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

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

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

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

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

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

propagation separation approach for local likelihood estimation the paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models including e g the regression density poisson and binary response model the method extends the adaptive weights smoothing aws procedure introduced in polzehl and spokoiny 2000 in context of image denoising the main idea of the method is to describe a greatest possible local neighborhood of every design point x i in which the local parametric assumption is justified by the data the method is especially powerful for model functions having large homogeneous regions and sharp discontinuities the performance of the proposed procedure is illustrated by numerical examples for density estimation and classification we also establish some remarkable theoretical nonasymptotic results on properties of the new algorithm this includes the propagation property which particularly yields the root n consistency of the resulting estimate in the homogeneous case we also state an oracle result which implies rate optimality of the estimate under usual smoothness conditions and a separation result which explains the sensitivity of the method to structural changes

partitioned time stepping for a parabolic two domain problem there have been many numerical simulations but few analytical results of stability and accuracy of algorithms for computational modeling of fluid fluid and fluid structure interaction problems where two domains corresponding to different fluids ocean atmosphere or a fluid and deformable solid blood flow are separated by an interface as a simplified model of the first examples this report considers two heat equations in omega 1 omega 2 subset mathbb r 2 adjoined by an interface i omega 1 cap omega 2 subset mathbb r the heat equations are coupled by a condition that allows energy to pass back and forth across the interface i while preserving the total global energy of the monolithic coupled problem to compute approximate solutions to the above problem only using subdomain solvers two first order in time fully discrete methods are presented the methods consist of an implicit explicit approach in which the action across i is lagged and a partitioned method based on passing interface values back and forth across i stability and convergence results are derived for both schemes numerical experiments that support the theoretical results are presented

the immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow in the immersed interface method a boundary immersed in a fluid is represented as a singular force in the navier stokes equations this paper presents an explicit approach for computing the singular force to enforce prescribed motion of a rigid boundary in an incompressible viscous flow the tangential component of the singular force is related to the surface vorticity and is calculated from the normal derivative of the velocity the normal component of the singular force is determined from a predictor and a corrector the predictor uses the normal derivative of the vorticity the corrector superposes a homogeneous solution to the pressure poisson equation to achieve the desired normal derivative of the pressure in the current immersed interface method the velocity and the pressure are solved using the mac scheme with the incorporation of jump conditions induced by the singular force and a discontinuous finite body force the body force is applied to obtain the rigid motion of the fluid enclosed by the boundary circular couette flow flow past a cylinder and flow around flappers are simulated to test the accuracy stability and efficiency of the method as well as the effect of the corrector with no stiff springs to model rigid boundaries the method is stable at relatively high reynolds numbers

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

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

an hybrid finite volume finite element method for variable density incompressible flows this paper is devoted to the numerical simulation of variable density incompressible flows modeled by the navier stokes system we introduce an hybrid scheme which combines a finite volume approach for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint the breakthrough relies on the definition of a suitable footbridge between the two methods through the design of compatibility condition in turn the method is very flexible and allows to deal with unstructured meshes several numerical tests are performed to show the scheme capabilities in particular the viscous rayleigh taylor instability evolution is carefully investigated

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

an efficient method for the incompressible navier stokes equations on irregular domains with no slip boundary conditions high order up to the boundary common efficient schemes for the incompressible navier stokes equations such as projection or fractional step methods have limited temporal accuracy as a result of matrix splitting errors or introduce errors near the domain boundaries which destroy uniform convergence to the solution in this paper we recast the incompressible constant density navier stokes equations with the velocity prescribed at the boundary as an equivalent system for the primary variables velocity and pressure equation for the pressure the key difference from the usual approaches occurs at the boundaries where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time this avoids the common difficulty of an apparently over determined poisson problem since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity the recast equations are ideal for numerical marching methods the new system can be discretized using a variety of methods including semi implicit treatments of viscosity and in principle to any desired order of accuracy in this work we illustrate the approach with a 2 d second order finite difference scheme on a cartesian grid and devise an algorithm to solve the equations on domains with curved non conforming boundaries including a case with a non trivial topology a circular obstruction inside the domain this algorithm achieves second order accuracy in the l norm for both the velocity and the pressure the scheme has a natural extension to 3 d

multiscale adaptive regression models for neuroimaging data neuroimaging studies aim to analyze imaging data with complex spatial patterns in a large number of locations called voxels on a two dimensional 2d surface or in a 3d volume conventional analyses of imaging data include two sequential steps spatially smoothing imaging data and then independently fitting a statistical model at each voxel however conventional analyses suffer from the same amount of smoothing throughout the whole image the arbitrary choice of smoothing extent and low statistical power in detecting spatial patterns we propose a multiscale adaptive regression model marm to integrate the propagation separation ps approach polzehl and spokoiny 2000 2006 with statistical modeling at each voxel for spatial and adaptive analysis of neuroimaging data from multiple subjects marm has three features being spatial being hierarchical and being adaptive we use a multiscale adaptive estimation and testing procedure maet to utilize imaging observations from the neighboring voxels of the current voxel to adaptively calculate parameter estimates and test statistics theoretically we establish consistency and asymptotic normality of the adaptive parameter estimates and the asymptotic distribution of the adaptive test statistics our simulation studies and real data analysis confirm that marm significantly outperforms conventional analyses of imaging data

a framework for uncertainty and validation of 3 d registrationmethods based on points and frames in this paper we propose and analyze several methods to estimate a rigid transformation from a set of 3 d matched points or matched frames which are important features in geometric algorithms we also develop tools to predict and verify the accuracy of these estimations the theoretical contributions are an intrinsic model of noise for transformations based on composition rather than addition a unified formalism for the estimation of both the rigid transformation and its covariance matrix for points or frames correspondences and a statistical validation method to verify the error estimation which applies even when no ground truth is available we analyze and demonstrate on synthetic data that our scheme is well behaved the practical contribution of the paper is the validation of our transformation estimation method in the case of 3 d medical images which shows that an accuracy of the registration far below the size of a voxel can be achieved and in the case of protein substructure matching where frame features drastically improve both selectivity and complexity

a two level variational multiscale method for convection dominated convection diffusion equations this paper studies the error in the efficient implementation of and time stepping methods for a variational multiscale method vms for solving convection dominated problems the vms studied uses a fine mesh c0 finite element space xh to approximate the concentration and a coarse mesh discontinuous vector finite element space lh for the large scales of the flux in the two scale discretization our tests show that these choices lead to an efficient vms whose complexity is further reduced if a locally l2 orthogonal basis for lh is used a fully implicit and a semi implicit treatment of the terms which link effects across scales are tested and compared the semi implicit vms was much more efficient the observed global accuracy of the most straightforward vms implementation was much better than the artificial diffusion stabilization and comparable to a streamline diffusion finite element method in our tests

essentially compact schemes for unsteady viscous incompressible flows a new fourth order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced the scheme is based on the vorticity stream function formulation it is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions it is also very efficient at every time step or runge kutta stage only two poisson like equations have to be solved the poisson like equations are amenable to standard fast poisson solvers usually designed for second order schemes detailed comparison with the second order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow this efficient fourth order scheme also made it possible to compute the driven cavity flow at reynolds number 106on a 10242grid at a reasonable cost fourth order convergence is proved under mild regularity requirements this is the first such result to our knowledge

a split step finite element method for incompressible navier stokes equations with high order accuracy up to the boundary abstract an efficient and accurate finite element algorithm is described for the numerical solution of the incompressible navier stokes ins equations the new algorithm that solves the ins equations in a velocity pressure reformulation is based on a split step scheme in conjunction with the standard finite element method the split step scheme employed for the temporal discretization of our algorithm completely separates the pressure updates from the solution of velocity variables when the pressure equation is formed explicitly the algorithm avoids solving a saddle point problem therefore our algorithm has more flexibility in choosing finite element spaces in contrast popular mixed finite element methods that solve the ins equations in the primitive variables or velocity divergence formulation lead to discrete saddle point problems whose solution depends on the choice of finite element spaces for velocity and pressure that is subject to the well known ladyzenskaja babuska brezzi lbb or inf sup condition for efficiency and robustness lagrange piecewise polynomial finite elements of equal order for both velocity and pressure are used accurate numerical boundary condition for the pressure equation is also investigated motivated by a post processing technique that calculates derivatives of a finite element solution with super convergent error estimates an alternative numerical boundary condition is proposed for the pressure equation at the discrete level the new numerical pressure boundary condition that can be regarded as a better implementation of the compatibility boundary condition improves the boundary layer errors of the pressure solution a normal mode analysis is performed using a simplified model problem on a uniform mesh to demonstrate the numerical properties of our methods convergence studies using p 1 elements support the analytical results and demonstrates that our algorithm with the new numerical boundary condition achieves the optimal second order accuracy for both velocity and pressure up to the boundary benchmark problems are also computed and carefully compared with existing studies finally as an example to illustrate that our approach can be easily adapted for higher order finite elements we solve the classical flow past a cylinder problem using p n finite elements with n 1

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

clinical dt mri estimation smoothing and fiber tracking with log euclidean metrics diffusion tensor magnetic resonance imaging dt mri or dti is an imaging modality that is gaining importance in clinical applications however in a clinical environment data have to be acquired rapidly often at the expense of the image quality this often results in dti datasets that are not suitable for complex postprocessing like fiber tracking we propose a new variational framework to improve the estimation of dt mri in this clinical context most of the existing estimation methods rely on a log gaussian noise gaussian noise on the image logarithms or a gaussian noise that do not reflect the rician nature of the noise in mr images with a low signal to noise ratio snr with these methods the rician noise induces a shrinking effect the tensor volume is underestimated when other noise models are used for the estimation in this paper we propose a maximum likelihood strategy that fully exploits the assumption of a rician noise to further reduce the influence of the noise we optimally exploit the spatial correlation by coupling the estimation with an anisotropic prior previously proposed on the spatial regularity of the tensor field itself which results in a maximum a posteriori estimation optimizing such a nonlinear criterion requires adapted tools for tensor computing we show that riemannian metrics for tensors and more specifically the log euclidean metrics are a good candidate and that this criterion can be efficiently optimized experiments on synthetic data show that our method correctly handles the shrinking effect even with very low snr and that the positive definiteness of tensors is always ensured results on real clinical data demonstrate the truthfulness of the proposed approach and show promising improvements of fiber tracking in the brain and the spinal cord

decoupled energy stable schemes for phase field vesicle membrane model we consider the numerical approximations of the classical phase field vesicle membrane models proposed a decade ago in du et al 2004 6 we first reformulate the model derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish the energy dissipation law then we develop a stabilized decoupled time discretization scheme for the coupled nonlinear system the scheme is unconditionally energy stable and leads to linear and decoupled elliptic equations to be solved at each time step stability analysis and ample numerical simulations are presented thereafter

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

an efficient numerical method for the equations of steady and unsteady flows of homogeneous incompressible newtonian fluid in the present paper we present some numerical methods to solve the equations of steady and unsteady flows such as those in the microcirculatory bed and large blood vessels arteries and veins respectively in the case of steady flows the method does not need neither any boundary conditions on pressure nor any small parameter and the main computation consists of solving some poisson equations in the case of unsteady flows the scheme uses a consistent neumann boundary condition for the pressure poisson equation at each time step a poisson and heat equation are solved for the pressure and each velocity component respectively the accuracy and efficiency of scheme are checked by a set of numerical tests

some current cfd issues relevant to the incompressible navier stokes equations abstract the goals of this paper are to carefully define a particular class of well set incompressible navier stokes problems in the continuum partial differential equation pde setting and to discuss some relevant and sometimes poorly understood issues related to these well posed pde problems both in the continuum world and in its computer counterpart emphasized are three common formulations used to generate computer codes the primitive variable velocity pressure formulation and two derived variable formulations pressure poisson equation and vorticity transport equation

complex ginzburg landau equation with nonlocal coupling a ginzburg landau type equation with nonlocal coupling is derived systematically as a reduced form of a universal class of reaction diffusion systems near the hopf bifurcation point and in the presence of another small parameter the reaction diffusion systems to be reduced are such that the chemical components constituting local oscillators are nondiffusive or hardly diffusive so that the oscillators are almost uncoupled while there is an extra diffusive component which introduces effective nonlocal coupling over the oscillators linear stability analysis of the reduced equation about the uniform oscillation is also carried out this revealed that new types of instability which can never arise in the ordinary complex ginzburg landau equation are possible and their physical implication is briefly discussed

stability and convergence of efficient navier stokes solvers via a commutator estimate for strong solutions of the incompressible navier stokes equations in bounded domains with velocity specified at the boundary we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure these schemes require no solution of stationary stokes systems nor any compatibility between velocity and pressure spaces to ensure an inf sup condition and are representative of a class of highly efficient computational methods that have recently emerged the proofs are simple based upon a new sharp estimate for the commutator of the laplacian and helmholtz projection operators this allows us to treat an unconstrained formulation of the navier stokes equations as a perturbed diffusion equation c 2006 wiley periodicals inc

a completely explicit scheme of cauchy problem in bslm for solving the navier stokes equations abstract this paper presents a backward semi lagrangian method bslm with third order convergence in both time and space for solving incompressible navier stokes equations the third order backward differentiation formula for the total time derivative and the projection method for the steady state stokes equation are used a fourth order difference scheme together with a local bi cubic interpolation is used to solve the resulted two governing equations for the velocity and pressure this paper mainly focuses on the development of an efficient scheme for solving the nonlinear cauchy problem of the characteristic curve we employ a modified linear multi step method of the implicit type based on the error correction strategy a novel contribution of this paper is the design of a completely explicit formula for the three foot points the proposed method is superior to existing methods in terms of computational costs and accuracy allowing the use of a large time step size the fully explicit formula for the foot points significantly improves the performance of a common bslm for a wide range of practical applications

spatial shrinkage estimation of diffusion tensors on diffusion weighted imaging data diffusion tensor imaging dti based on the diffusion weighted imaging dwi data acquired from magnetic resonance experiments has been widely used to analyze the physical structure of white matter fibers in the human brain in vivo the raw dwi data however carry noise this contaminates the diffusion tensor dt estimates and introduces systematic bias into the induced eigenvalues these bias components affect the effectiveness of fiber tracking algorithms in this article we propose a two stage spatial shrinkage estimation spske procedure to accommodate the spatial information carried in dwi data in dt estimation and to reduce the bias components in the corresponding derived eigenvalues to this end in the framework of the heteroscedastic linear model spske incorporates l 1 type penalization and the locally weighted least square function the theoretical properties of spske are explored the effectiveness of spske is further illustrated by simulation and real data examples supplementary materi

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 gradient stable scheme for a phase field model for the moving contact line problem in this paper an efficient numerical scheme is designed for a phase field model for the moving contact line problem which consists of a coupled system of the cahn hilliard and navier stokes equations with the generalized navier boundary condition 1 2 4 the nonlinear version of the scheme is semi implicit in time and is based on a convex splitting of the cahn hilliard free energy including the boundary energy together with a projection method for the navier stokes equations we show under certain conditions the scheme has the total energy decaying property and is unconditionally stable the linearized scheme is easy to implement and introduces only mild cfl time constraint numerical tests are carried out to verify the accuracy and stability of the scheme the behavior of the solution near the contact line is examined it is verified that when the interface intersects with the boundary the consistent splitting scheme 21 22 for the navier stokes equations has the better accuracy for pressure

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

the euler implicit explicit scheme for the 2d time dependent navier stokes equations with smooth or non smooth initial data this paper considers the stability and convergence results for the euler implicit explicit scheme applied to the spatially discretized two dimensional 2d time dependent navier stokes equations a galerkin finite element spatial discretization is assumed and the temporal treatment is implicit explict scheme which is implicit for the linear terms and explicit for the nonlinear term here the stability condition depends on the smoothness of the initial data u 0 h i e the time step condition is c 0 in the case of a 2 log h c 0 in the case of a 1 and th 2 c 0 in the case of a 0 for mesh size h and some positive constant c 0 we provide the h 2 stability of the scheme under the stability condition with a 0 1 2 and obtain the optimal h 1 l 2 error estimate of the numerical velocity and the optimal l 2 error estimate of the numerical pressure under the stability condition with a 1 2

phase equation with nonlinear excitation for nonlocally coupled oscillators abstract some reaction diffusion systems feature nonlocal interaction and near the point of hopf bifurcation can be represented as a system of nonlocally coupled oscillators phase of oscillations satisfies an evolution pde which takes different forms depending on the values of parameters in the simplest case the equation is effectively a diffusion equation which is excitation free however more complex forms are possible such as the nikolaevskii equation and the kuramoto sivashinsky equation incorporating linear excitation we analyse a situation when the phase equation is based on nonlinear excitation we derive conditions on the values of the parameters leading to the situation and show that the values satisfying the conditions exist

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

convergence analysis for a higher order scheme for the time dependent navier stokes equations abstract in this paper we discuss a third order fully discrete scheme for the time dependent navier stokes equations we prove that the third order temporal discrete scheme proposed by baker et al in 1 is convergent under a weaker condition some numerical experiments are shown to confirm the efficiency of the higher order scheme

an immersed interface method for simulating the interaction of a fluid with moving boundaries in the immersed interface method boundaries are represented as singular force in the navier stokes equations which enters a numerical scheme as jump conditions recently we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3d with second order spatial and temporal accuracy near the boundaries sheng xu z jane wang systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation siam j sci comput 2006 in press in this paper we implement the immersed interface method to incorporate these jump conditions in a 2d numerical scheme we study the accuracy efficiency and robustness of our method by simulating taylor couette flow flow induced by a relaxing balloon flow past single and multiple cylinders and flow around a flapping wing our results show that 1 our code has second order accuracy in the infinity norm for both the velocity and the pressure 2 the addition of an object introduces relatively insignificant computational cost 3 the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

a convergence study of a new partitioned fluid structure interaction algorithm based on fictitious mass and damping we develop analyze and validate a new method for simulating fluid structure interactions fsis which is based on fictitious mass and fictitious damping in the structure equation we employ a partitioned method for the fluid and structure motions in conjunction with sub iteration and aitken relaxation in particular the use of such fictitious parameters requires sub iterations in order to reduce the induced error in addition to the local temporal truncation error to this end proper levels of tolerance for terminating the sub iteration procedure have been obtained in order to recover the formal order of temporal accuracy for the coupled fsi problem these fictitious terms have a significant effect leading to better convergence rate and hence substantially smaller number of sub iterations through analysis we identify the proper range of these parameters which we then verify by corresponding numerical tests we implement the method in the context of spectral element discretization which is more sensitive than low order methods to numerical instabilities arising in the explicit fsi coupling however the method we present here is simple and general and hence applicable to fsi based on any other discretization we demonstrate the effectiveness of the method in applications involving 2d vortex induced vibrations viv and in 3d flexible arteries with structural density close to blood density we also present 3d results for a patient specific aneurysmal flow under pulsatile flow conditions examining in particular the sensitivity of the results on different values of the fictitious parameters

a 3d immersed interface method for fluid solid interaction abstract in immersed interface methods solids in a fluid are represented by singular forces in the navier stokes equations and flow jump conditions induced by the singular forces directly enter into numerical schemes this paper focuses on the implementation of an immersed interface method for simulating fluid solid interaction in 3d the method employs the mac scheme for the spatial discretization the rk4 scheme for the time integration and an fft based poisson solver for the pressure poisson equation a fluid solid interface is tracked by lagrangian markers intersections of the interface with mac grid lines identify finite difference stencils on which jump contributions to finite difference schemes are needed to find the intersections and to interpolate jump conditions from the lagrangian markers to the intersections parametric triangulation of the interface is used the velocity of the lagrangian markers is interpolated directly from surrounding mac grid nodes with interpolation schemes accounting for jump conditions numerical examples demonstrate that 1 the method has near second order accuracy in the infinity norm for velocity and the accuracy for pressure is between first and second order 2 the method conserves the volume enclosed by a no penetration boundary and 3 the method can efficiently handle multiple moving solids with ease

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

test of significance based on wavelet thresholding and neyman s truncation abstract traditional nonparametric tests such as the kolmogorov smirnov test and the cramer von mises test are based on the empirical distribution functions although these tests possess root n consistency they effectively use only information contained in the low frequencies this leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high frequency alternations the drawback can be repaired via smoothing based test statistics in this article we propose two such kind of test statistics based on the wavelet thresholding and the neyman truncation we provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations while maintaining the same capability in detecting smooth alternative densities as the traditional tests similar conclusions can be made for two sample nonparametric tests of distribution functions in that case the traditional linear rank tests such as th

accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term we present numerical schemes for the incompressible navier stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure poisson equation the pressure is treated explicitly in time completely decoupling the computation of the momentum and kinematic equations the result is a class of extremely efficient navier stokes solvers full time accuracy is achieved for all flow variables the key to the schemes is a neumann boundary condition for the pressure poisson equation which enforces the incompressibility condition for the velocity field irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint indeed we prove unconditional stability of the new formulation for the stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term systematic numerical experiments for the full navier stokes equations indicate that a second order implicit time discretization of the viscous term with the pressure and convective terms treated explicitly is stable under the standard cfl condition additionally various numerical examples are presented including both implicit and explicit time discretizations using spectral and finite difference spatial discretizations demonstrating the accuracy flexibility and efficiency of this class of schemes in particular a galerkin formulation is presented requiring only c0 elements to implement

an iteration free backward semi lagrangian scheme for solving incompressible navier stokes equations a backward semi lagrangian method based on the error correction method is designed to solve incompressible navier stokes equations the time derivative of the stokes equation is discretized with the second order backward differentiation formula for the induced steady stokes equation a projection method is used to split it into velocity and pressure fourth order finite differences for partial derivatives are used to the boundary value problems for the velocity and the pressure also finite linear systems for poisson equations and helmholtz equations are solved with a matrix diagonalization technique for characteristic curves satisfying highly nonlinear self consistent initial value problems the departure points are solved with an error correction strategy having a temporal convergence of order two the constructed algorithm turns out to be completely iteration free in particular the suggested algorithm possesses a good behavior of the total energy conservation compared to existing methods to assess the effectiveness of the method two dimensional lid driven cavity problems with large different reynolds numbers are solved the doubly periodic shear layer flows are also used to assess the efficiency of the algorithm

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

finite difference schemes for incompressible flow based on local pressure boundary conditions in this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible navier stokes equations in the velocity pressure formulation their use is especially well suited for the computation of moderate to large reynolds number flows we explore the similarities between the implementation and use of local pressure boundary conditions and local vorticity boundary conditions in the design of numerical schemes for incompressible flow in 2d in their respective formulations when these local numerical boundary conditions are coupled with a fully explicit convectively stable time stepping procedure the resulting methods are simple to implement and highly efficient unlike the vorticity formulation the use of the local pressure boundary condition approach is readily applicable to 3d flows the simplicity of the local pressure boundary condition approach and its easy adaptation to more general flow settings make the resulting scheme an attractive alternative to the more popular methods for solving the navier stokes equations in the velocity pressure formulation we present numerical results of a second order finite difference scheme on a nonstaggered grid using local pressure boundary conditions stability and accuracy of the scheme applied to stokes flow is demonstrated using normal mode analysis also described is the extension of the method to variable density flows

pod galerkin reduced order methods for combined navier stokes transport equations based on a hybrid fv fe solver abstract the purpose of this work is to introduce a novel pod galerkin strategy for the semi implicit hybrid high order finite volume finite element solver introduced in bermudez et al 2014 and busto et al 2018 the interest is into the incompressible navier stokes equations coupled with an additional transport equation the full order model employed in this article makes use of staggered meshes this feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes the reduced order model presented herein accounts for velocity pressure and a transport related variable the pressure term at both the full order and the reduced order level is reconstructed making use of a projection method more precisely a poisson equation for pressure is considered within the reduced order model results are verified against three dimensional manufactured test cases moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed

an implicit technique for solving 3d low reynolds number moving free surface flows this paper describes the development of an implicit finite difference method for solving transient three dimensional incompressible free surface flows to reduce the cpu time of explicit low reynolds number calculations we have combined a projection method with an implicit technique for treating the pressure on the free surface the projection method is employed to uncouple the velocity and the pressure fields allowing each variable to be solved separately we employ the normal stress condition on the free surface to derive an implicit technique for calculating the pressure at the free surface numerical results demonstrate that this modification is essential for the construction of methods that are more stable than those provided by discretizing the free surface explicitly in addition we show that the proposed method can be applied to viscoelastic fluids numerical results include the simulation of jet buckling and extrudate swell for reynolds numbers in the range 0 01 0 5

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

probability convexity and harmonic maps with small image i uniqueness and fine existence this paper uses probability theory to provide an approach to the dirichlet problem for harmonic maps the probabilistic tools used are those of manifold valued brownian motion and martingales probabilistic proofs are given of uniqueness continuous dependence on data and existence of generalized solutions finely harmonic maps for target domains which have convex geometry this class of domain includes all regular geodesic balls links are made with new results in other fields the martingale dirichlet problem riemannian centres of mass

spatially varying coefficient model for neuroimaging data with jump discontinuities motivated by recent work on studying massive imaging data in various neuroimaging studies we propose a novel spatially varying coefficient model svcm to capture the varying association between imaging measures in a three dimensional volume or two dimensional surface with a set of covariates two stylized features of neuorimaging data are the presence of multiple piecewise smooth regions with unknown edges and jumps and substantial spatial correlations to specifically account for these two features svcm includes a measurement model with multiple varying coefficient functions a jumping surface model for each varying coefficient function and a functional principal component model we develop a three stage estimation procedure to simultaneously estimate the varying coefficient functions and the spatial correlations the estimation procedure includes a fast multiscale adaptive estimation and testing procedure to independently estimate each varying coefficient function while preserving its edges among

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

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

diffusion tensor imaging structural adaptive smoothing abstract diffusion tensor imaging dti data is characterized by a high noise level thus estimation errors of quantities like anisotropy indices or the main diffusion direction used for fiber tracking are relatively large and may significantly confound the accuracy of dti in clinical or neuroscience applications besides pulse sequence optimization noise reduction by smoothing the data can be pursued as a complementary approach to increase the accuracy of dti here we suggest an anisotropic structural adaptive smoothing procedure which is based on the propagation separation method and preserves the structures seen in dti and their different sizes and shapes it is applied to artificial phantom data and a brain scan we show that this method significantly improves the quality of the estimate of the diffusion tensor by means of both bias and variance reduction and hence enables one either to reduce the number of scans or to enhance the input for subsequent analysis such as fiber tracking

chemical turbulence equivalent to nikolavskii turbulence we find evidence that a certain class of reaction diffusion rd systems can exhibit chemical turbulence equivalent to nikolaevskii turbulence we study an extended complex ginzburg landau cgl equation derived from this class of rd systems first we show numerically that the power spectrum of this cgl equation in the neighborhood of a codimension two turing char21 benjamin feir point is qualitatively quite similar to that of the nikolaevskii equation then we demonstrate that the nikolaevskii equation can in fact be obtained from this cgl equation through a phase reduction procedure

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

numerical solution of the incompressible navier stokes equations in primitive variables and velocity vorticity formulation a finite difference method is presented for the numerical solution of the navier stokes equations of motion of a viscous incompressible fluid in two dimensions in primitive variables and velocity vorticity formulation for the case of primitive variables using a staggered grid and introducing an auxiliary function of the coordinate system and considering the form of the initial equation on lines passing through the nodal point x 0 y 0 and parallel to the coordinate axes we can separate it into two parts that are finally reduced to ordinary linear differential equations one for each dimension discretization of these equations leads to a system of linear equations in n unknowns which is solved by an iterative technique and the method converges rapidly giving satisfactory results for the pressure variable we consider a pressure poisson equation with suitable neumann type boundary conditions in case of velocity vorticity formulation similar procedure is used for a collocated grid numerical results up to reynolds number 5000 confirming the accuracy of the proposed method are presented for configurations of interest such as poiseuille flow lid driven cavity flow in square domain and flow in a backward facing step in the presence of a pressure gradient

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105