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

exact projections and the lagrange galerkin method a realistic alternative to quadrature abstract the lagrange galerkin method has proved a very successful method in computational fluid dynamics cfd in practice but it has suffered from question marks about its stability when the integrals are approximated by quadratures in this paper we will prove instability for wider ranges of quadrature formulae than previously considered we will also introduce on approximate integration technique on triangular grids that cheaply recovers the unconditional stability of the lagrange galerkin method with no loss in accuracy the same technique can also be used for the exact projection of data from one arbitrary triangular grid to another different arbitrary triangular grid this could be used in grid adaptation algorithms or in triangular multigrid procedures

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

an efficient scheme for solving steady incompressible navier stokes equations abstract the steady incompressible navier stokes equations in a 2d driven cavity are solved in primitive variables by means of the multigrid method the pressure and the components of the velocity are discretized on staggered grids a block implicit relaxation technique is used to achieve a good convergence and a simplified fmg fas algorithm is proposed special focus on the finite differences scheme used to approach the convection terms is made and a large discussion with other schemes is given results in a square driven cavity are obtained for reynolds numbers as high as 15 000 on fine uniform meshes and the solution is in good agreement with other studies for re 5000 the secondary vortices are very well represented showing the robustness of the method for reynolds numbers higher than 5000 the loss of stability for the steady solution is discussed moreover some computations on a rectangular cavity of aspect ratio equal to two are presented in addition the method is very efficient as far as cpu time is concerned for instance the solution for re 1000 on a 128 128 grid is obtained within 24 s on a siemens vp 200

a fractional step solution method for the unsteady incompressible navier stokes equations in generalized coordinate systems abstract a fractional step method is developed for solving the time dependent three dimensional incompressible navier stokes equations in generalized coordinate systems the primitive variable formulation uses the pressure defined at the center of the computational cell and the volume fluxes across the faces of the cells as the dependent variables instead of the cartesian components of the velocity this choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell the governing equations are discretized by finite volumes using a staggered mesh system the solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a poisson equation this procedure combined with a consistent approximation of the geometric quantities is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain the favorable convergence properties of the poisson solver the momentum equations are solved by an approximate factorization method and a novel zebra scheme with four color ordering is devised for the efficient solution of the poisson equation several two and three dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method good agreement is obtained in all cases

one step l α stable temporal integration for the backward semi lagrangian scheme and its application in guiding center problems abstract we present a backward semi lagrangian method with third order temporal accuracy for solving the guiding center problem for solving highly nonlinear characteristic curves we propose an iteration free l stable method equipped with an error correction technique and a classical collocation method as a discretization of the poisson equation we adopt the fourth order centered differential scheme and the shortley weller discretization depending on the boundary of the domain and apply the local cubic interpolation polynomial for the evaluation of the density and the potential at the non grid foot points of the characteristics the novelty of the proposed method is its good stability as well as its outstanding computational cost compared to those of the recently developed high order adams moulton method furthermore based on several numerical tests we conclude that the proposed method exhibits excellent performance for long time stable simulations and allows for large temporal step sizes

a second order accurate projection method for the incompressible navier stokes equations on non graded adaptive grids we present an unconditionally stable second order accurate projection method for the incompressible navier stokes equations on non graded adaptive cartesian grids we employ quadtree and octree data structures as an efficient means to represent the grid we use the supra convergent poisson solver of c h min f gibou h ceniceros a supra convergent finite difference scheme for the variable coefficient poisson equation on fully adaptive grids cam report 05 29 j comput phys in press a second order accurate semi lagrangian method to update the momentum equation an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non graded grids we sample all the variables at the grid nodes producing a scheme that is straightforward to implement we propose two and three dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the l1 and l norms

a sharp interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations a method for reducing the spurious pressure oscillations observed when simulating moving boundary flow problems with sharp interface immersed boundary methods ibms is proposed by first identifying the primary cause of these oscillations to be the violation of the geometric conservation law near the immersed boundary we adopt a cut cell based approach to strictly enforce geometric conservation in order to limit the complexity associated with the cut cell method the cut cell based discretization is limited only to the pressure poisson and velocity correction equations in the fractional step method and the small cell problem tackled by introducing a virtual cell merging technique the method is shown to retain all the desirable properties of the original finite difference based ibm while at the same time reducing pressure oscillations for moving boundaries by roughly an order of magnitude

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

numerical solution of a pde model for a ratchet cap pricing with bgm interest rate dynamics abstract in this paper we present a new numerical method to price an interest rate derivative the financial product consists of a particular ratchet cap contract which contains a set of ratchet caplets for this purpose we first pose the pde pricing model for each ratchet caplet by means of feynman kac theorem the underlying interest rates are the forward libor rates the dynamics of which are assumed to follow the recently introduced bgm lmm market model for the set of pdes associated to the ratchet caplets pricing problems we propose a second order crank nicolson characteristics time discretization scheme combined with a finite element discretization in the interest rate variables in order to illustrate the performance of the numerical methods we present an academic test and a real example of a particular ratchet cap pricing in the second case a comparison between the results obtained by monte carlo simulation and the proposed method is presented

comparison of newton s and quasi newton s method solvers for the navier stokes equations the results of a computational evaluation of several newton s and quasi newton s method solvers are discussed and analyzed computer time and memory requirements for iterating a solution to the steady state are recorded for each method roe s flux difference splitting together with the spekreijse van albada continuous limiter is used for the spatial discretization sparse matrix inversions are performed using a modified version of the boeing real sparse library routines and the conjugate gradient squared algorithm the methods are applied to exact and approximate newton s method jacobian systems for flat plate and flat plate wedge type ge ometries results indicate that the quasi newton s method solvers do not exhibit quadratic convergence but can be more efficient than the exact newton s method in select cases consistent goal of computational fluid dynamics cfd research has been to improve the efficiency of numerical algorithms one measure of this efficiency is the log of the residual which varies linearly with iteration count for typical cfd techniques these methods can be expected to converge at the same rate no matter how close they are to the final solution an exception to this behavior is the quadratic convergence newton s method for which the convergence rate actually increases as the final solution is approached when quadratic convergence has been attained the number of cor rect digits in the solution doubles with each iteration this behavior has been illustrated by several researchers who have obtained solutions of the potential equation 1 the euler equa tions 2 5 and the navier stokes equations 2 4 6 13 unfortu nately the number of iterations is a less important measure of the efficiency of a code than the total computation time quadratic convergence rates are typically accompanied by significant increases in algorithm complexity and hence in creased per iteration computational requirements it was the intent of this research to explore how the conver gence rate and per iteration efficiency vary for similar new ton s and quasi newton s method solvers so that the most efficient computational strategy can be identified to accom plish this goal the quadratic convergence newton s method solver developed by orkwis and mcrae6 8 was modified to create several quasi newton s method solvers approxima tions were made to the form of the jacobian and to the jacobian matrix inversion process the resulting schemes are evaluated based on their robustness and efficiency for two test problems

the low frequency axisymmetric modal sound radiation efficiency of an elastically supported annular plate abstract this paper deals with the sound radiation efficiency of a vibrating thin elastically supported annular plate embedded into a flat rigid baffle the free axisymmetric time harmonic vibrations have been considered for a single mode it has been assumed that the influence of the air column above the plate on the plate s vibrations is negligible first the sound radiation efficiency has been formulated as an integral further rigorous mathematical manipulations have been carried out based on the theory of summation of multiple expansion series containing the hypergeometric functions as a result the formulations have been expressed as some fast convergent expansion series containing only the bessel and struve functions of integer order and the spherical bessel functions the presented formulations of sound radiation efficiency of an elastically supported annular plate are useful for numerical calculations within the low frequency range what is important for practical reasons the formulations are valid for axisymmetric boundary conditions and they enable changing the values of boundary stiffness constants consequently the analysis of influence of the plate s edge attachment on the sound radiation efficiency has been performed the limiting transitions have also been performed from formulations valid for the elastically supported annular plates to the formulations valid for annular plates with classical boundary conditions clamped simply supported and free at one edge or at both edges

newton krylov algorithm with a loosely coupled turbulence model for aerodynamic flows a fast newton krylov algorithm is presented that solves the turbulent navier stokes equations on unstructured 2 d grids the model of spalart and allmaras provides the turbulent viscosity and is loosely coupled to the mean flow equations it is often assumed that the turbulence model must be fully coupled to obtain the full benefit of an inexact newton algorithm we demonstrate that a loosely coupled algorithm is effective and has some advantages such as reduced storage requirements and smoother transient oscillations a transonic single element case converges to 1 1012 in 90 s on recent commodity hardware whereas the lift coefficient is converged to three figures in one quarter of that time

numerical analysis of a second order pure lagrange galerkin method for convection diffusion problems part ii fully discretized scheme and numerical results we analyze a second order pure lagrange galerkin method for variable coefficient convection diffusion possibly degenerate diffusion equations with mixed dirichlet robin boundary conditions in a previous paper the proposed second order pure lagrangian time discretization scheme was introduced and analyzed for the same problem more precisely the l infty h 1 stability and l infty h 1 error estimates of order o delta t 2 had been obtained moreover for the particular case of incompressible flows stability inequalities with constants independent of the final time have been stated in the present paper l infty h 1 error estimates of order o delta t 2 o h k are obtained for the fully discretized pure lagrange galerkin method to prove these results we use some properties obtained in the previous paper finally numerical tests are presented that confirm the theoretical results

newton s method solver for high speed viscous separated flowfields a new method for calculating the 2 d laminar navier stokes equations is presented the method uses newton s method for nonlinear systems of equations to find steady state solutions the navier stokes equations are approximated by finite differences using roe s flux difference splitting second order accuracy is attained by using spekreijse s interpolation with van albada s limiter the exact newton s method jacobian matrix is inverted by using recent sparse matrix routines the symbolic manipulation package macsyma is used to develop and write the fortran code

a second order characteristics finite element scheme for natural convection problems in this paper a second order characteristics finite element scheme is applied to the numerical solution of natural convection problems firstly after recalling the mathematical model a second order time discretization of the material time derivative is introduced next fully discretized schemes are proposed by using finite element methods numerical results for the two dimensional problem of buoyancy driven flow in a square cavity with differentially heated side walls are given and compared with a reference solution

newton s method solver for the axisymmetric navier stokes equations a new two dimensional newton s method navier stokes solver is extended to the axisymmetric form of the equations roe s flux difference splitting is applied to the axisymmetric equations using glaister s approach a geometric conservation law correction term is added to insure satisfaction of the uniform stream reproduction criteria higher order accuracy is attained through the use of spekreijse s interpolation procedure with van albada s contiouous limiter harten and hyman s minimum eigenvalue correction is also included

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

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

a higher order compact finite difference algorithm for solving the incompressible navier stokes equations on the basis of the projection method a higher order compact finite difference algorithm which possesses a good spatial behavior is developed for solving the 2d unsteady incompressible navier stokes equations in primitive variable the present method is established on a staggered grid system and is at least third order accurate in space a third order accurate upwind compact difference approximation is used to discretize the non linear convective terms a fourth order symmetrical compact difference approximation is used to discretize the viscous terms and a fourth order compact difference approximation on a cell centered mesh is used to discretize the first derivatives in the continuity equation the pressure poisson equation is approximated using a fourth order compact difference scheme constructed currently on the nine point 2d stencil new fourth order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine point 2d stencil for the assessment of the effectiveness and accuracy of the method particularly its spatial behavior a problem with analytical solution and another one with a steep gradient are numerically solved finally steady and unsteady solutions for the lid driven cavity flow are also used to assess the efficiency of this algorithm copyright c 2011 john wiley sons ltd

high order time discretization for backward semi lagrangian methods we introduce different high order time discretization schemes for backward semi lagrangian methods these schemes are based on multi step schemes like adams moulton and adams bashforth schemes combined with backward finite difference schemes we apply these methods to transport equations for plasma physics applications and for the numerical simulation of instabilities in fluid mechanics in the context of backward semi lagrangian methods this time discretization strategy is particularly efficient and accurate when the spatial error discretization becomes negligible and allows to use large time steps

studies of numerical methods for the plane navier stokes equations abstract in the present report and in our most recent work we have concentrated on three basic formulations primitive variables vorticity stream function and stream function biharmonic formulation with each such formulation there are a variety of difference equations that could be used and then there are a large number of iteration schemes that could be employed to solve these difference equations the ultimate goal of course is to find the best combination of all these techniques

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

an adaptive level set approach for incompressible two phase flows we present a numerical method using the level set approach for solving incompressible two phase flow with surface tension in the level set approach the free surface is represented as the zero level set of a smooth function this has the effect of replacing the advection of density which has steep gradients at the free surface with the advection of the level set function which is smooth in addition the free surface can merge or break up with no special treatment we maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects in this work we couple the level set scheme to an adaptive projection method for the incompressible navier stokes equations in order to achieve higher resolution of the free surface with a minimum of additional expense we present two dimensional axisymmetric and fully three dimensional results of air bubble and water drop computations

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

numerical analysis of convection diffusion reaction problems with higher order characteristics finite elements part ii fully discretized scheme and quadrature formulas in this paper a higher order lagrange galerkin discretization method is analyzed when applied to a variable coefficient convection possibly degenerated diffusion reaction equation with mixed dirichlet robin boundary conditions in a previous paper a bermu dez m r nogueiras and c va zquez siam j numer anal to appear the proposed second order time discretization scheme has been rigorously introduced for exact and approximated characteristics moreover the l infty l 2 stability property and l infty l 2 error estimates of order o delta t 2 have been obtained as a continuation of that work consistency error estimates of order o delta t 2 h k are obtained for the fully discretized lagrange galerkin scheme moreover adequate quadrature formulas are proposed for the practical implementation of the method with particular finite element spaces finally some numerical tests illustrate the theoretical results and the performance of the combination of second order lagrange galerkin schemes with quadrature formulas

approximations for the bessel and struve functions on presente des approximations polynomiales et fractions rationnelles a precision minimax pour les fonctions de bessel de premiere et seconde espece d ordre zero et un et pour les fonctions de struve du meme ordre

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

an error corrected euler method for solving stiff problems based on chebyshev collocation in this paper we present error corrected euler methods for solving stiff initial value problems which not only avoid unnecessary iteration process that may be required in most implicit methods but also have such a good stability as all implicit methods possess the proposed methods use a chebyshev collocation technique as well as an asymptotical linear ordinary differential equation of first order derived from the difference between the exact solution and the euler s polygon these methods with or without the jacobian are analyzed in terms of convergence and stability in particular it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the jacobian numerical tests are given to support the theoretical analysis as evidences

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

an adaptive mesh projection method for viscous incompressible flow many fluid flow problems of practical interest particularly at high reynolds number are characterized by small regions of complex and rapidly varying fluid motion surrounded by larger regions of relatively smooth flow efficient solution of such problems requires an adaptive mesh refinement capability to concentrate computational effort where it is most needed we present in this paper a fractional step version of chorin s projection method for incompressible flow with adaptive mesh refinement which is second order accurate in both space and time convection terms are handled by a high resolution upwind method which provides excellent resolution of small scale features of the flow while a multilevel iterative scheme efficiently solves the parabolic and elliptic equations associated with viscosity and the projection numerical examples demonstrate the performance of the method on two dimensional problems involving vortex spindown with viscosity and inviscid vortex merger

convergence on error correction methods for solving initial value problems higher order semi explicit one step error correction methods ecm for solving initial value problems are developed ecm provides the excellent convergence o h 2 p 2 one wants to get without any iteration processes required by most implicit type methods this is possible if one constructs a local approximation having a residual error o h p on each time step as a practical example we construct a local quadratic approximation further it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second order methods which can be improved into a semi explicit type ecm of the order of accuracy 6 the stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non stiff problems it should be remarked that the ecm approach developed here does not yield explicit methods but semi implicit methods of the rosenbrock type both ecm and rosenbrock s methods require to solve a few linear systems at each integration step but the ecm approach involves 2p 2 evaluations of the jacobian matrix per integration step whereas the rosenbrock method demands one evaluation only however it is much easier to get high order methods by using the ecm approach

a high order characteristics method for the incompressible navier stokes equations abstract we analyze a high order characteristics method for the navier stokes equations we focus on the cases of the first second and third order in time schemes with finite element spatial discretization a numerical comparison between the first and second order schemes is done for steady or transient states flows

sharp interface cartesian grid method i an easily implemented technique for 3d moving boundary computations a cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three dimensional immersed boundaries the embedded boundaries are represented using level sets and treated in a sharp manner without the use of source terms to represent boundary effects the narrow band distance function field in the level set boundary representation facilitates implementation of the finite difference flow solver the resulting algorithm is implemented in a straightforward manner in three dimensions and retains global second order accuracy the accuracy of the finite difference scheme is established and shown to be comparable to finite volume schemes that are considerably more difficult to implement moving boundaries are handled naturally the pressure solver is accelerated using an algebraic multigrid technique adapted to be effective in the presence of moving embedded boundaries benchmarking of the method is performed against available numerical as well as experimental results

unconditional stability and error estimates of modified characteristics fems for the navier stokes equations the paper is concerned with the unconditional stability and convergence of characteristics type methods for the time dependent navier stokes equations we present optimal error estimates in l 2 l2 and h 1 h1 norms for a typical modified characteristics finite element method unconditionally while all previous works require certain time step restrictions the analysis is based on an iterated characteristic time discrete system with which the error function is split into a temporal error and a spatial error with a rigorous analysis to the characteristic time discrete system we prove that the difference between the numerical solution and the solution of the time discrete system is tau independent where tau denotes the time stepsize thus numerical solution in w 1 infty w1 is bounded and optimal error estimates can be obtained in a traditional way numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time dependent navier stokes equations the approach used in this paper can be easily extended to many other characteristics based methods

geometric integration over irregular domains with application to level set methods we present a geometric approach for calculating integrals over irregular domains described by a level set function this procedure can be used to evaluate integrals over a lower dimensional interface and may be used to evaluate the contribution of singular source terms this approach produces results that are second order accurate and robust to the perturbation of the interface location on the grid moreover since we use a cell wise approach this procedure can be easily extended to quadtree and octree grids we demonstrate the second order accuracy and the robustness of the method in two and three spatial dimensions

a cartesian grid method for solving the two dimensional streamfunction vorticity equations in irregular regions we describe a method for solving the two dimensional navier stokes equations in irregular physical domains our method is based on an underlying uniform cartesian grid and second order finite difference finite volume discretizations of the streamfunction vorticity equations geometry representing stationary solid obstacles in the flow domain is embedded in the cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells along the embedded boundary we determine a distribution of vorticity sources needed to impose the no slip flow conditions this distribution appears as a right hand side term in the discretized fluid equations and so we can use fast solvers to solve the linear systems that arise to handle the advective terms we use the high resolution algorithms in clawpack we show that our stokes solver is second order accurate for steady state solutions and that our full navier stokes solver is between first and second order accurate and reproduces results from well studied benchmark problems in viscous fluid flow finally we demonstrate the robustness of our code on flow in a complex domain

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

very high order compact finite difference schemes on non uniform grids for incompressible navier stokes equations this article presents a family of very high order non uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible navier stokes equations the high order compact schemes on non uniform grids developed in shukla and zhong r k shukla x zhong derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation j comput phys 204 2005 404 for linear model equations are extended to the full navier stokes equations in the vorticity and streamfunction formulation two methods for the solution of helmholtz and poisson equations using high order compact schemes on non uniform grids are developed the schemes are constructed so that they maintain a high order of accuracy not only in the interior but also at the boundary second order semi implicit temporal discretization is achieved through an implicit backward differentiation scheme for the linear viscous terms and an explicit adam bashforth scheme for the non linear convective terms the boundary values of vorticity are determined using an influence matrix technique the resulting discretized system with boundary closures of the same high order as the interior is shown to be stable when applied to the two dimensional incompressible navier stokes equations provided enough grid points are clustered at the boundary the resolution characteristics of the high order compact finite difference schemes are illustrated through their application to the one dimensional linear wave equation and the two dimensional driven cavity flow comparisons with the benchmark solutions for the two dimensional driven cavity flow thermal convection in a square box and flow past an impulsively started cylinder show that the high order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary

mathematical analysis and numerical methods for a pde model of a stock loan pricing problem in this paper the mathematical analysis of a model for pricing stock loan contracts when the accumulative dividend yield associated to the stock is returned by the lender to the borrower on redemption is carried out more precisely the model is formulated in terms of an obstacle problem associated to a kolmogorov equation and the existence and uniqueness in the set of solutions with polynomial growth are obtained also some regularity properties of the solution are analyzed next for the numerical solution of the problem the combination of crank nicolson lagrange galerkin with the augmented lagrangian active set method is described finally some numerical examples illustrate the theoretical properties of the optimal redeeming boundary previously stated in the literature

a second order time accurate ale lagrange galerkin method applied to wind engineering and control of bridge profiles abstract in this paper we propose a second order time accurate ale formulation of lagrange galerkin methods for the transient solution of navier stokes equations for an incompressible fluid in a moving domain following a previous work rev europeenne elem finis 8 1999 659 we show that this time discretization based on a backward differentiation formula and the construction of characteristic paths in a grid relative velocity field leads to second order time accuracy numerical results are compared with those given by a first order approach the numerical diffusion is dramatically reduced which makes the use of turbulence model quasi mandatory for realistic wind engineering applications the software developed was also used to solve elementary control problems the solution method is based on the linearization of the discretized problem including the method of characteristics the formulation and implementation were successfully completed and will be detailed elsewhere

two coupled particle finite volume methods using delaunay voronoi meshes for the approximation of vlasov poisson and vlasov maxwell equations this paper deals with the approximation of vlasov poisson and vlasov maxwell equations we present two coupled particle finite volume methods which use the properties of delaunay voronoi meshes these methods are applied to benchmark calculations and engineering problems such as simulation of electron injector devices

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

multistep characteristic method for incompressible flow in porous media in this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media the discretization involves a three level time scheme based on the characteristic method and a five point finite difference scheme for space discretization we prove that the convergence is of order o h 2 dt 2 which is in contrast to the convergence of order o h dt proved for a singlestep characteristic with the same space discretization numerical experiments demonstrate the stability and second order convergence of the scheme

a versatile sharp interface immersed boundary method for incompressible flows with complex boundaries a sharp interface immersed boundary method for simulating incompressible viscous flow past three dimensional immersed bodies is described the method employs a multi dimensional ghost cell methodology to satisfy the boundary conditions on the immersed boundary and the method is designed to handle highly complex three dimensional stationary moving and or deforming bodies the complex immersed surfaces are represented by grids consisting of unstructured triangular elements while the flow is computed on non uniform cartesian grids the paper describes the salient features of the methodology with special emphasis on the immersed boundary treatment for stationary and moving boundaries simulations of a number of canonical two and three dimensional flows are used to verify the accuracy and fidelity of the solver over a range of reynolds numbers flow past suddenly accelerated bodies are used to validate the solver for moving boundary problems finally two cases inspired from biology with highly complex three dimensional bodies are simulated in order to demonstrate the versatility of the method

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

a second order characteristic finite element scheme for convection diffusion problems a new characteristic finite element scheme is presented for it is of second order accuracy in time increment symmetric and unconditionally stable optimal error estimates are proved in the framework of l 2 theory numerical results are presented for two examples which show the advantage of the scheme

numerical analysis of convection diffusion reaction problems with higher order characteristics finite elements part i time discretization in this paper a higher order characteristics time discretization scheme is analyzed for a variable coefficient convection possibly degenerate diffusion reaction equation with mixed dirichlet robin boundary conditions first the proposed second order time discretization scheme is rigorously introduced for exact and approximate characteristics next under not very restrictive hypotheses on the data the l infty l 2 stability is proved and l infty l 2 error estimates of order o delta t 2 are obtained lagrange galerkin schemes using different finite elements spaces will be analyzed in the second part of this work to appear in siam j numer anal where quadrature formulas are proposed for practical implementation

a stable projection method for the incompressible navier stokes equations on arbitrary geometries and adaptive quad octrees we present a numerical method for solving the incompressible navier stokes equations on non graded quadtree and octree meshes and arbitrary geometries the viscosity is treated implicitly through a finite volume approach based on voronoi partitions while the convective term is discretized with a semi lagrangian scheme thus relaxing the restrictions on the time step a novel stable implementation of the projection step is introduced making use of the marker and cell layout for the data the solver is validated numerically in two and three spatial dimensions

approximation of the struve function h1 occurring in impedance calculations the problem of the rigid piston radiator mounted in an infinite baffle has been studied widely for tutorial as well as for practical reasons the resulting theory is commonly applied to model a loudspeaker in the audio frequency range a special function the struve function h 1 z occurs in the expressions for the rigid piston radiator this struve function is not readily available in programs such as matlab or mathcad nor in computer languages such as fortran and c therefore a simple and effective approximation of h 1 z which is valid for all z is developed some examples of the application of the struve function in acoustics are presented

an efficient fluid solid coupling algorithm for single phase flows we present a simple and efficient fluid solid coupling method in two and three spatial dimensions in particular we consider the numerical approximation of the navier stokes equations on irregular domains and propose a novel approach for solving the hodge projection step on arbitrary shaped domains this method is straightforward to implement and leads to a symmetric positive definite linear system for both the projection step and for the implicit treatment of the viscosity we demonstrate the accuracy of our method in the l 1 and l norms and present its removing the errors associated with the conventional rasterization type discretizations we apply this method to the simulation of a flow past a cylinder in two spatial dimensions and show that our method can reproduce the known stable and unstable regimes as well as correct lift and drag forces we also apply this method to the simulation of a flow past a sphere in three spatial dimensions at low and moderate reynolds number to reproduce the known steady axisymmetric and non axisymmetric flow regimes we further apply this algorithm to the coupling of flows with moving rigid bodies

pricing pension plans under jump diffusion models for the salary in this paper we consider the valuation of a defined benefit pension plan in the presence of jumps in the underlying salary and including the possibility of early retirement we will consider that the salary follows a jump diffusion model thus giving rise to a partial integro differential equation pide after posing the model we propose the appropriate numerical methods to solve the pide problem these methods mainly consists of lagrange galerkin discretizations combined with augmented lagrangian active set techniques and with the explicit treatment of the integral term finally we compare the numerical results with those ones obtained with monte carlo techniques

a 2d compact fourth order projection decomposition method a 2d fourth order compact direct scheme projection decomposition method for solving incompressible viscous flows in multi connected rectangular domains is devised in each subdomain the governing navier stokes equations are discretized by using fourth order compact schemes in space and second order scheme in time the coupling between subdomains is based on a direct non overlapping multidomain method it allows to solve each helmholtz poisson problem resulting of a projection method in complex geometries the major difficulty of the poisson neumann problem solvability is addressed and correctly treated the present numerical method is checked through some classical numerical experiments first the second order accuracy in time and the fourth order accuracy in space are shown by matching with the analytical solution of the taylor problem the method is also tested by simulating the flow in a 2d lid driven cavity the utility of the compact scheme projection decomposition method approach is further illustrated by two other benchmark problems viz the flow over a backward facing step and the laminar flow past a square prism the present results are in good agreement with the experimental data and other numerical solutions available in the literature