the truncated complex moment problem let 2n denote a sequence of complex numbers 00 01 10 0 2n 2n 0 00 0 ij ji and let k denote a closed subset of the complex plane c the truncated complex k moment problem for entails determining whether there exists a positive borel measure on c such that ij zizj d 0 i j 2n and supp k for k kp a semi algebraic set determined by a collection of complex polynomials p pi z z mi 1 we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix m n and the localizing matrices mpi we prove that there exists a rankm n atomic representing measure for 2n supported in kp if and only if m n 0 and there is some rankpreserving extension m n 1 for which mpi n ki 0 where deg pi 2ki or 2ki 1 1 i m

on a fem method for a linearized version of the oldroyd s problem abstract we present a study of a finite element method fem for the approximation of a linearized version of the oldroyd s problem the method is based on a work in zamm 65 1985 449 concerning the continuous problem on a splitting of the momentum equation and on a modification of the streamline upwind petrov galerkin method in comput meth appl mech engrg 45 1984 285 it is also formally connected to the modified elastic viscous split stress method in j non newtonian fluid mech 60 1995 27 this method allows us to improve some numerical results expected for more standard methods in particular we obtain convergence results in the case of a null newtonian fraction of the viscosity

active control and drag optimization for flow past a circular cylinder the main objective of this article is to investigate computational methods for the active control and drag optimization of incompressible viscous flow past cylinders using the two dimensional navier stokes equations as the flow model the computational methodology relies on the following ingredients space discretization of the navier stokes equations by finite element approximations time discretization by a second order accurate two step implicit explicit finite difference scheme calculation of the cost function gradient by the adjoint equation approach and minimization of the cost function by a quasi newton method a la bfgs the above methods have been applied to boundary control by rotation of the flow around a circular cylinder and show 30 to 60 drag reduction compared to the fixed cylinder configuration for reynolds numbers in the range from 200 to 1000

cubic column relations in truncated moment problems for the truncated moment problem associated to a complex sequence 2n ij i j z i j 2n to have a representing measure it is necessary for the moment matrix m n to be positive semidefinite and for the algebraic variety v to satisfy rankm n cardv as well as a consistency condition the riesz functional vanishes on every polynomial of degree at most 2n that vanishes on v in previous work with l fialkow and h m moller the first named author proved that for the extremal case rankm n cardv positivity and consistency are sufficient for the existence of a representing measure in this paper we solve the truncated moment problem for cubic column relations in m 3 of the form z3 itz uz u t r we do this by checking consistency for u t in the open cone determined by 0 u t 2 u we first prove that the algebraic variety has exactly 7 points and rankm 3 7 we then apply the above mentioned result to obtain a concrete computable necessary and sufficient condition for the existence of a representing measure

non extremal sextic moment problems abstract positive semidefiniteness recursiveness and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems also positive semidefiniteness combined with another necessary condition consistency is a sufficient condition in the case of extremal moment problems i e when the rank of the moment matrix denoted by r and the cardinality of the associated algebraic variety denoted by v are equal however these conditions are not sufficient for non extremal sextic or higher order truncated moment problems in this paper we settle three key instances of the non extremal i e r v sextic moment problem as follows when r 7 positive semidefiniteness consistency and the variety condition guarantee the existence of a 7 atomic representing measure when r 8 we construct two determining algorithms corresponding to the cases v 9 and v to accomplish this we generalize the rank reduction technique developed in previous work where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

on existence and uniqueness of solutions to a boussinesq system with nonlinear and mixed boundary conditions abstract we study a boussinesq system in a bounded domain with an outlet boundary portion where fluid can leave or re enter on this boundary part we consider a do nothing condition for the fluid flow and a new artificial condition for the heat transfer that couples nonlinearly the fluid velocity and temperature the latter can be further adjusted if convective or conductive phenomena are dominant we prove existence and in some cases uniqueness of weak solutions to stationary and evolutionary problems by a fixed point strategy under suitable assumptions on the data a variety of numerical tests shows the improved performance of the new artificial condition with respect to other standard choices in the literature

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

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

robust model identification of actuated vortex wakes we present a low order modeling technique for actuated flows based on the regularization of an inverse problem the inverse problem aims at minimizing the error between the model predictions and some reference simulations the parameters to be identified are a subset of the coefficients of a polynomial expansion which models the temporal dynamics of a small number of global modes these global modes are found by proper orthogonal decomposition which is a method to compute the most representative elements of an existing simulation database in terms of energy it is shown that low order control models based on a simple galerkin projection and usual calibration techniques are not viable they are either ill posed or they give a poor approximation of the solution as soon as they are used to predict cases not belonging to the original solution database in contrast numerical evidence shows that the method we propose is robust with respect to variations of the control laws applied thus allowing the actual use of such models for control

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 numerical method for the study of nonlinear stability of axisymmetric flows based on the vector potential we develop in this paper a numerical method to simulate three dimensional incompressible flows based on a decomposition of the flow into an axisymmetric part in terms of the stream function and the circulation and a non axisymmetric part in terms of a potential vector function the method is specially suited for the study of nonlinear stability of axially symmetric flows because one may follow neatly the raising of the different non axisymmetric modes their nonlinear development and their nonlinear interaction the numerical technique combines finite differences on a non uniform grid in the axial direction a chebyshev spectral collocation technique in the radial direction and a fourier spectral method in the azimuthal direction for the non axisymmetric vector potential as an example to check the efficiency and accuracy of the method we apply it to the flow inside a rotating circular pipe and compare the resulting travelling waves with previous stability results for this problem for different values of the reynolds and the swirl numbers

a zonal galerkin free pod model for incompressible flows abstract a domain decomposition method which couples a high and a low fidelity model is proposed to reduce the computational cost of a flow simulation this approach requires to solve the high fidelity model in a small portion of the computational domain while the external field is described by a galerkin free proper orthogonal decomposition pod model we propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models this zonal approach can be used to study multi body configurations or to perform detailed local analyses in the framework of shape optimisation problems the efficiency of the method to perform predictive low cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem

probleme des moments sur un compact de rn et decomposition de polynomes a plusieurs variables on donne la solution du probleme des moments pour un compact d interieur non vide quelconque de r n on utilise cette resolution pour obtenir un resultat de decomposition des polynomes strictement positifs sur un compact d interieur non vide

simulation of 2d external viscous flows by means of a domain decomposition method two dimensional external viscous flows are numerically approximated by means of a domain decomposition technique which combines a vortex method and a finite differences method the vortex method is used in the flow region which is dominated by convective effects whereas the finite differences method is used in the flow region where viscous diffusion effects are dominant meanwhile a new vortex method which is suitable for approximating first order linear hyperbolic problems supplemented with dirichlet boundary conditions is presented comparison with numerical and experimental data show that the method is well adapted for calculating two dimensional external flows at moderate reynolds numbers

artificial compressibility characteristics based schemes for variable density incompressible multispecies flows part ii multigrid implementation and numerical tests the paper presents an investigation of the accuracy and efficiency of artificial compressibility characteristics based cb schemes for variable density incompressible flows the cb schemes have been implemented in conjunction with a multigrid method for accelerating numerical convergence and a fourth order explicit runge kutta method for the integration of the governing equations in time the implementation of the cb schemes is obtained in conjunction with first second and third order interpolation formulas for calculating the variables at the cell faces of the computational volume the accuracy and efficiency of the schemes are examined against analytical and experimental results for diffusion broadening in two and three dimensional microfluidic channels a problem that has motivated the development of the present methods moreover unsteady inviscid simulations have been performed for variable density mixing layer the computations revealed that accuracy and efficiency depend on the cb scheme design the best multigrid convergence rates were exhibited by the conservative cb scheme which is obtained by the fully conservative formulation of the variable density incompressible equations

on a posteriori estimates for a linearized oldroyd s problem we study and give some a posteriori estimates for finite element approximation of a linearized oldroyd s problem for the approximation of viscoelastic fluid flows the major difficulty lies in the coupling between two different kinds of equations a hyperbolic equation associated to the constitutive equation and stokes problem these estimates are then obtained combining two different techniques used for each kind of equation the first technique is given in first iciam paris 1987 for stokes s equations and the second in c r acad sci paris ser i 322 1996 493 for friedrichs s system

a spectral time discretization for flows with dominant periodicity an accurate and efficient treatment of periodic and quasi periodic flows based on the temporal fourier decomposition of the navier stokes equations is suggested a numerical implementation for a laminar afterbody wake in a 2d channel is presented this implementation is formulated in primitive variables and uses an ordinary second order accurate finite volume space discretization combined with a standard pressure correction procedure a multistep time marching scheme for numerical and physical transients is developed for flows with a variable dominant period a period correction algorithm is used the transients characterizing the instability development are simulated the numerical results obtained for the afterbody wake confirm the expectations concerning the efficiency and high time accuracy of the method moreover the method provides direct access to quantities difficult to obtain by other methods such as the envelope and the angular velocity variation of the unstable mode

artificial compressibility characteristics based schemes for variable density incompressible multi species flows part i derivation of different formulations and constant density limit the paper presents various formulations of characteristics based schemes in the framework of the artificial compressibility method for variable density incompressible flows in contrast to constant density incompressible flows where the characteristics based variables reconstruction leads to a single formulation in the case of variable density flows three different schemes can be obtained henceforth labeled as transport conservative and hybrid schemes the conservative scheme results in pseudo compressibility terms in the multi species density reconstruction it is shown that in the limit of constant density the transport scheme becomes the original characteristics based scheme for incompressible flows but the conservative and hybrid schemes lead to a new characteristics based variant for constant density flows the characteristics based schemes are combined with second and third order interpolation for increasing the computational accuracy locally at the cell faces of the control volume numerical experiments for constant density flows reveal that all the characteristics based schemes result in the same flow solution but they exhibit different convergence behavior the multigrid implementation and numerical studies for variable density flows are presented in part ii of this study

physical formulation and numerical algorithm for simulating n immiscible incompressible fluids involving general order parameters we present a family of physical formulations and a numerical algorithm based on a class of general order parameters for simulating the motion of a mixture of n n 2 immiscible incompressible fluids with given densities dynamic viscosities and pairwise surface tensions the n phase formulations stem from a phase field model we developed in a recent work based on the conservations of mass momentum and the second law of thermodynamics the introduction of general order parameters leads to an extremely strongly coupled system of n 1 phase field equations on the other hand the general form enables one to compute the n phase mixing energy density coefficients in an explicit fashion in terms of the pairwise surface tensions we show that the increased complexity in the form of the phase field equations associated with general order parameters in actuality does not cause essential computational difficulties our numerical algorithm reformulates the n 1 strongly coupled phase field equations for general order parameters into 2 n 1 helmholtz type equations that are completely de coupled from one another this leads to a computational complexity comparable to that for the simplified phase field equations associated with certain special choice of the order parameters we demonstrate the capabilities of the method developed herein using several test problems involving multiple fluid phases and large contrasts in densities and viscosities among the multitude of fluids in particular by comparing simulation results with the langmuir de gennes theory of floating liquid lenses we show that the method using general order parameters produces physically accurate results for multiple fluid phases

implicit fully mesh less method for compressible viscous flow calculations a dual time implicit mesh less scheme is developed for solution of governing viscous flow equations the computational efficiency of the method is enhanced by adopting accelerating techniques such as local time stepping and residual smoothing the taylor series least square method is used for approximation of derivatives at each node which leads to a central difference spatial discretization the capabilities of the method are demonstrated by flow computations about standard cases at subsonic and transonic laminar flow conditions results are presented which indicate good agreements with the alternative numerical and experimental data the computational time is considerably reduced when using the proposed mesh less method compared with the explicit mesh less and finite volume counterparts using the same distribution of points

bdf like methods for nonlinear dynamic analysis we present several time integration algorithms of second order accuracy that are numerically simple and effective for nonlinear elastodynamic problems these algorithms are based on a general four step scheme that has a resemblance to the backward differentiation formulas we also present an extension to the composite strategy of the bathe method appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity and less dissipative members of each algorithm have been identified we demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions and test these algorithms with several three dimensional nonlinear elastodynamic problems involving large deformations and rotations employing st venant kirchhoff and compressible neo hookean hyperelastic material models these tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well known instability

multidimensional upwinding for incompressible flows based on characteristics in this paper the multidimensional characteristic based upwind scheme mcb which has been recently introduced by the authors is applied to two another benchmark problems namely flow in a channel with a backward facing step and two dimensional steady and unsteady flows past a circular cylinder extension of mcb scheme for calculating convective fluxes on non cartesian grids is presented here for the flow over backward facing step obtained results were compared against well known experimental data and the results show high accuracy of mcb scheme and faster convergence rate with respect to conventional cb scheme in the case of flow over circular cylinder the flow at steady and transient regimes is investigated by mcb scheme again the results obtained by mcb are compared to the other results in the literature and show good agreement with them also rapid convergence rate of mcb was observed in this case too it is concluded that the genuinely multidimensional characteristic based mcb scheme has been introduced earlier by the authors is a robust and powerful scheme for modeling incompressible viscous flows for achieving the high accuracy and remarkable advantage in convergence rate with respect to conventional characteristic based schemes

the core variety of a multisequence in the truncated moment problem abstract let m i i z n i m 0 0 denote a real n dimensional multisequence of finite degree m the truncated moment problem concerns the existence of a positive borel measure supported in r n such that 0 1 i r n x i d i z n i m we associate to 2 d an algebraic variety in r n called the core variety v v the core variety contains the support of each representing measure we show that if v is nonempty then 2 d 1 has a representing measure moreover if v is a nonempty compact or determining set then 2 d has a representing measure we also use the core variety to exhibit a sequence with positive definite moment matrix and positive riesz functional which fails to have a representing measure

a posteriori estimators for mixed finite element approximations of a fluid obeying the power law we study a posteriori error estimation in the approximation by the finite element method of a fluid which obeys the power law 2 d u r 2d u p f u 0 1 r 2 abstract and calculable a posteriori estimators are given in two fields and in a three fields version of this problem furthermore we give some examples of fe spaces satisfying the inf sup condition which relates the discrete space of tensors and the discrete space of velocities needed for this formulation

an energy stable scheme for incompressible navier stokes equations with periodically updated coefficient matrix abstract we present an energy stable scheme for simulating the incompressible navier stokes equations based on the generalized positive auxiliary variable gpav framework in the gpav reformulated system the original nonlinear term is replaced by a linear term plus a correction term where the correction term is put under control by an auxiliary variable the proposed scheme incorporates a pressure correction type strategy into the gpav procedure and it satisfies a discrete energy stability property the scheme entails the computation of two copies of the velocity and pressure within a time step by solving an individual de coupled linear equation for each of these field variables upon discretization the pressure linear system involves a constant coefficient matrix that can be pre computed while the velocity linear system involves a coefficient matrix that is updated periodically once every k 0 time steps in the current work where k 0 is a user specified integer the auxiliary variable being a scalar valued number is computed by a well defined explicit formula which guarantees the positivity of its computed values it is observed that the current method can produce accurate simulation results at large or fairly large time step sizes for the incompressible navier stokes equations the impact of the periodic coefficient matrix update on the overall cost of the method is observed to be small in typical numerical simulations several flow problems have been simulated to demonstrate the accuracy and performance of the method developed herein

boundary conditions for a divergence free velocity pressure formulation of the navier stokes equations new sets of boundary conditions for the velocity pressure formulation of the incompressible navier stokes equations are derived the boundary conditions have the same form on both inflow and outflow boundaries and lead to a divergence free solution moreover the specific form of the boundary condition makes it possible derive a symmetric positive definite equation system for the internal pressure numerical experiments support the theoretical conclusions

a fast immersed boundary method for external incompressible viscous flows using lattice green s functions a new parallel computationally efficient immersed boundary method for solving three dimensional viscous incompressible flows on unbounded domains is presented immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original peskin s immersed boundary method unlike peskin s method boundary forces are regarded as lagrange multipliers that are used to satisfy the no slip condition the incompressible navier stokes equations are discretized on an unbounded staggered cartesian grid and are solved in a finite number of operations using lattice green s function techniques these techniques are used to automatically enforce the natural free space boundary conditions and to implement a novel block wise adaptive grid that significantly reduces the run time cost of solutions by limiting operations to grid cells in the immediate vicinity and near wake region of the immersed surface these techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half explicit runge kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation incompressibility constraint and no slip constraint linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation free nested projection technique the algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems e g discrete poisson problems that are compatible with recent parallel fast multipole methods for difference equations numerical experiments on low aspect ratio flat plates and spheres at reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation

fast multigrid method for solving incompressible hydrodynamic problems with free surfaces we develop of a finite volume multigrid euler scheme for solving three dimensional fully nonlinear ship wave problems the flowfield and the a priori unknown free surface location are calculated by coupling the free surface kinematic and dynamic equations with the equations of motion for the bulk flow the evolution of the free surface boundary condition is linked to the evolution of the bulk flow via a novel iteration strategy that allows temporary leakage through the surface before the solution is converged the method of artificial compressibility is used to enforce the incompressibility constraint for the bulk flow a multigrid algorithm is used to accelerate convergence to a steady state

an immersed boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

reduced order subscales for pod models in this work the reduced order subscales for proper orthogonal decomposition models are presented the basic idea consists in splitting the full order solution into the part which can be captured by the reduced order model and the part which cannot the subscales for which a model is required the proposed model for the subscales is defined as a linear function of the solution of the reduced order model the coefficients of this linear function are obtained by comparing the solution of the full order model with the solution of the reduced order model for the same initial conditions which for convenience are evaluated in the snapshots used to train the original reduced order model the difference between both solutions are the subscales for which a model can be built using a least squares procedure the subscales are then introduced as a correction in the reduced order model resulting in an important improvement in accuracy the enhanced reduced order model is tested in several numerical examples these practical cases show that the use of the subscales leads to more accurate solutions successfully corrects errors introduced by hyper reduction and allows to solve complex flow problems using a reduced number of degrees of freedom

a fast lattice green s function method for solving viscous incompressible flows on unbounded domains a computationally efficient method for solving three dimensional viscous incompressible flows on unbounded domains is presented the method formally discretizes the incompressible navier stokes equations on an unbounded staggered cartesian grid operations are limited to a finite computational domain through a lattice green s function technique this technique obtains solutions to inhomogeneous difference equations through the discrete convolution of source terms with the fundamental solutions of the discrete operators the differential algebraic equations describing the temporal evolution of the discrete momentum equation and incompressibility constraint are numerically solved by combining an integrating factor technique for the viscous term and a half explicit runge kutta scheme for the convective term a projection method that exploits the mimetic and commutativity properties of the discrete operators is used to efficiently solve the system of equations that arises in each stage of the time integration scheme linear complexity fast computation rates and parallel scalability are achieved using recently developed fast multipole methods for difference equations the accuracy and physical fidelity of solutions are verified through numerical simulations of vortex rings

a high order characteristics upwind fv method for incompressible flow and heat transfer simulation on unstructured grids abstract this paper presents a new unstructured grid upwind finite volume algorithm for accurate numerical simulation of incompressible flows and convection heat transfer on unstructured grids it is an upwind method at both the differential equation level and discretized equation level based on the method of characteristics this is made possible with the introduction of chorin s j comput phys 2 1967 12 26 artificial compressibility formulation to the incompressible flow equations flow variables are calculated along characteristics and their initial values are interpolated based on the signs of the corresponding characteristic speed in addition an upwind biased interpolation method of third order accuracy is used for interpolating flow variables on unstructured grids with these inherent upwinding techniques for evaluating convection fluxes at control volume surfaces no artificial viscosity is required and the method is accurate and less sensitive to the grid orientation thus suitable for unstructured grid application the use of the finite volume method combined with unstructured grids makes the scheme very flexible in dealing with very complex boundary geometries while remaining to be fully conservative at the cell as well as global level the discretized equations are solved by an explicit multistage runge kutta time stepping scheme which is found to be efficient in terms of cpu and memory overheads a computer code has been developed which uses the numerical methods presented and a highly efficient edge based data structure for flux calculation and data storage a number of well documented test cases including 2d laminar flow in a channel with a backward facing step forced and natural convection flows have been calculated in order to validate the code and evaluate the performance of the numerical algorithm good agreement between the computed and measured results have been obtained for all the cases to demonstrate the accuracy of the third order scheme flow in another channel with a backward facing step studied by gartling int j numer meth fluids 1990 11 is simulated with both the third order and first order schemes it is shown that the third order scheme can produce a velocity profile across the channel that has perfect agreement with that of gartling local citation whilst the first order scheme cannot even predict the separation near the upper wall the proposed numerical methods are also used to calculate 3d incompressible inviscid flows and results show that the method is accurate and robust for general 3d simulation the 3d navier stokes code is later used to study a lid driven cavity flow at a reynolds number of 400 the unstructured grid used has 97 336 nodes and 455 625 tetrahedral elements with strong grid clustering near the walls the convergence rate is found to be satisfactory which shows that the method is also robust for 3d navier stokes simulation the results obtained agree well with those reported by jiang et al comput meth appl mech eng 114 109 1994 results obtained using the second order central scheme with artificial viscosity are also presented for comparison it is observed that the third order characteristics scheme is more accurate than the latter thus confirming its third order accuracy

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

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

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

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

a lagged implicit segregated data reconstruction procedure to treat open boundaries the problem of treating open boundaries is still a challenging one applying fully developed condition is constrained to long enough domains without having enough physical evidence about what happens on boundaries the domain extent could not be shortened and computational costs could not be reduced from the advent of free open boundary conditions they were confined to mixed finite element procedures recent works have extended their application to coupled finite volume solvers based on the shape function data reconstruction a wider class of flow solvers available however rely on the segregated procedure where the velocity components and pressure are solved in succession moreover many finite volume algorithms do not use the shape function reconstruction in this work by proposing a lagged implicit procedure we have extended the application of the open boundary condition to these wider classes of flow solvers the proposed extension is a combination of lagged implicit data reconstruction and overall mass conservation enforcement which is easily applicable to any segregated and coupled flow solver to validate the compatibility of this extension benchmark problem of backward facing step is solved on successively truncated domains where open boundary may pass through recirculation zones results show that the proposed extension works fine for that problem it reduced the computational domain length and hence memory by a factor of 4 6 and the required computational time by a factor of 21 flow passing a cylinder is also solved which proves that the method could be applied to external flow problems as well

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

a stokes residual backflow stabilization method applied to physiological flows in computational fluid dynamics the presence of incoming flow at open boundaries backflow might often yield unphysical oscillations and instabilities issues even for moderate reynolds numbers it is widely accepted that this problem is caused by the incoming convective energy at the open boundary which cannot be controlled a priori when the velocity at the boundary is unknown in this work we propose a stabilized finite element formulation for the incompressible navier stokes equations in which the stabilization term is based on the residual of a weak stokes problem normal to the open boundary driven by an approximated boundary pressure gradient in particular the viscous term introduces additional dissipation which controls the incoming convective energy this method has the advantage as it does not require modifications or extensions of the computational domain moreover it does not require a priori assumptions on the shape of the boundary velocity field we illustrate our approach through several numerical examples relevant to blood and respiratory flows including womersley flows and realistic geometries coming from medical imaging the performance of the simulations is compared to recently reported approaches

the complex moment problem and subnormality a polar decomposition approach abstract it has been known that positive definiteness does not guarantee a bisequence to be a complex moment however it turns out that positive definite extendibility does theorems 1 and 22 and this is the main theme of this paper the main tool is generally understood polar decomposition to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context theorems 9 and 29 all this enables us to prove characterizations of subnormality of unbounded operators having invariant domain theorems 37 and 39 and their further applications theorems 41 and 43 and a description of the complex moment problem on real algebraic curves theorems 52 and 56 the latter question is completed in the appendix in which we relate the complex moment problem to the two dimensional real one with emphasis on real algebraic sets

wall bounded multiphase flows of n immiscible incompressible fluids we present an effective method for simulating wall bounded multiphase flows consisting of n n2 immiscible incompressible fluids with different densities viscosities and pairwise surface tensions the n phase physical formulation is based on a modified thermodynamically consistent phase field model that is more general than in a previous work and it is developed by considering the reduction consistency if some of the fluid components were absent from the system we propose an n phase contact angle boundary condition that is reduction consistent between n phases and m phases 2mn1 we also present a numerical algorithm for solving the n phase governing equations together with the contact angle boundary conditions developed herein extensive numerical experiments are presented for several flow problems involving multiple fluid components and solid wall boundaries to investigate the wettability effects with multiple types of contact angles in particular we compare simulation results with the de gennes theory for the contact angle effects on the liquid drop spreading on wall surfaces and demonstrate that our method produces physically accurate results

positivity of riesz functionals and solutions of quadratic and quartic moment problems abstract we employ positivity of riesz functionals to establish representing measures or approximate representing measures for truncated multivariate moment sequences for a truncated moment sequence y we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set k r n if and only if the associated riesz functional l y is k positive for a determining set k we prove that if l y is strictly k positive then y admits a representing measure supported in k as a consequence we are able to solve the truncated k moment problem of degree k in the cases i n k 2 4 and k r 2 ii n 1 k 2 and k is defined by one quadratic equality or inequality in particular these results solve the truncated moment problem in the remaining open cases of hilbert s theorem on sums of squares

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

boundary conditions for open boundaries for the incompressible navier stokes equation abstract in this paper we investigate new boundary conditions for the incompressible time dependent navier stokes equation especially inflow and outflow conditions are considered the equations are linearized around a constant flow so that we can use laplace fourier technique to investigate the strength of boundary layers at open boundaries such layers are unphysical and new boundary conditions are proposed so that these boundary layers are suppressed we also show that the boundary conditions we propose do not produce divergence furthermore they give solutions that do not grow in time as long as the forcing function in the system does not we also discuss the numerical treatment of boundaries when fourth order accurate finite difference operators are used to approximate spatial derivatives using higher order methods introduces eigensolutions with boundary layer thickness of the same order of magnitude as the grid size these eigensolutions have to be suppressed in order to not destroy the fourth order accuracy of the method numerical results for the non linear navier stokes equation together with the new boundary conditions are presented these calculations confirm that the results for the linearized problem hold for the non linear problem as well

energy stable boundary conditions based on a quadratic form applications to outflow open boundary problems in incompressible flows abstract we present a set of new energy stable open boundary conditions for tackling the backflow instability in simulations of outflow open boundary problems for incompressible flows these boundary conditions are developed through two steps i devise a general form of boundary conditions that ensure the energy stability by re formulating the boundary contribution into a quadratic form in terms of a symmetric matrix and computing an associated eigen problem and ii require that upon imposing the boundary conditions from the previous step the scale of boundary dissipation should match a physical scale these open boundary conditions can be re cast into the form of a traction type condition and therefore they can be implemented numerically using the splitting type algorithm from a previous work the current boundary conditions can effectively overcome the backflow instability typically encountered at moderate and high reynolds numbers in general they give rise to a non zero traction on the entire open boundary unlike previous related methods which only take effect in the backflow regions of the boundary extensive numerical experiments in two and three dimensions are presented to test the effectiveness and performance of the presented methods and simulation results are compared with the available experimental data to demonstrate their accuracy

an efficient algorithm for incompressible n phase flows abstract we present an efficient algorithm within the phase field framework for simulating the motion of a mixture of n n 2 immiscible incompressible fluids with possibly very different physical properties such as densities viscosities and pairwise surface tensions the algorithm employs a physical formulation for the n phase system that honors the conservations of mass and momentum and the second law of thermodynamics we present a method for uniquely determining the mixing energy density coefficients involved in the n phase model based on the pairwise surface tensions among the n fluids our numerical algorithm has several attractive properties that make it computationally very efficient i it has completely de coupled the computations for different flow variables and has also completely de coupled the computations for the n 1 phase field functions ii the algorithm only requires the solution of linear algebraic systems after discretization and no nonlinear algebraic solve is needed iii for each flow variable the linear algebraic system involves only constant and time independent coefficient matrices which can be pre computed during pre processing despite the variable density and variable viscosity of the n phase mixture iv within a time step the semi discretized system involves only individual de coupled helmholtz type including poisson equations despite the strongly coupled phase field system of fourth spatial order at the continuum level v the algorithm is suitable for large density contrasts and large viscosity contrasts among the n fluids extensive numerical experiments have been presented for several problems involving multiple fluid phases large density contrasts and large viscosity contrasts in particular we compare our simulations with the de gennes theory and demonstrate that our method produces physically accurate results for multiple fluid phases we also demonstrate the significant and sometimes dramatic effects of the gravity density ratios pairwise surface tensions and drop sizes on the n phase configurations and dynamics the numerical results show that the method developed herein is capable of dealing with n phase systems with large density ratios large viscosity ratios and pairwise surface tensions and that it can be a powerful tool for studying the interactions among multiple types of fluid interfaces

high order finite difference and multigrid methods for spatially evolving instability in a planar channel a computational study of the spatial instability of planar poiseuille flow is presented a fourth order finite difference with a fully implicit time marching scheme is developed on a staggered grid a semi coarsening multigrid method is applied to accelerate convergence for the implicit scheme at each time step and a line distributive relaxation is developed as a fast solver which is very robust and efficient for anisotropic grids a new treatment for outflow boundary conditions makes the buffer area as short as one wavelength the computational results demonstrate high accuracy in terms of agreement with linear theory and excellent efficiency in the sense that cost is comparable to and usually less than explicit schemes

proper orthogonal decomposition closure models for turbulent flows a numerical comparison this paper puts forth two new closure models for the proper orthogonal decomposition reduced order modeling of structurally dominated turbulent flows the dynamic subgrid scale model and the variational multiscale model these models which are considered state of the art in large eddy simulation together with the mixing length and the smagorinsky closure models are tested in the numerical simulation of 3d turbulent flow past a circular cylinder at re 1000 five criteria are used to judge the performance of the proper orthogonal decomposition reduced order models the kinetic energy spectrum the mean velocity the reynolds stresses the root mean square values of the velocity fluctuations and the time evolution of the pod coefficients all the numerical results are benchmarked against a direct numerical simulation based on these numerical results we conclude that the dynamic subgrid scale and the variational multiscale models are the most accurate

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

well posed boundary conditions for the navier stokes equations in this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations with those well posedness can be proven for a general initial boundary value problem the procedure is exemplified on the linearized navier stokes equations in two and three space dimensions on a general domain

preconditioned multigrid methods for unsteady incompressible flows a highly efficient numerical approach based on multigrid and preconditioning methods is developed for modeling 3d steady and time dependent incompressible flows thek turbulence model is used to estimate the effects of turbulence the model equations are solved together with the n s equations in a strongly coupled way and acceleration techniques like the multigrid method are also used for the turbulence model equations for unsteady problems a dual time stepping procedure is adopted to satisfy the divergence free constraint and to obtain a time accurate solution to improve the performance of this approach for small physical time steps a modification to residual smoothing parameters is proposed the numerical algorithm and the turbulence model are validated first by calculating unsteady inviscid flow around an oscillating cylinder unsteady laminar flow past a circular cylinder and steady high reynolds number turbulent flow over a 6 1 prolate spheroid then the three dimensional time dependent turbulent flow over a spheroid when it is undergoing a pitch up maneuver is calculated and compared with experimental data

numerical treatment of defective boundary conditions for the navier stokes equations we present a formulation for accommodating defective boundary conditions for the incompressible navier stokes equations where only averaged values are prescribed on measurable portions of the boundary in particular we consider the case where the flow rate is imposed on several domain sections this methodology has an interesting application in the numerical simulation of flow in blood vessels when only a reduced set of boundary data are generally available for the upstream and downstream sections

limits of positive flat bivariate moment matrices the bivariate moment problem for a sequence 6 of degree 6 remains unsolved but we prove that if the associated 10 10 moment matrix m3 satisfies m3 0 and rank m3 6 then admits a sequence of approximate representing measures and 5 has a representing measure more generally let fd denote the closure of the positive flat moment matrices of degree 2d in n variables each matrix in fd admits computable approximate representing measures and in 2013 jiawang nie and the author began to study concrete conditions for membership in this class let 2d i i zn i 2d 0 0 denote a real n dimensional sequence of degree 2d if the corresponding moment matrix md md is the limit of a sequence of positive flat moment matrices m k d i e m k d 0 and rank m k d rank m k d 1 then i md 0 ii rank md d 1 dim r x1 xn d 1 and iii 2d 1 admits a representing measure we extend our earlier results by proving conversely that for n 2 if md satisfies certain positivity and rank conditions related to i iii then md is the limit of positive flat moment matrices

on continuation of inviscid vortex patches abstract this investigation concerns solutions of the steady state euler equations in two dimensions featuring finite area regions with constant vorticity embedded in a potential flow using elementary methods of the functional analysis we derive precise conditions under which such solutions can be uniquely continued with respect to their parameters valid also in the presence of the kutta condition concerning a fixed separation point our approach is based on the implicit function theorem and perturbation equations derived using shape differentiation methods these theoretical results are illustrated with careful numerical computations carried out using the steklov poincare method which show the existence of a global manifold of solutions connecting the point vortex and the prandtl batchelor solution each of which satisfies the kutta condition

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

efficient and non reflecting far field boundary conditions for incompressible flow calculations abstract traditionally the artificial compressibility ac method of chorin is used for simulation of low speed and incompressible flows in this method the difficulty of continuity and momentum equations decoupling is removed by adding an artificial time derivative of pressure to the continuity equation for the first time a fully two dimensional upwind scheme was presented for ac equations by using characteristic structure of equations by the authors in this paper a new remote boundary calculation method is presented by using the idea of characteristics for these equations instead of simple boundary conditions which usually are employed for incompressible flows the flow quantities at the far field boundaries are evaluated by compatibility relations of characteristic equations this is implemented by assuming a row of ghost cells outside of the far field boundary and using the flux calculation method based on characteristics similar to the inside of computational domain the idea in conjunction with multidimensional characteristic based scheme was tested for incompressible flow around circular cylinder in comparison with conventional far field boundary condition and showed good improvements in the terms of accuracy and convergence speed

augmented lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three dimensional finite element simulations of blood flow in three dimensional blood flow simulations of the cardiovascular system velocity and pressure fields in the computational domain are highly affected by outlet boundary conditions this fact has motivated the development of novel methods to couple three dimensional computational domains with one dimensional numerical models or alternatively with zero dimensional or one dimensional analytic models in all such methods described to date whether they are explicit or implicit the relationship between flow and pressure at the outlet boundary is enforced weakly this coupling does not include any constraints on the shape of the velocity profiles nor on the distribution of pressure at the interface as a result there remain some classes of problems that are at best difficult to solve and at worst intractable with current numerical methods for simulating blood flow these include problems with significant flow reversal during part of the cardiac cycle or geometric complexity in the proximity of the outlet of the computational domain we have implemented a novel method to resolve these challenging problems whereby an augmented lagrangian method is used to enforce constraints on the shape of the velocity profile at the interface between the upstream computational domain and the downstream analytic domain these constraints on the shape of the velocity profile are added to the coupled multidomain method in order to implicitly couple the computational domain with downstream analytic models in this study an axisymmetric profile is imposed after ensuring that each constrained outlet boundary is circular we demonstrate herein that including constraints on the shape of the velocity profile does not affect velocity and pressure fields except in the immediate vicinity of the constrained outlet boundaries furthermore this new method enables the solution of problems which diverged with an unconstrained method

solving moment problems by dimensional extension polynomial inequalities the second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part for decomposing a polynomial which is positive on such a semi algebraic set into a canonical sum of squares of rational functions times explicit multipliers let n 1 be a flxed integer due to the fact that for n 1 not every nonnegative polynomial in r n can be written as a sum of squares of polynomials see for instance 2 x6 3 the moment problems in n variables are more di cult than the classical one variable problems this very intriguing territory has been investigated by many authors see 2 7 12 and their references although characterizations for measures whose support lies in an arbitrary generally unbounded semi algebraic set do not seem to exist the present paper starts from an idea of the second author see 19 about solving moment problems by a change of basis via an embedding of r n into a submanifold of a higher dimensional euclidean space rougly speaking we prove that certain n 1 dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover these extensions parametrize all possible solutions of the moment problem to be more speciflc let fi z r n x fi d x fi 2 z n

time integration schemes for the unsteady incompressible navier stokes equations several a stable integration schemes for the solution of the unsteady navier stokes equations for an incompressible fluid are investigated the numerical procedure follows the method of lines which leads to the solution of a non linear system of ordinary differential equations after the discretization of the spatial derivatives the non linear system is solved iteratively using the artificial compressibility method combined with an implicit upwind scheme thorough numerical studies are performed to evaluate the characteristics of the integration schemes the tests are conducted on two flow problems representative of transient and periodic motions

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

single node second order boundary schemes for the lattice boltzmann method abstract based on our recently developed maxwell iteration for the lattice boltzmann method we propose a class of single node boundary schemes for dirichlet boundary conditions of the navier stokes equations the schemes all have second order accuracy for both straight and curved boundaries the accuracy and stability of two specific schemes are examined through several numerical experiments the results validate the second order accuracy and show that a boundary scheme with a convex combination of distribution functions has better stability

solution of the truncated moment problem with variety y x 3 we show that positivity consistency and the variety condition which are sufficient to solve the truncated moment problem on planar curves of degree 2 are not sufficient for curves of higher degree using new partly algorithmic conditions based on positive moment matrix extensions we present a concrete solution to the truncated moment problem on the curve y x 3 we also use moment matrix extensions to solve in a less concrete sense truncated moment problems on curves of the form y g x and yg x 1 g x leading to degree bounded weighted sum of squares representations for polynomials which are strictly positive on such curves

enablers for robust pod models this paper focuses on improving the stability as well as the approximation properties of reduced order models rom based on proper orthogonal decomposition pod the rom is obtained by seeking a solution belonging to the pod subspace and that at the same time minimizes the navier stokes residuals we propose a modified rom that directly incorporates the pressure term in the model the rom is then stabilized making use of a method based on the fine scale equations an improvement of the pod solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations the methods proposed are tested on the two dimensional confined square cylinder wake flow in laminar regime

on the moment problem of closed semi algebraic sets the moment problem for compact semi algebraic sets has been solved in s1 see pd section 6 4 for a refinement of the result in the terminology explained below this means that each defining sequence f of a compact semialgebraic set kf has property smp on the other hand for many noncompact semi algebraic sets for instance sets containing a cone of dimension two km ps the moment problem is not solvable only very few noncompact semi algebraic sets classes of real algebraic curves so km ps and cylinder sets mc are known to have a positive solution of the moment problem in this paper we study semi algebraic setskf such that there exist polynomials h1 hn which are bounded on the set kf our main result theorem 1 reduces the moment problem for the set kf to the moment problem for the fiber sets kf c where c is real algebraic variety x r h1 x 1 hn x n from this theorem new classes of non compact closed semi algebraic sets are obtained for which the moment problem has an affirmative solution combined with a result of v powers and c scheiderer ps it follows that tube sets around certain real algebraic curves have property smp see theorem 9 let f f1 fk be a finite set of polynomials fj r x r x1 xd and let k kf be the associated closed semi algebraic subset defined by

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

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 nonlinear multigrid method for the three dimensional incompressible navier stokes equations a nonlinear multigrid method is developed for solving the three dimensional navier stokes equations in conjunction with the artificial compressibility formulation the method is based on the full multigrid fmg full approximation storage fas algorithm and is realized via an unsteady type procedure according to which the equations are not solved exactly on the coarsest grid but some pseudo time iterations are performed on the finer grids and some on the coarsest grid the multigrid method is implemented in conjunction with a third order upwind characteristics based scheme for the discretization of the convection terms and the fourth order runge kutta scheme for time integration the performance of the method is investigated for three dimensional flows in straight and curved channels as well as flow in a cubic cavity the multigrid acceleration is assessed in contrast to the single grid and mesh sequencing algorithms the effects of various multigrid components on the convergence acceleration such as prolongation operators as well as pre and postrelaxation iterations are also investigated

parallel multigrid computation of unsteady incompressible viscous flows using a matrix free implicit method and high resolution characteristics based scheme abstract a three dimensional parallel unstructured non nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated the finite volume navier stokes solver is based on the artificial compressibility approach with a high resolution method of characteristics based scheme for handling convection terms the unsteady flow is calculated with a matrix free implicit dual time stepping scheme the parallelization of the multigrid solver is achieved by multigrid domain decomposition approach mg dd using single program multiple data spmd and multiple instruction multiple data mimd programming paradigm there are two parallelization strategies proposed in this work first strategy is a one level parallelization strategy using geometric domain decomposition technique alone second strategy is a two level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques message passing interface mpi and openmp standard are used to communicate data between processors and decompose loop iterations arrays respectively the parallel multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid driven cavity flow a maximum speedup of 22 5 could be achieved on 32 processors for instance the lid driven cavity flow of re 1000 the results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements a detailed study of the time step size and number of pseudo sub iterations per time step required for simulating unsteady flow are presented in this paper

the spatial operator in the incompressible navier stokes oseen and stokes equations abstract we investigate the spatial operator in the incompressible navier stokes oseen and stokes equations and show how to avoid singularities associated with null spaces by choosing specific boundary conditions the theoretical results are derived for a general form of energy stable boundary conditions and applied to a few commonly used ones the analysis is done on a system that simultaneously covers the nonlinear incompressible navier stokes the oseen and the stokes equations when the spectrum of the spatial operator is investigated we restrict the analysis to the oseen and stokes equations the continuous analysis carries over to the discrete setting by using the summation by parts framework

a duality proof of tchakaloff s theorem abstract tchakaloff s theorem establishes the existence of a quadrature rule of prescribed degree relative to a positive compactly supported measure that is absolutely continuous with respect to lebesgue measure on r d subsequent extensions were obtained by mysovskikh and by putinar we provide new proofs and partial extensions of these results based on duality techniques utilized by stochel we also obtain new uniqueness criteria in the truncated complex moment problem

an analogue of the riesz haviland theorem for the truncated moment problem abstract let 2 n i i 2 n denote a d dimensional real multisequence let k denote a closed subset of r d and let p 2 n p r x 1 x d deg p 2 n corresponding to the riesz functional l l p 2 n r is defined by l a i x i a i i we say that l is k positive if whenever p p 2 n and p k 0 then l p 0 we prove that admits a k representing measure if and only if l admits a k positive linear extension l p 2 n 2 r this provides a generalization from the full moment problem to the truncated moment problem of the riesz haviland theorem we also show that a semialgebraic set solves the truncated moment problem in terms of natural degree bounded positivity conditions if and only if each polynomial strictly positive on that set admits a degree bounded weighted sum of squares representation

the standard upwind compact difference schemes for incompressible flow simulations compact difference schemes have been used extensively for solving the incompressible navier stokes equations however the earlier formulations of the schemes are of central type called central compact schemes ccs which are dispersive and susceptible to numerical instability to enhance stability of ccs the optimal upwind compact schemes oucs are developed recently by adding high order dissipative terms to ccs in this paper it is found that oucs are essentially not of the upwind type because they do not use upwind biased but central type of stencils furthermore oucs are not the most optimal since orders of accuracy of oucs are at least one order lower than the maximum achievable orders new upwind compact schemes called standard upwind compact schemes sucs are developed in this paper in contrast to oucs sucs are constructed based completely on upwind biased stencils and hence can gain adequate numerical dissipation with no need for introducing optimization calculations furthermore sucs can achieve the maximum achievable orders of accuracy and hence be more compact than oucs more importantly sucs have prominent advantages on combining the stable and high resolution properties which are demonstrated from the global spectral analyses and typical numerical experiments