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

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

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

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

model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

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

a finite difference galerkin formulation for the incompressible navier stokes equations abstract the development of a new computational method for solving the incompressible navier stokes equations in primitive variable form is presented it is found that certain finite difference approximations for these equations can be transformed into an equivalent system which efficiently determines the discrete velocity field and which completely eliminates the pressure two such difference schemes for two dimensional problems are examined and some preliminary numerical results are discussed for the steady driven cavity problem

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

stable and accurate pressure approximation for unsteady incompressible viscous flow how to properly specify boundary conditions for pressure is a longstanding problem for the incompressible navier stokes equations with no slip boundary conditions an analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the laplacian and leray projection operators here we make use of this formula to a improve the accuracy of computing pressure in two kinds of existing time discrete projection methods implicit in viscosity only and b devise new higher order accurate time discrete projection methods that extend a slip correction idea behind the well known finite difference scheme of kim and moin we test these schemes for stability and accuracy using various combinations of c 0 finite elements for all three kinds of time discretization one can obtain third order accuracy for both pressure and velocity without a time step stability restriction of diffusive type furthermore two kinds of projection methods are found stable using piecewise linear elements for both velocity and pressure

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

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

a quadratic spline based interface quasi reconstruction algorithm for accurate tracking of two phase flows a new quadratic spline based interface quasi reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature the fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous plic approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves the conversion is achieved by a sequence of predictor corrector operations enforcing function c 0 and derivative c 1 continuity at the cell boundaries using simple analytical expressions for the continuity requirements the efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations these benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes only in situations where the interface geometry is primarily flat the rate of convergence becomes linear with the mesh size the flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location as a consequence it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface this has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium finally the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

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 cartesian grid embedded boundary method for hyperbolic conservation laws we present a second order godunov algorithm to solve time dependent hyperbolic systems of conservation laws on irregular domains our approach is based on a formally consistent discretization of the conservation laws on a finite volume grid obtained from intersecting the domain with a cartesian grid we address the small cell stability problem associated with such methods by hybridizing our conservative discretization with a stable nonconservative discretization at irregular control volumes and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation the resulting method is second order accurate in l 1 for smooth problems and is robust in the presence of large amplitude discontinuities intersecting the irregular boundary

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

the localization sequence for the algebraic k theory of topological k theory we verify a conjecture of rognes by establishing a localization cofiber sequence of spectra k mathbb z to k ku to k ku to sigma k mathbb z for the algebraic k theory of topological k theory we deduce the existence of this sequence as a consequence of a devissage theorem identifying the k theory of the waldhausen category of finitely generated finite stage postnikov towers of modules over a connective a infty ring spectrum r with the quillen k theory of the abelian category of finitely generated pi 0 r modules

quasi similarity solutions for one phase stefan problems with time dependent boundary conditions we consider the one dimensional one phase stefan problems with temperature or thermal flux boundary conditions exact and closed form solutions for these two free boundary problems are very limited and highly restricted to particular initial and boundary conditions and so are the similarity solutions in general only numerical methods allow one to completely solve such nonlinear problems in this paper the quasi similarity solutions are defined and for each problem an analytical approximation of the interfacial position is given as function of the time dependent boundary datum

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

efficient implementation of thinc scheme in this paper we improve the multi dimensional thinc tangent of hyperbola for interface capturing scheme f xiao y honma t kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer meth fluid 48 2005 1023 the thinc scheme is a vof volume of fluid type method in the original thinc scheme one dimensional thinc scheme was straightforwardly used for multi dimensional cases in this paper we propose the wlic weighed line interface calculation method to extend the thinc scheme to multi dimension in the wlic method the interface is reconstructed by taking an average of line interfaces along x y and z coordinates with the weights calculated from surface normal the wlic method can extend the thinc scheme to multi dimension while maintaining simplicity of implementation and achieve a higher accuracy than the original thinc scheme the wlic method can readily extend the thinc scheme to three dimension

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 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

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

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

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

higher order temporal integration for the incompressible navier stokes equations in bounded domains author s minion ml saye ri abstract 2018 elsevier inc this paper compares and contrasts higher order semi implicit temporal integration strategies for the incompressible navier stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations particular focus is placed on the imposition of boundary conditions in the semi implicit formulation the accurate treatment of the pressure term and the smoothness of the numerical solution for different formulations the main result presented here is the formulation and numerical validation of a single step semi implicit method called spectral deferred pressure corrections which can in theory obtain arbitrary formal order of accuracy in time numerical results demonstrate up to eighth order accuracy scenarios where optimal temporal accuracy is attained and scenarios where order reduction is observed due to time dependent boundary conditions

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 stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

discrete exterior calculus discretization of incompressible navier stokes equations over surface simplicial meshes a conservative discretization of incompressible navier stokes equations is developed based on discrete exterior calculus dec a distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product the governing equations are first rewritten using the exterior calculus notation replacing vector calculus differential operators by the exterior derivative hodge star and wedge product operators the discretization is then carried out by substituting with the corresponding discrete operators based on the dec framework numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured triangular meshes and first order accuracy for otherwise unstructured meshes by construction the method is conservative in that both mass and vorticity are conserved up to machine precision the relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step

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

approximate analytical solutions for a shrinking core model for the discharge of a lithium iron phosphate electrode by the adomian decomposition method abstract in this paper we solve the mathematical model that describes the variation of lithium concentration in a lithium iron phosphate lifepo 4 particle during the process of lithium intercalation into the particle with shrinking core during the discharge process the model is composed of a second order linear partial differential equation satisfied by the distribution function of lithium concentration with an unknown moving boundary function and an ordinary partial differential equation satisfied by these two unknown functions an approximate analytic solution for the partial differential equation with undetermined parameters is first given by the adomian decomposition method adm and then we require it to satisfy the moving boundary condition to determine these parameters in order to obtain the solution for the model we need not transform the moving boundary into a fixed boundary as in prior research our new approach in solving the model shows that the adm is an efficient method for solving moving boundary problems based on the algorithm provided by the adm we decompose the complex operation of solving the model into a sequence of sub operations which are easily implemented by using the numerical and symbolic operations in matlab by completing these sub operations we obtain an accurate expression of the approximate analytic solution for engineering simulations

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

model category structures on chain complexes of sheaves in this paper we try to determine when the derived category of an abelian category is the homotopy category of a model structure on the category of chain complexes we prove that this is always the case when the abelian category is a grothendieck category as has also been done by morel but this model structure is not very useful for dening derived tensor products we therefore consider another method for constructing a model structure and apply it to the category of sheaves on a well behaved ringed space the resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces

physically motivated domain decomposition for singularly perturbed equations a domain decomposition algorithm suitable for the efficient and accurate solution of a parabolic reaction convection diffusion equation with small parameter multiplying the diffusion term is presented convergence is established via maximum principle arguments the equation arises in the modeling of laminar transonic flow decomposition into subdomains is accomplished via singular perturbation analysis which dictates regions where certain reduced equations may be solved in place of the full equation effectively preconditioning the problem this paper concentrates on the theoretical basis of the method establishing local and global a priori error bounds

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

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

finite element approximations of singular parabolic problems an efficient implementation of finite element methods for free boundary parabolic problems in general two dimensional space domains is presented the stefan problem the hele shaw problem and the porous medium equation are included backward differences or linearization techniques are used for the time discretization of the problem the performances of these schemes are discussed with several numerical tests

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

a geometrical area preserving volume of fluid advection method a new class of algorithms that preserve mass exactly for incompressible flows on a cartesian mesh are presented they amount to piecewise linear area preserving mappings of tessellations of the plane they are equivalent to volume of fluid vof advection methods which are decomposed into an eulerian implicit scheme in one direction followed by a lagrangian explicit step in the other one it is demonstrated that mass conservation is exact for incompressible flows and that there are no undershoots or overshoots of the volume fraction which thus always remains constrained between 0 and 1

algebraic k theory and abstract homotopy theory abstract we decompose the k theory space of a waldhausen category in terms of its dwyer kan simplicial localization this leads to a criterion for functors to induce equivalences of k theory spectra that generalizes and explains many of the criteria appearing in the literature we show that under mild hypotheses a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of k theory spectra

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

a cell centered adaptive projection method for the incompressible euler equations we present an algorithm to compute adaptive solutions for incompressible flows using block structured local refinement in both space and time this method uses a projection formulation based on a cell centered approximate projection which allows the use of a single set of cell centered solvers because of refinement in time additional steps are taken to accurately discretize the advection and projection operators at grid refinement boundaries using composite operators which span the coarse and refined grids this ensures that the method is approximately freestream preserving and satisfies an appropriate form of the divergence constraint

a method for solving moving boundary problems a new method for solving moving boundary problems or stefan problems is proposed two associated heat conduction problems which have fixed domains and have the same boundary and initial conditions as the original problem are introduced each of these includes a moving source as a supplementary condition the new problem has no moving boundary and can be studied by many known methods and techniques one dimensional problems are analyzed in detail for exploratory purposes two examples one of which is the neumann problem are discussed

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

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

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 new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

the k theory of assemblers abstract in this paper we introduce the notion of an assembler which formally encodes cutting and pasting data an assembler has an associated k theory spectrum in which 0 is the free abelian group of objects of the assembler modulo the cutting and pasting relations and in which the higher homotopy groups encode further geometric invariants the goal of this paper is to prove structural theorems about this k theory spectrum including analogs of quillen s localization and devissage theorems we demonstrate the uses of these theorems by analyzing the assembler associated to the grothendieck ring of varieties and the assembler associated to scissors congruence groups of polytopes

an improved particle correction procedure for the particle level set method the particle level set method d enright r fedkiw j ferziger i mitchell a hybrid particle level set method for improved interface capturing j comput phys 183 2002 83 116 can substantially improve the mass conservation property of the level set method by using lagrangian marker particles to correct the level set function in the under resolved regions in this study the limitations of the particle level set method due to the errors introduced in the particle correction process are analyzed and an improved particle correction procedure is developed based on a new interface reconstruction scheme moreover the zero level set is anchored as the level set functions are reinitialized hence the additional particle correction after the level set reinitialization is avoided with this new scheme a well defined zero level set can be obtained and the disturbances to the interface are significantly reduced consequently the particle reseeding operation will barely result in the loss of interface characteristics and can be applied as frequently as necessary to demonstrate the accuracy and robustness of the proposed method two extreme particle reseeding strategies one without reseeding and the other with reseeding every time step are applied in several benchmark advection tests and the results are compared with each other three interfacial flow cases a 2d surface tension driven oscillating droplet a 2d gas bubble rising in a quiescent liquid and a 3d drop impact onto a liquid pool are simulated to illustrate the advantages of the current method over the level set and the original particle level set methods with regard to the smoothness of geometric properties and mass conservation in real physical applications

uncoupled temporally second order accurate implicit solver of incompressible navier stokes equations an efficient method with temporal second order accuracy for solving the incompressible navier stokes equations in generalized coordinate systems is developed efficiency is obtained by extending an existing fractional step semi implicit solver into an implicit one with implicit approximation of the convection terms a novel three level linearization scheme is proposed to decouple the momentum equations from one another in addition to the fractional step approach that decouples the continuity equation from the momentum equations consequently all of the governing equations are uncoupled without degrading temporal accuracy or stability and without the need for iterations to solve the nonlinear system of equations at each time level the proposed decoupling technique can be used for other systems of nonlinear and time dependent partial differential equations moreover only minor modifications are required to apply the method to existing two level schemes several test cases confirm that the proposed decoupled scheme has temporal second order accuracy and allows the use of large time steps

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

a well conditioned augmented system for solving navier stokes equations in irregular domains an augmented method based on a cartesian grid is proposed for the incompressible navier stokes equations in irregular domains the irregular domain is embedded into a rectangular one so that a fast poisson solver can be utilized in the projection method unlike several methods suggested in the literature that set the force strengths as unknowns which often results in an ill conditioned linear system we set the jump in the normal derivative of the velocity as the augmented variable the new approach improves the condition number of the system for the augmented variable significantly using the immersed interface method we are able to achieve second order accuracy for the velocity numerical results and comparisons to benchmark tests are given to validate the new method a lid driven cavity flow with multiple obstacles and different geometries are also presented

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

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

a diffuse interface model for axisymmetric immiscible two phase flow a diffuse interface model is considered for solving axisymmetric immiscible two phase flow with surface tension the navier stokes ns equations are modified by the addition of a continuum forcing the interface between the two fluids is considered as the half level set of a mass concentration c which is governed by the cahn hilliard ch equation a fourth order degenerate nonlinear parabolic diffusion equation in this work we develop a nonlinear multigrid method to solve the ch equation with degenerate mobility and couple this to a projection method for the incompressible ns equations the diffuse interface method can deal with topological transitions such as breakup and coalescence smoothly without ad hoc cut and connect or other artificial procedures we present results for rayleigh s capillary instability up to forming satellite drops the results agree well with the linear stability theory

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions

derived l adic zeta functions abstract let k 0 v k be the grothendieck group of k varieties campbell and zakharevich have constructed a higher algebraic k theory spectrum k v k such that 0 k v k k 0 v k in this paper we construct non trivial classes in the higher homotopy groups of k v k when k is finite or a subfield of c to do this we give a recipe for lifting motivic measures k 0 v k k 0 e to maps of spectra k v k k e we consider two special cases the classical local zeta function thought of as a homomorphism k 0 v f q k 0 end q l and the compactly supported euler characteristic thought of as a homomorphism k 0 v c k 0 q we use lifts of these motivic measures to prove that the grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map s k v k is nontrivial in higher dimensions when k is finite or a subfield of c and moreover that when k is finite this map is not surjective on higher homotopy groups