vries equation kdv formula is a completely integrable hamiltonian system of infinite dimension with phase space the sobolev spacehn s1 r n 1 hamiltonian h q s1 1 2 xq x 2 q x 3 dx and poisson structure x the functionq 0 is an elliptic fixed point we prove that for anyn 1 the korteweg de vries equation and thus the entire kdv hierarchy admits globally defined real analytic action angle variables as a consequence it follows that in a neighborhood ofq 0 inh1 s1 r the kdv hamiltonian h and similarly any hamiltonian in the kdv hierarchy admits a convergent birkhoff normal form to the best of our knowledge this is the first such example in infinite dimension moreover using the constructed action angle variables we analyze the regularity properties of the hamiltonian vectorfield of kdv

error growth and phase lag analysis for high courant numbers abstract the type of numerical scheme used in numerical computation of fluid flow largely affects the accuracy of the solution in order to obtain accurate numerical solution in a transient analysis implicit schemes are known to be unconditionally stable for linear systems while non linearity in advection systems often limits the stability nevertheless the implicit schemes for non linear systems allow larger courant numbers than explicit schemes however these assertions have been addressed based on a simple numerical error growth analysis which does not account for dispersion and phase lag errors even in linear systems thus motivated by the correct error dynamics given in sengupta et al 2007 it has been shown here that stable and accurate numerical solutions are possible for higher courant numbers while their low values do not always lead to stable and accurate solutions here higher courant numbers can be achieved by finer grid spacing keeping the time step same time integration is performed by implicit scheme for which numerical stability is guaranteed and thus a higher courant number implies higher resolution of the solution this has been proven through a test case of the propagation of acoustic wave in water by considering the latter to be a compressible medium by solving unsteady navier stokes equations this observation is elucidated by evaluating the error growth with first order in time and second order in space schemes with the latter represented by fully implicit form at t n 1 t the results show that for high courant numbers the value of error growth is very close to unity and confirms to the needed neutral stability of the obtained numerical solution furthermore the value of phase error nearly vanishes and reveals that numerical waves travel with nearly the same speed as physical waves

shear rate projection schemes for non newtonian fluids abstract the operator splitting approach applied to the navier stokes equations gave rise to various numerical methods for the simulations of the dynamics of fluids the separate work of chorin and temam on this subject gave birth to the so called projection methods the most basic of those schemes the incremental and non incremental variant see guermond et al 2006 induce an artificial neumann boundary condition on the pressure the so called rotational incremental pressure correction scheme proposed by timmermans et al 1996 gives a consistent equation for the pressure in case of a newtonian fluids with homogeneous viscosity in this work we propose a family of projection methods for generalized newtonian fluids based on an extension of the rotational projection scheme called shear rate projections these methods produce consistent pressure when applied to generalized newtonian fluids accuracy of the methods will be illustrated using a manufactured solution numerical experiments for the flow past a cylinder with a carreau rheological model will also be presented

a scalable consistent second order sph solver for unsteady low reynolds number flows abstract smoothed particle hydrodynamics sph has successfully been used to study a variety of cases involving nearly inviscid flows where conservation properties allow for good physical approximation despite poor theoretical approximation properties of differential operators when used to study unsteady low reynolds number flow with large dissipation conservation alone cannot ensure quality of approximation and the traditional approach is inconsistent an alternative formulation has recently become popular making use of an approximate splitting scheme to ensure a divergence free velocity field however this scheme relies on an inconsistent discretization of the laplacian that diverges as particles become disordered under flow we present an incremental pressure correction scheme and combination of existing differential operator renormalizations that are able to achieve second order accuracy in time and space a brief review of sph approximation theory is provided to highlight the necessity of these renormalizations in implementing an approximate factorization scheme we demonstrate that when fast algebraic multigrid preconditioners are used to solve the resulting linear systems the scheme results in a consistent approximation that is scalable and amenable to parallelization several validation cases are presented for which a speedup of several orders of magnitude is achieved over traditional sph approaches finally an interface has been developed between the particle library lammps and the sparse linear algebra libraries in trilinos providing a massively parallel 3d sph capability scaling results for up to 134 million particles on 32 768 cores are presented along with a demonstration of the capability to simulate complex 3d geometries these results show that the added complexity of applying the necessary consistency corrections actually provides a factor of four speed up per linear solver iteration versus the uncorrected case despite the additional cost of constructing the corrections the resulting library provides a method that is consistent efficient and second order in both space and time while maintaining the flexibility of classical sph for single phase flows

stability and convergence of second order time discrete projection method for the linearized oldroyd model abstract in this paper we consider the second order time discrete projection method for the linearized oldroyd model based on the time iterative discrete scheme by the projection method the original problem is decoupled into two linear subproblems and each subproblem can be solved easily unconditional stability and the corresponding convergence results of the numerical solutions are derived our main results are that the convergence orders in time for the velocity in l 2 norm is second order and for the pressure in h 1 norm is first order finally some numerical examples are provided to verify the performances of the developed numerical method

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

convergence analysis of the fem approximation of the first order projection method for incompressible flows with and without the inf sup condition in this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible navier stokes equations using a finite element approximation for the space discretization we consider two situations in the first one the analysis relies on the satisfaction of the inf sup condition for the velocity pressure finite element spaces after that we study a fully discrete fractional step method using a poisson equation for the pressure in this case the velocity pressure interpolations do not need to accomplish the inf sup condition and in fact we consider the case in which equal velocity pressure interpolation is used optimal convergence results in time and space have been obtained in both cases

on convergence of chorin s projection method to a leray hopf weak solution the projection method to solve the incompressible navier stokes equations was first studied by chorin math comput 1969 in the framework of a finite difference method and temam arch ration mech anal 1969 in the framework of a finite element method chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions assuming existence of a c 5 solution while temam demonstrated an abstract argument to obtain a leray hopf weak solution in problems on a bounded domain with the no slip boundary condition in the present paper the authors extend chorin s result with full details to obtain convergent finite difference approximation of a leray hopf weak solution to the incompressible navier stokes equations on an arbitrary bounded lipschitz domain of mathbb r 3 with the no slip boundary condition and an external force we prove unconditional solvability of our implicit scheme and strong l 2 convergence up to a subsequence under the scaling condition h 3 alpha le tau no upper bound is necessary where h tau are space time discretization parameters respectively and alpha in 0 2 is any fixed constant the results contain a compactness method based on a new interpolation inequality for step functions