a fourth order compact scheme for heat transfer problem in porous media in this paper we have extended the recently proposed fourth order compact scheme by pandit et al 2007 which was used earlier only for the convection diffusion equations without considering reaction and nonhomogeneous source terms and designed to compute flow in a two sided lid driven differentially heated square cavity filled with a fluid saturated porous medium the governing navier stokes n s equations in streamfunction vorticity form of brinkman extended darcy model including the energy transport equation are all solved as a coupled system of equations for the five field variables consisting of streamfunction vorticity two velocities and temperature moreover local entropy generation distributions are determined based on the obtained dimensionless velocity and temperature values the derivative source term present in vorticity equation has been treated as fourth order compact using pade scheme the details for the derivation of difference relations at boundaries to generate accurate and stable solutions are also given we have computed the results for three different cases depending on the direction of moving walls to assess the numerical accuracy of our proposed scheme one pertinent test problem with known exact solution is used the higher order compact scheme adopted in the present study yields consistent performance for variation of key parameters e g richardson number r i darcy number d a grashof number g r and fixed prandtl number p r 0 7 the present results are compared with numerical results available in the literature and excellent match is observed in all the cases establishing the efficiency and accuracy of the extended combined formulations of our fourth order compact scheme and pade schemes

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

convergence of a compact scheme for the pure streamfunction formulation of the unsteady navier stokes system this paper is devoted to the analysis of a new compact scheme for the navier stokes equations in pure streamfunction formulation numerical results using that scheme have been reported in m ben artzi e j comput phys 205 2005 pp 640 664 the scheme discussed here combines the stephenson scheme for the biharmonic operator and ideas from box scheme methodology consistency and convergence are proved for the full nonlinear system instead of customary periodic conditions the case of boundary conditions is addressed it is shown that in one dimension the truncation error for the biharmonic operator is o h 4 at interior points and o h at near boundary points in two dimensions the truncation error is o h 2 at interior points due to the cross terms and o h at near boundary points hence the scheme is globally of order four in the one dimensional periodic case and of order two in the two dimensional periodic case but of order 3 2 for one and two dimensional nonperiodic boundary conditions we emphasize in particular that there is no special treatment of the boundary thus allowing robust use of the scheme the finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications

an efficient compact difference scheme for solving the streamfunction formulation of the incompressible navier stokes equations recently a new paradigm for solving the steady navier stokes equations using a streamfunction velocity formulation was proposed by gupta and kalita m m gupta j c kalita a new paradigm for solving navier stokes equations streamfunction velocity formulation j comput phys 207 2005 52 68 which avoids difficulties inherent in the conventional streamfunction vorticity and primitive variable formulations it is discovered that this formulation can reached second order accurate and obtained accuracy solutions with little additional cost for a couple of fluid flow problems in this paper an efficient compact finite difference approximation named as five point constant coefficient second order compact 5pcc soc scheme is proposed for the streamfunction formulation of the steady incompressible navier stokes equations in which the grid values of the streamfunction and the values of its first derivatives velocities are carried as the unknown variables the derivation approach is simple and can be easily used to derive compact difference schemes for other similar high order elliptic differential equations numerical examples including the lid driven cavity flow problem and a problem of flow in a rectangular cavity with the hight width ratio of 2 are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme the results obtained are compared with ones by different available numerical methods in the literature the present scheme not only shows second order accurate but also proves more effective than the existing second order compact scheme of the streamfunction formulation in the aspect of computational cost

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

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

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 fast direct solver for the biharmonic problem in a rectangular grid we present a fast direct solver methodology for the dirichlet biharmonic problem in a rectangle the solver is applicable in the case of the second order stephenson scheme j w stephenson j comput phys 55 1984 pp 65 80 as well as in the case of a new fourth order scheme which is discussed in this paper and is based on the capacitance matrix method b l buzbee and f w dorr siam j numer anal 11 1974 pp 1136 1150 p bjorstad siam j numer anal 20 1983 pp 59 71 the discrete biharmonic operator is decomposed into two components the first is a diagonal operator in the eigenfunction basis of the laplacian to which the fft algorithm is applied the second is a low rank perturbation operator given by the capacitance matrix which is due to the deviation of the discrete operators from diagonal form the sherman morrison formula g h golub and c f van loan matrix computations 3rd ed john hopkins university press baltimore md 1996 is applied to obtain a fast solution of the resulting linear system of equations

a pure compact scheme for the streamfunction formulation of navier stokes equations a pure streamfunction formulation is introduced for the numerical simulation of the two dimensional incompressible navier stokes equations the idea is to replace the vorticity in the vorticity streamfunction evolution equation by the laplacian of the streamfunction the resulting formulation includes the streamfunction only thus no inter function relations need to be invoked a compact numerical scheme which interpolates streamfunction values as well as its first order derivatives is presented and analyzed a number of numerical experiments are presented including driven and double driven cavities where the reynolds numbers are sufficiently large leading to symmetry breaking of asymptotic solutions

convergence analysis of a colocated finite volume scheme for the incompressible navier stokes equations on general 2d or 3d meshes we study a colocated cell centered finite volume method for the approximation of the incompressible navier stokes equations posed on a 2d or 3d finite domain the discrete unknowns are the components of the velocity and the pressure all of them colocated at the center of the cells of a unique mesh such a configuration is known to lead to stability problems hence the need for a stabilization technique which we choose of the brezzi pitka ranta type the scheme features two essential properties the discrete gradient is the transpose of the divergence terms and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields as a consequence the scheme is proved to be unconditionally stable and convergent for the stokes problem and for the transient and the steady navier stokes equations in this latter case for a given sequence of approximate solutions computed on meshes the size of which tends to zero we prove up to a subsequence the l 2 convergence of the components of the velocity and in the steady case the weak l 2 convergence of the pressure the proof relies on the study of space and time translates of approximate solutions which allows the application of kolmogorov s theorem the limit of this subsequence is then shown to be a weak solution of the navier stokes equations numerical examples are performed to obtain numerical convergence rates in both the linear and nonlinear cases

a compact difference scheme for the biharmonic equation in planar irregular domains we present a finite difference scheme applicable to general irregular planar domains to approximate the biharmonic equation the irregular domain is embedded in a cartesian grid in order to approximate delta 2 phi at a grid point we interpolate the data on the irregular stencil by a polynomial of degree six the finite difference scheme is delta 2 q phi 0 0 where q phi is the interpolation polynomial the interpolation polynomial is not uniquely determined we present a method to construct such an interpolation polynomial and prove that our construction is second order accurate for a regular stencil m ben artzi j p croisille and d fishelov siam j sci comput 31 2008 pp 303 333 shows that the proposed interpolation polynomial is fourth order accurate we present some suitable numerical examples

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 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

multigrid solution of automatically generated high order discretizations for the biharmonic equation in this work we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil the solution and its first derivatives are carried as unknowns at the grid points dirichlet boundary conditions are thus incorporated naturally since the approximations use the 9 point compact stencil no special formulas are needed near the boundaries both second order and fourth order discretizations are derived the fourth order approximation produce more accurate results than the 13 point classical stencil or the commonly used system of two second order equations coupled with the boundary condition the method suffers from slow convergence when classical iteration methods such as gauss seidel or sor are employed in order to alleviate this problem we propose several multigrid techniques that exhibit grid independent convergence and solve the biharmonic equation in a small amount of computer time test results from three different problems including stokes flow in a driven cavity are reported

meshfree point collocation method for the stream vorticity formulation of 2d incompressible navier stokes equations abstract meshfree point collocation method is developed for the stream vorticity formulation of two dimensional incompressible navier stokes equations particular emphasis is placed on the novel formulation of effective vorticity condition on no slip boundaries the moving least square approximation is employed to construct shape functions in conjunction with the framework of point collocation method the derivatives of an arbitrary function can be obtained by the linear combination of these shape functions which enables the vorticity boundary condition to vary linearly with the boundary velocity and the stream function together with the second order partial differential equations for the stream and vorticity functions this boundary condition of vorticity provides the efficient meshfree point collocation scheme for the two dimensional incompressible flow the accuracy and stability for the proposed scheme are demonstrated through a new type of application problem with complex geometry in addition to several typical examples of steady state flow simulation

a compact streamfunction velocity scheme on nonuniform grids for the 2d steady incompressible navier stokes equations in this paper an effective compact finite difference approximation which carries streamfunction and its first derivatives velocities as the unknown variables for the streamfunction velocity formulation of the steady two dimensional incompressible navier stokes equation is developed on non uniform orthogonal cartesian grids to solve the resulting system of equations a multigrid iterative strategy on nonuniform grids is introduced by using the interpolation techniques numerical experiments involving two test problems with analytical solutions and the lid driven square cavity flow problem are carried out to display the superiority of the currently developed method on nonuniform grid numerical results show that the present method on nonuniform grids gets as similarly efficient convergence rate as on uniform grids viz second order accuracy and the resolution of the computed solutions for the problems with the sharp changes can be significantly improved when the nonuniform grid strategy is utilized the backward facing step flow is also calculated by the present method to exhibit the capability to simulate the distant field using fewer grid points the solution for the natural convection problem reveals further the wide applications of the present method not only in the flow problems but also in the heat transfer problems all of these numerical results demonstrate the accuracy and efficiency of the currently proposed schemes

hopf bifurcation of the unsteady regularized driven cavity flow abstract a numerical simulation of the unsteady incompressible flow in the unit cavity is performed by using a chebyshev tau approximation for the space variables the high accuracy of the spectral methods and the condensed distribution of the chebyshev collocation points near the boundary enable us to obtain reliable results for high reynolds numbers with a moderate number of modes it is found that the flow converges to a stationary state for reynolds numbers re up to 10 000 for reynolds numbers larger than a critical value 10 000

random lsc functions an ergodic theorem an ergodic theorem for random lsc lower semicontinuous functions is obtained by relying on a scalarization of such functions in the process kolmogorov s extension theorem for random lsc functions is established applications to statistical estimation problems composite materials and stochastic optimization problems are briefly noted

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

a domain decomposition method for approximating the conformal modules of long quadrilaterals this paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals the method has been studied already by us and also by d gaier and w k hayman but only in connection with a special class of quadrilaterals viz quadrilaterals where a the defining domain is bounded by two parallel straight lines and two jordan arcs and b the four specified boundary points are the four corners where the arcs meet the straight lines our main purpose here is to explain how the method may be extended to a wider class of quadrilaterals than that indicated above

a central difference scheme for a pure stream function formulation of incompressible viscous flow we present a numerical scheme for incompressible viscous flow formulated as an equation for the stream function the pure stream function formulation obviates the difficulty associated with vorticity boundary conditions the resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver the advection of vorticity is implemented with a high resolution central scheme that remains stable and accurate in the presence of large gradients the accuracy and robustness of the method are demonstrated for high reynolds number flows in a lid driven cavity

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

an efficient transient navier stokes solver on compact nonuniform space grids in this paper we propose an implicit higher order compact hoc finite difference scheme for solving the two dimensional 2d unsteady navier stokes n s equations on nonuniform space grids this temporally second order accurate scheme which requires no transformation from the physical to the computational plane is at least third order accurate in space which has been demonstrated with numerical experiments it efficiently captures both transient and steady state solutions of the n s equations with dirichlet as well as neumann boundary conditions the proposed scheme is likely to be very useful for the computation of transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variation numerical results are presented and compared with analytical as well as established numerical data excellent comparison is obtained in all the cases

an upwind compact difference scheme for solving the streamfunction velocity formulation of the unsteady incompressible navier stokes equation abstract in this paper an upwind compact difference method with second order accuracy both in space and time is proposed for the streamfunction velocity formulation of the unsteady incompressible navier stokes equations the first derivatives of streamfunction velocities are discretized by two type compact schemes viz the third order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth order symmetric compact scheme is employed for the biharmonic term as a result a five point constant coefficient second order compact scheme is established in which the computational stencils for streamfunction only require grid values at five points at both n th and n 1 th time levels the new scheme can suppress non physical oscillations moreover the unconditional stability of the scheme for the linear model is proved by means of the discrete von neumann analysis four numerical experiments involving a test problem with the analytic solution doubly periodic double shear layer flow problem lid driven square cavity flow problem and two sided non facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme the present scheme not only shows the good numerical performance for the problems with sharp gradients but also proves more effective than the existing second order compact scheme of the streamfunction velocity formulation in the aspect of computational cost

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