importance of convection and damping on rates of convergence for the lax wendroff method it is well known that in solving steady state problems using hyperbolic time stepping methods the intent is to drive the transients to zero as quickly as possible in this paper the convergence to steady state of the lax wendroff method applied to solving the equations of gas dynamics is analyzed for the laval nozzle problem by comparing the relative rates of damping and convection using a linearized eigenmode analysis this analysis is developed for the simpler isenthalpic system and then extended to the full euler equations finally this allows a comparison between these systems for both models useful analytical information can be gleaned about the transient behavior of these systems especially in regard to quantifying the competitive factors affecting the removal of unsteady waves

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

direct solution of the vorticity stream function ordinary differential equations by a chebyshev approximation abstract a numerical method for solving the coupled vorticity stream function equations in one dimension with an exact noniterative determination of the vorticity boundary values is presented the one dimensional form considered can represent the fourier modes of a two dimensional problem the equations are separated by replacing the derivative specifications for the stream function at the boundary points with equivalent conditions of an integral type for the vorticity a spectral approximation by means of chebyshev polynomials is considered the numerical properties of the algorithm are investigated against a few analytical examples which demonstrate the accuracy of the proposed method

finite volume solutions of convection diffusion test problems the cell vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics more recently one of us has proposed an extension for viscous flows the purpose of the present paper is two fold first we have applied this scheme to a well known convection diffusion model problem involving flow round a 180 degrees bend which highlights some of the issues concerning the application of the boundary conditions in such cell based schemes the results are remarkably good when the boundary conditions are applied in an appropriate manner in our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one dimensional problem our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra convergence phenomena enjoyed by the proposed methods 25 refs 9 figs 1 tab

noniterative time stepping schemes with adaptive truncation error control for the solution of richards equation 1 noniterative implicit time stepping schemes with adaptive temporal truncation error control are developed for the solution of the pressure form of richards equation first and second order linearizations of an adaptive backward euler thomas gladwell formulation are introduced and are shown to constrain the temporal truncation errors near a user prescribed tolerance and maintain adequate mass balance numerical experiments demonstrate that accurate noniterative linearizations achieve cost effective solutions of problems where soils are described by highly nonlinear and nonsmooth constitutive functions for these problems many conventional iterative solvers fail to converge the noniterative formulations are considerably more efficient than analogous time stepping schemes with iterative solvers the second order noniterative scheme is found to be more efficient than the first order noniterative scheme the proposed adaptive noniterative algorithms can be easily incorporated into existing backward euler software which is widely used for richards equation and other nonlinear pdes

resolution of the navier stokes equations in vorticity velocity formulation via control techniques abstract the velocity vorticity formulation of the navier stokes equations is turned into an optimal control problem subject to linear constraints which consist of laplacian equations for the velocity and for the vorticity various optimization algorithms can be used for the numerical resolution of the optimal control problem in question in particular the use of a conjugate gradient method is discussed

a galerkin implementation of the generalized helmholtz decomposition of vorticity formulations abstract vorticity formulations for the incompressible navier stokes equations have certain advantages over primitive variable formulations including the fact that the number of equations to be solved is reduced however the accurate implementation of the boundary conditions seems to continue to be an impediment to the acceptance and use of numerical methods based on vorticity formulations velocity boundary conditions can be implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields as described by the generalized helmholtz decomposition ghd this can be accomplished in one of two ways by either solving for boundary vorticity leading to a dirichlet boundary condition for the vorticity equation or solving for boundary vortex sheet strengths leading to a neumann condition in the past vortex sheet strengths have often been determined by solving an over specified set of linear equations the over specification arose because integral constraints were imposed on the vortex sheet strengths these integral constraints are not necessary and typically are included to mitigate errors in determining the vortex sheet strengths themselves further the constraints overspecify the linear system requiring least squares solution techniques to more accurately satisfy both components of the velocity boundary conditions a galerkin formulation is applied to the generalized helmholtz decomposition this formulation implicitly satisfies an integral constraint that is more general than many of the integral constraints that have been explicitly imposed two implementations of the galerkin ghd are considered in the current work one based on determining the boundary vorticity and one based on determining the boundary vortex sheet strengths a finite element method fem is implemented to solve the vorticity equation along with the boundary data generated from the ghd

application of the finite difference heterogeneous multiscale method to the richards equation 1 this paper extends the finite difference heterogeneous multiscale method fdhmm to simulate transient unsaturated water flow problems in random porous media the numerical method is based on the use of two different schemes for the original equation at different grid levels which allows numerical results at a lower cost than solving the original equations the main feature of fdhmm is that the necessary data for the macroscopic model are supplied by solving the microscopic model on a sparse spatial domain the generated code is verified by applying the linearization model of the richards equation considering two different constitutive relationships this method is applied to several test examples with different soil textures and boundary conditions both the dirichlet and the periodic boundary conditions are considered for solving the local microscopic model when the water flow in heterogeneous unsaturated soils is simulated by fdhmm the numerical experiments demonstrate that fdhmm can effectively simulate the transient unsaturated water flow in the specific soils the numerical experiments also demonstrate that fdhmm can achieve accurate global mass balance and is a globally convergent algorithm and the spatial correlation length of random coefficients under the specific standard deviation has relatively little influence on the accuracy of the method

adaptive backward euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow an automatic time stepping scheme with embedded error control is developed and applied to the moisture based richards equation the algorithm is based on the first order backward euler scheme and uses a numerical estimate of the local truncation error and an efficient time step selector to control the temporal accuracy of the integration local extrapolation equivalent to the use of an unconditionally stable thomas gladwell algorithm achieves second order temporal accuracy at minimal additional costs the time stepping algorithm also provides accurate initial estimates for the iterative non linear solver numerical tests confirm the ability of the scheme to automatically optimize the time step size to match a user prescribed temporal error tolerance an important merit of the proposed method is its conceptual and computational simplicity it can be directly incorporated into existing or new software based on the backward euler scheme currently prevalent in subsurface hydrologic modelling and markedly improves their performance compared with simple fixed or heuristic time step selection the generality of the approach also makes possible its use for solving pdes in other engineering applications where strong non linearity stability or implementation considerations favour a simple and robust low order method or where there is a legacy of backward euler codes in current use copyright 2001 john wiley sons ltd

finite analytic numerical method for unsteady two dimensional navier stokes equations abstract the main purpose of this paper is to develop a finite analytic fa numerical solution for unsteady two dimensional navier stokes equations the fa method utilizes the analytic solution in a small local element to formulate the algebraic representation of partial differential equations in this study the combination of linear and exponential functions that satisfy the governing equation is adopted as the boundary function thereby improving the accuracy of the finite analytic solution two flows one a starting cavity flow and the other a vortex shedding flow behind a rectangular block are solved by the fa method the starting square cavity flow is solved for reynolds numbers of 400 1000 and 2000 to show the accuracy and stability of the fa solution the fa solution for flow over a rectangular block h h 4 predicts the strouhal number for reynolds numbers of 100 and 500 to be 0 156 and 0 125 details of the flow patterns are given in addition to streamlines and vorticity distribution rest streamlines are given to illustrate the vortex motion downstream of the block

the accuracy of cell vertex finite volume methods on quadrilateral meshes for linear first order hyperbolic equations in two dimensions we restate the cell vertex finite volume scheme as a finite element method on structured meshes consisting of distorted quadrilaterals the global error is shown to be of second order in various mesh dependent norms provided that the quadrilaterals are close to parallelograms in the sense that the distance between the midpoints of the diagonals is of the same order as the measure of the guadrilateral on tensor product nonuniform meshes the cell vertex scheme coincides with the familiar box scheme in this case second order accuracy is shown without any additional assumption on the regularity of the mesh which explains the insensitivity of the cell vertex scheme to mesh stretching in the coordinate directions observed in practice 17 refs

the kinematic laplacian equation method a novel procedure for the navier stokes equations in the vorticity velocity formulation is presented the time evolution of the vorticity is solved as an ode problem on each node of the spatial discretization using at each step of the time discretization the spatial solution for the velocity field provided by a new pde expression called the kinematic laplacian equation kle this complete decoupling of the two variables in a vorticity in time velocity in space split algorithm reduces the number of unknowns to solve in the time integration process and also favors the use of advanced ode algorithms enhancing the efficiency and robustness of time integration the issue of the imposition of vorticity boundary conditions is addressed as well as the details of the implementation of the kle by isoparametric finite element discretization we shall see some validation results of the kle method applied to the classical case of a circular cylinder in impulsive started pure translational steady motion at several reynolds numbers in the range 5

a box scheme for transcritical flow the accurate computer simulation of river and pipe flow is of great importance in the design of urban drainage networks the use of implicit numerical schemes allows the time step to be chosen on the basis of accuracy rather than stability offering a potential computational saving over explicit methods the highly successful box scheme is an implicit method which can be used to model a wide range of subcritical and supercritical flows however care must be taken over the modelling of transcritical flows since unless the correct internal boundary conditions are imposed the scheme becomes unstable the necessity of accurately tracking all the critical interfaces and treating them accordingly can be algorithmically complex and in practice the underlying mathematical model is often modified to ensure that the flow remains essentially subcritical such a modification however inevitably leads to additional errors and incorrect qualitative behaviour can be observed in this paper we show how the technique of residual distribution can be successfully implemented in order to accurately model unsteady transcritical flow without the need to know a priori which regions of the computational domain correspond to subcritical and supercritical flow when used in conjunction with a form of artificial smoothing the resulting method generates very high resolution results even for transcritical problems involving shocks as can be seen in the numerical results

convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions we prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions the result is applied to the discontinuous galerkin method due to cockburn hou and shu

mass conservative numerical solutions of the head based richards equation numerical procedures for efficient mass conservative solutions of the head based form of the richards equation are presented mass conservative solutions are shown to result when the capacity coefficient c is formulated by equating the storage term and its chain rule expansion in their discretized forms equivalence in the storage term expansion is maintained in finite difference models when c is evaluated with a standard chord slope approximation this scheme is shown to produce excellent global mass balance accuracy in simulations of vertical moisture infiltration an analogous approach to the expansion of the storage term using finite elements results in element dependent expressions of c application of this approach produces mass balance accuracy with errors less than 1 but also exhibits slow convergence in the consistent form a nontraditional finite element procedure is presented which maintains equivalence in the storage term expansion when c is evaluated with the standard chord slope approximation this scheme exhibits excellent mass balance accuracy in either the consistent or lumped forms without significant loss in computational efficiency

numerical study for differential constitutive equations with polymer melts by using a hybrid finite element volume method this paper covers the numerical investigation of complex flow tube tooling cable coating using the single equation representation of the pom pom sxpp model here the hybrid scheme consists of a taylor galerkin finite element discretisation and a cell vertex fluctuation distribution finite volume approach is employed for this method the momentum continuity equations have been solved by a fractional staged taylor galerkin pressure correction procedure and invoke a cell vertex fluctuation distribution finite volume scheme for the constitutive law stress response and pressure drop across the flow are investigated with respect to variation in weissenberg number and solvent fraction investigation of complex flow tube tooling cable coating using the single equation representation of the pom pom sxpp model influence of variation in weissenberg number on stress response and pressure drop across the flow evaluate free surface movement by using the time dependent prediction method which is termed phan thien dh dt scheme

analysis of finite difference schemes for unsteady navier stokes equations in vorticity formulation in this paper we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible navier stokes equations in vorticity stream function formulation the no slip boundary condition for the velocity is converted into local vorticity boundary conditions thom s formula wilkes formula or other local formulas in the earlier literature can be used in the second order method while high order formulas such as briley s formula can be used in the fourth order compact difference scheme proposed by e and liu the stability analysis of these long stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula the main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function we choose to analyze the second order scheme with wilkes formula in detail in this case we can avoid the complicated technique necessitated by the strang type high order expansions as a consequence our analysis results in almost optimal regularity assumption for the exact solution the above methodology is very general we also give a detailed analysis for the fourth order scheme using a 1 d stokes model

fresh look at floating shock fitting a fast implicit upwind procedure for the two dimensional euler equations is described that allows accurate computations of shocked flows on nonadapted meshes away from shocks the second order accurate upwinding is based on the split coefficient matrix scm method in the presence of shocks the difference stencils are modified using a floating shock fitting technique rapid convergence to steady state solutions is attained with a diagonalized approximate factorization af algorithm results are presented for riemann s problem for a regular shock reflection at an inviscid wall for supersonic flow past a cylinder and for a transonic airfoil all computed shocks are ideally sharp and in excellent agreement with other numerical results or exact solutions most importantly this has been accomplished on unusually crude meshes without any attempt to align grid lines with shock fronts or to cluster grid lines around shocks

experimental evidence of randomness and nonuniqueness in unsaturated outflow experiments designed for hydraulic parameter estimation single transient outflow experiments are commonly conducted for inverse estimation of unsaturated hydraulic parameters we assess the validity of this procedure through several repeated experiments on the same sample a medium sand contained in a pressure cell outflow was induced by one or multiple step changes in bottom boundary suction such that there were replicates for each of several step levels in suction we observed that experiments with small initial step changes were poorly reproducible even though our setup allowed reproduction of almost identical initial saturation for each run experiments with large step changes were well reproducible but the outflow response was virtually the same for different step levels neither type of observation was predicted by a theoretical sensitivity analysis of the richards equation given the minimal inaccuracy in our experiments inverse estimation yielded incompatible apparent hydraulic parameters for different flow conditions our results imply experimental limitations of the inversion procedure

a mass conservative switching algorithm for modeling fluid flow in variably saturated porous media numerical solutions of flow equation in fluid content based form or in fluid pressure head based form are often tradeoffs between speed accuracy and convenience the fluid content based form can be solved quite rapidly with low cpu time and perfect mass balance however it cannot be used in saturated regions as diffusivity function becomes infinite and strictly becomes invalid in composite layered and real heterogeneous porous materials due to singularity and discontinuity in fluid content profile this formulation also gives misleading impression that gradient in fluid content causes the flow of fluid in porous materials where in reality gravity and fluid pressure potential gradient produce the motion the pressure head based form on the other hand is more flexible but due to its highly nonlinear nature is much more time consuming and produces poor global mass balance for dry initial conditions very fine spatial and temporal discretizations are needed to maintain mass balance property for these scenarios the mixed form of the flow equation partially solves these issues as it maintains acceptable mass balance and is applicable to layered heterogeneous and composite fractured foundations however it is only applicable in unsaturated zones in this study a switching algorithm was proposed and implemented in which the mass conservative mixed form and the pressure head based form were respectively used in the unsaturated and saturated zones of an initial boundary value flow problem involving a variably saturated porous medium the algorithm showed excellent agreement with a reference solution obtained on a very fine spatiotemporal mesh the simulator was then calibrated with several real world large scale experimental datasets in all cases the proposed algorithm exhibited close agreements with the experimental time space series the algorithm poses excellent mass balance property and can easily be used in both saturated and unsaturated regions without special treatment of fluid content discontinuities in heterogeneous and layered porous media the proposed algorithm can also be extended to simulate multiphase and multidimensional flow problems

the efficient generation of simple two dimensional adaptive grids we present an efficient adaptive procedure for solving steady partial differential equations based on the simple two dimensional quadrilateral grid generator of huang and sloan siam j sci comput 15 1994 pp 776 797 the mesh equations are derived using an equidistribution principle and are discretized in the physical domain this allows the application of the adaptive grid generator to finite element and finite volume methods to efficiently solve the nonlinear algebraic equations describing the adapted grid we propose an alternating line gauss seidel relaxation procedure the beneficial effect of adaptive grids is demonstrated by coupling the grid generator to finite volume and finite difference approximations of advection and advection diffusion test problems for some examples the proposed method is shown to be 50 times more efficient than previously used solution procedures

vorticity boundary conditions and boundary vorticity generation for two dimensional viscous incompressible flows abstract in this paper we present boundary conditions appropriate for the vorticity formulation of the two dimensional incompressible viscous navier stokes equations these boundary conditions are incorporated into a finite difference scheme and the resulting method is of the vorticity creation type i e vorticity is generated at the boundary to ensure that the tangential velocity boundary condition is satisfied the results of computations with this finite difference method are presented for flow past a circular cylinder a difference scheme and computational results for a model problem the prandtl boundary layer equations describing flow over a semi infinite flat plate are also presented

finite volumes and finite elements two good friends in this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic convective diffusion and fluid flow problems the paper shows that both procedures share a number of features like mesh discretization and approximation moreover it is shown that in many cases both techniques are completely equivalent

a fast algorithm for direct simulation of particulate flows using conforming grids this paper presents a development of the direction splitting algorithm for flows in complex geometries proposed in 2 to the case of flows containing rigid particles the main novelty of this method is that the grid is fit to the time dependent domain at each time step which allows for an optimal spatial approximation of the navier stokes equations in general this is a very hard task for multidimensional problems however it is significantly simplified if the momentum equations are discretized by a direction splitting scheme such a scheme requires only solving a set of one dimensional problems and the grid can be easily fit to the boundary in the direction of each of these problems here we use a mac discretization stencil but the same idea can be applied to other discretizations the equations of motion of each particle are discretized explicitly and the computed particle velocities are imposed as dirichlet boundary conditions for the navier stokes equations on the adapted grid the pressure is extended within the particles in a fictitious domain fashion but the divergence stencil is fit to the particle boundaries in this paper the direction splitting algorithm with boundary fitting is compared to pure fictitious domain methods

nonoscillatory solution of the steady state inviscid burgers equation by mathematical programming abstract numerical solution of the steady state inviscid burgers equation u2 0 on 0 1 u 0 g0 u 1 g1 g0 g1 is considered to obtain the physically relevant discontinuous solution of this problem the equation is singularly perturbed by adding a small amount of viscosity u u2 0 a cell entered finite difference scheme that uses two points for the inviscid part and four points for the viscous part is proposed the overdetermined nonlinear system that results is solved by a homotopy based on the newton linearizer and a linear programming algorithm used to solve l problems difficulties in capturing interior layers centered at node points are observed however the computational results for interior layers centered between node points and for boundary layers show accurate nonoscillatory solutions with discontinuities captured in one cell on both coarse and fine grids the amount of artificial viscosity required to obtain these solutions is small 10 9 compared to the amount of artificial viscosity required by standard methods a partial proof of pointwise and l1 0 1 convergence of the linear spline interpolant of the numerical solution to the inviscid solution h 0 and h 0 is given for one case

upwind iteration methods for the cell vertex scheme in one dimension this paper describes and analyses a series of methods for solving the algebraic equations obtained from the cell vertex finite volume discretisation in one dimension the objective is to explore the possibilities for improved iteration methods that may be applied to cell vertex discretisations of the navier stokes equations in higher dimensions in general there is no natural one to one correspondence between cellbased residuals and nodal unknowns for this system in order to devise iteration schemes it is therefore necessary to provide a mapping between cells and nodes the family of methods introduced here is based on the application of standard iterative techniques to a nodal residual formed of a combination of neighbouring cell based residuals it includes the familiar lax wendroff iteration upwind iteration schemes and marching schemes capable of attaining convergence rates independent of the number of algebraic equations the aim in each case is to set to zero the residual for each cell apart from exceptional cells such as those containing shocks the final results show that matrix based upwind iteration methods using cell residuals modified to take account of critical points and applying several local iterations converge in around 15 iterations

hyperbolicity and optimal coordinates for the three dimensional supersonic euler equations the two dimensional 2 d steady euler equations for supersonic flow in cartesian coordinates x y is hyperbolic except on the singular line u a where a is the speed of sound and u is the x component of velocity however the cauchy problem for x x circ with initial data prescribed at x x circ is well posed only when u a for the 2 d case the optimal coordinates have been found w h hui and d l chu comput fluid dynamics 4 1996 pp 403 426 to be the orthogonal system consisting of streamlines and their orthogonal lines this gives the most robust and accurate computation for supersonic flow it is commonly thought e g c y loh and m s liou j comput phys 113 1994 pp 224 248 that the system of three dimensional 3 d steady euler equations for supersonic flow is hyperbolic the present paper shows that it is hyperbolic only when the velocity in the marching direction is supersonic in which case it also simultaneously guarantees the well posedness of the cau

a consistent boundary condition for vorticity streamfunction simulation of wall bounded vortex flow this paper was motivated by a computational effort to model the aircraft wake vortex hazard near airports those flows can usually be approximated as a pair of two dimensional incompressible vortices interacting with an infinite solid wall the present numerical model uses a vorticity streamfunction formulation to simulate the two dimensional problem since vorticity generation near the wall alters the evolution of the secondary and tertiary vortices directly accurate modeling of the wall region is critical a local relationship between vorticity and streamfunction for a wall boundary condition has been established from a new point of view by specifying the streamfunction at the boundary grid lifted away from the solid wall the new method has utilized taylor series expansions near the wall boundary although the resulting expressions are only first order accurate the critical importance of physical and mathematical consistency has been achieved numerical test cases are discussed in the paper

an analysis of a superconvergence result for a singularly perturbed boundary value problem we give a new proof that the el mistikawy and werle finite difference scheme is uniformly second order accurate for a nonselfadjoint singularly perturbed boundary value problem to do this we use exponential finite elements and a discretized green s function the proof is direct gives the nodal errors explicitly in integral form and involves much less computation than in previous proofs of the result

finite element solution of the unsteady navier stokes equations by a fractional step method abstract a finite element method is presented for the numerical simulation of time dependent incompressible viscous flows the method is based on a fractional step approach to the time integration of the navier stokes equations in which only the incompressibility condition is treated implicitly this leads to a computational scheme of extremely simple algorithmic structure that is particularly attractive for cost effective solutions of large scale problems numerical results indicate the versatility and effectiveness of the proposed method

a variational spectral method for the two dimensional stokes problem abstract a spectral galerkin scheme for solving the two dimensional stationary stokes problem is considered in vorticity stream function formulation the stokes problem is reduced to a biharmonic one a new formulation for the decomposition of the biharmonic operator is introduced main difficulty in this decomposition is to find the trace of the vorticity using the adjoint of the composition of trace operator and green s operator a variational form is obtained for computing the boundary value of the vorticity the discrete variational form preserves the properties such as coercivity and continuity of the continuous case which are important for our convergence analysis convergence of our scheme is proved and numerical experiments are performed for the steady driven cavity problem

galerkin spectral method for the vorticity and stream function equations a galerkin legendre spectral method for the solution of the vorticity and stream function equations in uncoupled form under no slip conditions in a square domain is presented which fully exploits the separation of variables in the two elliptic problems benefits from a nonsingular influence matrix and is able to solve the singular driven cavity problem modulo gibbs phenomenon even without regularizing the boundary condition at the corners

pseudo unsteady difference schemes for discontinuous solutions of steady state one dimensional fluid dynamics problems abstract this paper presents two unsteady difference schemes for one dimensional steady state shock solutions starting from the pseudo unsteady equations the so called explicit upwind scheme is actually an explicit forward time centered space difference scheme with an appropriate artificial viscosity term which depends on the directions of the characteristics this scheme considered as an iterative scheme has good convergence properties and leads to steady state numerical shocks each with at most one point of transition regardless of the position of the shock with respect to the mesh the mesh can be refined locally and computation continued locally to make the profile of the shock sharper to increase the rate of convergence the corresponding implicit scheme is presented results of numerical tests for a shock tube and a convergent divergent nozzle are given

integral conditions for the pressure in the computation of incompressible viscous flows abstract the problem of finding the correct conditions for the pressure in the time discretized navier stokes equations when the incompressibility constraint is replaced by a poisson equation for the pressure is critically examined it is shown that the pressure conditions required in a nonfractional step scheme to formulate the problem as a system of split second order equations are of an integral character and similar to the previously discovered integral conditions for the vorticity the novel integral conditions for the pressure are used to derive a finite element method which is very similar to that developed by glowinski and pironneau and is the finite element counterpart of the influence matrix method of kleiser and schumann

truncation error analysis of the finite volume method for a model steady convective equation abstract a fundamental error relation inherent in the gauss divergence theorem provided the basis for studying the truncation error of the finite volume method with cell center formulation for a model steady convective equation the consistency of the classical first and second order upwind schemes are proved to be seriously dependent on the grid distribution if the simple nature of the finite volume method is retained it is found that a consistent scheme for convective terms can not avoid the dissipation due to grid nonuniformity the geometrical interpretation of the diffusive terms such as vzo are undertaken classical finite volume methods which average an undefined 0 from its neighbouring nodes are shown to introduce false fluxes whenever grid nonuniformity is present a simple modification is proposed to cancel the false fluxes two test cases for the laplace equation illustrate the applicability of the proposed scheme however even with this modified scheme the consistency problem of the diffusion term is found to be seriously dependent on the grid uniformity

some new procedures for numerical solution of variably saturated flow problems persistent difficulties that arise in forming numerical solutions of variably saturated flow problems include controlling the stability of the nonlinear equation solvers and devising a reliable yet efficient method for determining the positions of seepage surfaces new techniques for addressing these problems are applied to a subdomain finite element discretization of the governing flow equations a series of test problems demonstrates that the techniques are reliable and efficient for a wide variety of problems

an analysis of a cell vertex finite volume method for a parabolic convection diffusion problem we examine a cell vertex finite volume method which is applied to a model parabolic convection diffusion problem by using techniques from finite element analysis local errors away from all layers are obtained in a seminorm that is related to but weaker than the l 2 norm

vorticity conditioning in the computation of two dimensional viscous flows abstract the problem of evaluating the boundary values of the vorticity in the calculation of two dimensional viscous flows is considered it is shown that the splitting of the fourth order equation for the stream function into two second order problems implies specific integral conditions which fix the abstract projection of the vorticity field with respect to the linear manifold of the harmonic functions these conditions are a direct consequence of the boundary conditions on the velocity and ensure satisfaction of physically essential conservation laws for the vorticity the discrete analogue of the projection conditions produces as many algebraic equations as the number of boundary points and requires the solution of an equal number of dirichlet problems in the particular case of stationary linearized equations stokes equations a direct i e noniterative method of solution is obtained steady and unsteady computational schemes relying on the projection conditions on the vorticity are introduced and extensive numerical results of finite difference calculations of the driven cavity model problem are reported and discussed

numerical solution of steady navier stokes problems using integral representations steady state incompressible navier stokes and continuity equations are recast into integral representations for the velocity and the vorticity vector using the concept of fundamental solutions integrals are discretized using finite fourier series approximations an efficient iterative procedure is established for the numerical solution of internal flow problems in two dimensions the iteration loop in this procedure is composed of a kinetic step and a kinematic step explicit computations are utilized in each step boundary vorticity values required for the kinetic step are determined from constraint equations obtained from the kinematic aspect of the problem expressed in the integral representation form several flow problems are studied numerically for various reynolds numbers flow structures are determined from numerical results obtained a bachelor s flow model of constant vorticity is verified numerically for high reynolds number closed streamline flows inside a circle

a finite volume scheme with shock fitting for the steady euler equations abstract analysis of two alternative finite volume formulations in respect of accuracy on non uniform meshes and number of spurious modes leads to a preference for the more compact cell vertex scheme over the cell centre scheme the resulting equations are solved iteratively by using a lax wendroff procedure as a smoother for a multigrid algorithm then application of boundary conditions in a natural way leads rapidly to all individual residuals being driven close to zero except at shocks at shocks the residuals should not be zero and a shock fitting procedure is introduced to avoid this inconsistency sharp accurate solutions on a relatively coarse mesh are obtained for a channel low problem in which the zierep singularity is displayed and for the naca 0012 aerofoil

truncation error and stability analysis of iterative and non iterative thomas gladwell methods for first order non linear differential equations the consistency and stability of a thomas gladwell family of multistage time stepping schemes for the solution of first order non linear differential equations are examined it is shown that the consistency and stability conditions are less stringent than those derived for second order governing equations second order accuracy is achieved by approximating the solution and its derivative at the same location within the time step useful flexibility is available in the evaluation of the non linear coefficients and is exploited to develop a new non iterative modification of the thomas gladwell method that is second order accurate and unconditionally stable a case study from applied hydrogeology using the non linear richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non iterative formulation

robust solution of richards equation for nonuniform porous media capillary pressure saturation relative permeability relations described using the van genuchten 1980 and mualem 1976 models for nonuniform porous media lead to numerical convergence difficulties when used with richards equation for certain auxiliary conditions these difficulties arise because of discontinuities in the derivative of specific moisture capacity and relative permeability as a function of capillary pressure convergence difficulties are illustrated using standard numerical approaches to simulate such problems we investigate constitutive relations interblock permeability nonlinear algebraic system approximation methods and two time integration approaches an integral permeability approach approximated by hermite polynomials is recommended and shown to be robust and economical for a set of test problems which correspond to sand loam and clay loam media

mixed covolume methods for quasi linear second order elliptic problems we consider covolume methods for the mixed formulations of quasi linear second order elliptic problems covolume methods for the mixed formulations of linear elliptic problem was first considered by russell rigorous block centered discretizations on irregular grids improved simulation of complex reservoir systems tech report 3 project report reservoir simulation research corporation tulsa ok 1995 and tested extensively in cai et al comput geosci 1 1997 pp 289 315 jones a mixed finite volume element method for accurate computation of fluid velocities in porous media ph d thesis university of colorado denver 1995 the analysis was carried out by chou and kwak siam j numer anal 37 2000 pp 758 771 for linear symmetric problems where they showed optimal error estimates in l2 norm for the pressure and in h div norm for the velocity in this paper we extend their results to quasi linear problems by following milner s argument math comp 44 1985 pp 303 320 through an adaptation of the duality argument of douglas and roberts math comp 44 1985 pp 39 52 for mixed covolume methods