on spectral multigrid methods for the time dependent navier stokes equations a new splitting scheme is proposed for the numerical solution of the time dependent incompressible navier stokes equations by spectral methods a staggered grid is used for the pressure improved intermediate boundary conditions are employed in the split step for the velocity and spectral multigrid techniques are used for the solution of the implicit equations

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

a comparison of time discretization schemes for two timescale problems in geophysical fluid dynamics abstract in many geophysical fluid modeling applications there exist two very different time scales essentially fast waves and slow vortices at the largest planetary scales inertial gravity waves propagate through the fluid at phase speeds much faster than particle velocities while at small scales fast acoustic modes coexist with much slower vortex motions if numerical models are to be affordable in this context schemes which do not explicitly resolve the fast motion must be devised although this has traditionally been done for strictly numerical reasons the result is physically equivalent to a temporal sub grid scale model examples of such schemes examined here are the semi implicit split explicit and linear propagator methods all of these methods fail to resolve the fast motion accurately but in very different ways recent progress in the field of rotating stratified turbulence is employed on a simplified two time scale version of the boussinesq equations in order to determine how well these schemes represent the evolution of the slow variables it is found that the split explicit and linear propagator methods exhibit spurious interactions between the slow modes and their numerically retarded fast modes in certain conditions when t is large the semi implicit method is able to maintain a frequency separation but it mimics the true evolution best when time filtering is applied

on the accuracy of fractional step methods for conservation laws in two dimensions in this paper it is shown that on the assumption of the initial data belonging to bv bf r 2 cap l infty bf r 2 the convergence rates in l 1 bf r 2 for both godunov s spatial splitting method and strang s splitting method for scalar conservation laws in two dimensional spaces are of o delta t 1 2 where delta t is the splitting time step the author illustrates by numerical experiments that for some two dimensional riemann problems the rate is of the order o delta t it is also shown by counterexamples that if the initial data belongs to l infty bf r 2 no error estimate of the type o delta t beta for any beta 0 is valid

the accuracy of the fractional step method we analyze the accuracy of the fractional step method of kim and moin j comput phys 59 1985 pp 308 323 for the incompressible navier stokes equations we show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method we also show that the pressure in any projection method can be at best first order accurate our analysis is simpler and more direct than the previous analyses of this method we also show that there is no numerical boundary layer for velocity or pressure but there is one for the auxiliary pressure variable

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

global c regularity for second order quasilinear elliptic euqations in divergence form on considere des equations elliptiques quasilineaires d ordre 2 sous forme divergence sous des conditions aux limites de dirihlet

scaling of constitutive relationships in unsaturated heterogeneous media a numerical investigation numerical models of groundwater flow in the unsaturated zone require relative conductivity and moisture retention curves at a discretization scale much larger than that at which these constitutive relationships are measured this study seeks to identify simple scaling laws linking parameters of block scale constitutive relationships with the parameters of their measurement scale counterparts represented here using gardner russo models the two parameters of these models saturated conductivity and pore size distribution parameter are viewed as cross correlated regionalized variables with isotropic or anisotropic spatial correlation structures and are simulated using the sequential gaussian method upscaled constitutive curves for steady gravity drainage through three dimensional heterogeneous domains are then constructed by repeating numerical flow experiments for a series of applied infiltration rates and plotting the resulting upscaled relative conductivities and average water saturations against average matric potentials the experiments reveal clear log linear relationships between upscaled relative conductivities and average matric potentials the slopes of which are used to define an upscaled pore size distribution parameter the reciprocal of this quantity which represents an upscaled characteristic capillary length is very well approximated by a spatial power average of its measurement scale values furthermore the averaging exponent is found to be the same as that associated with the upscaling of saturated conductivities over the same flow domains when average matric potentials are normalized using this upscaled characteristic capillary length relative conductivity and moisture saturation values for widely different flow field realizations collapse along well defined curves however for wet conditions the form of these curves differs from that of the models assumed at the measurement scale

avoiding order reduction when integrating reaction diffusion boundary value problems with exponential splitting methods abstract in this paper we suggest a technique to avoid order reduction in time when integrating reaction diffusion boundary value problems under non homogeneous boundary conditions with exponential splitting methods more precisely we consider lie trotter and strang splitting methods and dirichlet neumann and robin boundary conditions beginning from an abstract framework in banach spaces a thorough error analysis after full discretization is performed and some numerical results are shown which corroborate the theoretical results

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

discontinuous parameter estimates with least squares estimators we discuss weighted least squares estimates of ill conditioned linear inverse problems where weights are chosen to be inverse error covariance matrices least squares estimators are the maximum likelihood estimate for normally distributed data and parameters but here we do not assume particular probability distributions weights for the estimator are found by ensuring its minimum follows a g 2 distribution previous work with this approach has shown that it is competitive with regularization methods such as the l curve and generalized cross validation gcv 20 in this work we extend the method to find diagonal weighting matrices rather than a scalar regularization parameter diagonal weighting matrices are advantageous because they give piecewise smooth least squares estimates and hence are a mechanism through which least squares can be used to estimate discontinuous parameters this is explained by viewing least squares estimation as a constrained optimization problem results with diagonal weighting matrices are given for a benchmark discontinuous inverse problem from 13 in addition the method is used to estimate soil moisture from data collected in the dry creek watershed near boise idaho parameter estimates are found that combine two different types of measurements and weighting matrices are found that incorporate uncertainty due to spatial variation so that the parameters can be used over larger scales than those that were measured

stability analysis of the numerical method of characteristics applied to a class of energy preserving hyperbolic systems part ii nonreflecting boundary conditions abstract we show that imposition of non periodic in place of periodic boundary conditions bc can alter stability of modes in the method of characteristics moc employing certain ordinary differential equation ode numerical solvers thus using non periodic bc may render some of the moc schemes stable for most practical computations even though they are unstable for periodic bc this fact contradicts a statement found in some textbooks and known as part of the babenko gelfand criterion that an instability detected by the von neumann analysis for a given numerical scheme implies an instability of that scheme with arbitrary i e non periodic bc we explain the mechanism behind this contradiction which lies in a certain property of the scheme s eigenmodes that is assumed by the babenko gelfand criterion but does not hold for eigenmodes of some of the moc based schemes we also show that and explain why for the moc employing some other ode solvers stability of the modes may indeed not be improved by non periodic bc as the babenko gelfand criterion implies

numerical analyses of operator splitting algorithms for the two dimensional advection diffusion equation abstract holly and preissmann s hp scheme has been the basis of a large number of operator splitting algorithms for the solution of the advection diffusion equation however these algorithms including hp are first order accurate in time due to splitting errors error analyses of these algorithms incorporating splitting error and errors resulting from numerical solutions of the split advection and diffusion equations are lacking in this paper error analysis of a second order accurate adaptation of the hp scheme ahp is presented for the two dimensional advection diffusion equation a modified ahp scheme mahp is suggested to remove the ad hoc nature of boundary conditions during the diffusion step of computations in both hp and ahp as boundary conditions specified for the advection diffusion equation are not applicable to the split equations second order accurate boundary conditions for the split advection and diffusion equations are derived an analysis of numerical dispersion and dissipation associated with the numerical procedure for the advection equation is presented the analysis establishes a criterion so that computational errors are small in two dimensional advection dominated transport problems several numerical examples are presented to verify the numerical analyses presented in the paper in addition a review of the current status of operator splitting algorithms for the advection diffusion equation is presented the objective of the review is to identify the issues that have not been addressed in the previous studies

numerical methods based on additive splittings for hyperbolic partial differential equations we derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds these techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales some of which are sometimes treated exactly the efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator this is analyzed and a technique is discussed for reducing this error through a simple change of variables a procedure for generating the appropriate boundary data for the intermediate solutions is also presented

a comparison of fourier pseudospectral methods for the solution of the korteweg de vries equation abstract various fourier pseudospectral methods are used to approximate the korteweg de vries equation the methods differ in their treatment of the time discretisation the methods are compared for computational efficiency using the 1 and 2 soliton test problems they are also compared from an accuracy viewpoint using the zabusky kruskal recurrence problem

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

implementation of the three dimensional turning bands random field generator numerical techniques to generate replicates of spatially correlated random fields are often used to synthesize sets of highly variable physical quantities in stochastic models of naturally heterogeneous systems within the realm of hydrologic research for example such tools are widely used to develop hypothetical rainfall distributions hydraulic conductivity fields fracture set properties and other surface or subsurface flow parameters the turning bands method is one such algorithm which generates two and three dimensional fields by combining values found from a series of one dimensional simulations along lines radiating outward from a coordinate origin previous work with two dimensional algorithms indicates that radial lines evenly spaced about the unit circle lead to enhanced convergence properties the same can be said for the three dimensional models but it is more difficult to choose an arbitrary number of evenly spaced lines about the unit sphere the current investigation shows that the use of larger numbers of randomly oriented lines 100 can enhance the performance of the three dimensional algorithm this improved performance is needed to effectively simulate problems characterized by full three dimensionality and or anisotropy in either monte carlo or single realization applications use of a large number of lines will also reduce the presence of a distortion effect manifested as linelike patterns in the field increased computational costs can be reduced by employing a fast fourier transform technique to generate the line processes

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

intermediate boundary conditions for time split methods applied to hyperbolic partial differential equations determination des conditions aux limites intermediaires appropriees pour les methodes de decomposition appliquees aux equations aux derivees partielles hyperboliques

the convergence of two numerical schemes for the navier stokes equations this paper deals with some convergence stability results concerning two numerical methods for solving the incompressible nonstationary navier stokes equations the algorithms are of a particular kind in what regards time discretization more precisely of the peaceman rachford and the strang type resp and have been obtained by modifying slightly the numerical treatment of the nonlinear terms in other schemes due to glowinski et al 1980 we first describe the full discretization of the homogeneous dirichlet problem using a general external approximation of the spatial functional spaces involved a particular and simple choice of such an approximation is the standardp 2 lagrange finite element for the velocity field when the fluid is bidimensional then we establish and prove convergence and stability and make some comments on the numerical treatment of other generally nonhomogeneous boundary conditions the theoretical results show that the schemes are at least conditionally stable and convergent which justifies the success of glowinski s methods

multilevel schemes for the shallow water equations in this paper we study a number of multilevel schemes for the numerical solution of the shallow water equations new schemes and new perspectives of known schemes are examined we consider the case of periodic boundary conditions spatial discretization is obtained using a fourier spectral galerkin method for the time integration two strategies are studied the first one is based on scale separation and we choose the time scheme explicit or semi implicit as a function of the spatial scales multilevel schemes the second approach is based on a splitting of the operators and we choose the time integration method as a function of the operator considered multistep or fractional schemes the numerical results obtained are compared with the explicit reference scheme leap frog scheme and with the semi implicit scheme leap frog scheme with crank nicholson scheme for the gravity terms both computed with a similar mesh the drawback of the explicit reference scheme being the numerical stability constraint on the time step and the drawback of the semi implicit scheme being the dispersive error the aim with the new schemes is to obtain schemes with less dispersive error than the semi implicit scheme and with better stability properties than the explicit reference scheme the numerical results obtained show that the schemes proposed allow one to reduce the dispersive error and to increase the numerical stability at reduced cost

generalized multiscale finite element method for multicontinua unsaturated flow problems in fractured porous media abstract in this paper we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media the mathematical model is described by the system of richards equations for each continuum that is coupled by the specific transfer term to illustrate the idea of our approach we consider a dual continua background model with discrete fractures networks that is generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media we present fine grid approximation based on the finite element method and discrete fracture model dfm approach in this model we construct an unstructured fine grid that takes into account complex fracture geometries for two and three dimensional formulations due to construction of the unstructured grid the fine grid approximation leads to the very large system of equations for reduction of the discrete system size we develop a multiscale method for coarse grid approximation of the coupled problem using generalized multiscale finite element method gmsfem in this method we construct coupled multiscale basis functions that are used to construct highly accurate coarse grid approximation the multiscale method allowed us to capture detailed interactions between multiple continua the adaptive approach is investigated where we consider two approaches for multiscale basis functions construction 1 based on the spectral characteristics of the local problems and 2 using simplified multiscale basis functions we investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations we present a comparison of the relative error for different number of basis functions and for adaptive approach numerical results illustrate that the presented method provides accurate solution of the unsaturated multicontinua problem on the coarse grid with huge reduction of the discrete system size

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

an operator splitting algorithm for coupled one dimensional advection diffusion reaction equations abstract an operator splitting algorithm for a system of one dimensional advection diffusion reaction equations describing the transport of non conservative pollutants is presented in this paper the algorithm is a strang type splitting procedure incorporating contributions from the inhomogeneous terms by the duhamel s principle the associated homogeneous equations are split into advection diffusion and reaction equations and solved by a backward method of characteristic a finite element method and an explicit runge kutta method respectively the boundary conditions applicable to the split equations are derived numerical analyses of the algorithm consisting of the stability the accuracy and the convergence of the solution procedure are presented the composite algorithm is second order accurate in time and space and conditionally stable the numerical characteristics of the algorithm are demonstrated by several examples

higher order muscl scheme for transport equation originating in a neuronal model we consider a transport equation with mixed boundary conditions dirichlet and neumann originating in a neuronal model this equation contains point wise delay and advanced argument the objective of this paper is to construct and analyze higher order muscl scheme to find a numerical solution the developed scheme is stable and convergent the importance of this scheme is that it is valid for large as well as for small values of point wise delay and advance some test examples are included to examine the behavior of the solution and to verify the order of convergence the effect of point wise delay and advance arguments on the solution is shown graphically

algorithms for solving richards equation for variably saturated soils previous work has shown that a one dimensional water content based finite difference algorithm can offer improved cpu efficiency in modeling infiltration into very dry heterogeneous soil when compared to similar pressure based algorithms the usefulness of the water content based method is limited however since it cannot be applied to saturated soils in this paper we develop two new methods which retain the advantages of water content based methods while still permitting fully saturated conditions in the first method we define a new variable for the transformed richards equation which has the characteristics of water content when soil is unsaturated and of pressure when soil is at or near saturation in addition to the transformed richards equation method an improved pressure based method which uses flux updating is presented both methods are implemented in algorithms using a preconditioned conjugate gradient method equation solver the performance of the algorithms is compared to that of an algorithm based on a mixed form of richards equation using modified picard iteration two test cases are examined the first test case examines two dimensional infiltration into very dry heterogeneous soil which remains unsaturated throughout the simulation period the second test case examines a two dimensional developing perched water table results indicate that the new methods retain the advantages of the water content based method for fully unsaturated heterogeneous problems while allowing for fully saturated conditions for the unsaturated problem the new algorithms were from 16 to 59 times faster than the algorithm based on the mixed form of richards equation using modified picard iteration for the saturated problem the new algorithms were from 6 to 25 times faster than the mixed base algorithm results also indicate that the performance of the mixed base algorithm could be significantly improved if a more effective adaptive time step were developed

a modified gautschi s method without order reduction when integrating boundary value nonlinear wave problems abstract in this paper we analyse the order reduction which turns up when integrating nonlinear wave problems with non homogeneous and time dependent boundary conditions with the well known gautschi s method moreover a technique is suggested to avoid that order reduction so that the classical local order 4 and global order 2 are recovered on the other hand the usual approximation for the derivative which is used together with this method is also analysed and a substantial improvement is suggested some numerical results are shown which corroborate the performed analysis

a robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media we study a nonlinear degenerate convection diffusion model problem having an application in groundwater aquifer and petroleum reservoir simulation the true solution typically possesses low regularity and therefore special numerical techniques for its approximation are needed we design a robust efficient and reliable linear relaxation approximation scheme we prove the convergence of iterations at each time step in the h 1 omega norm finally the convergence of the approximate solution in corresponding functional spaces to its exact counterpart for the parabolic problem is shown

statistical analysis of spatial variability in unsaturated flow parameters core scale estimates of soil parameters of the gardner russo and van gemachten models of the hydraulic conductivity and water retention functions were obtained for 417 undisturbed soil cores taken from a wall of a trench 20 m long 2 5 m deep using a procedure based on inverse problem methodology these estimates were used to evaluate the first two statistical moments of the underlying random space functions rsfs using the restricted maximum likelihood estimation procedure coupled with the weighted least squares procedure to estimate parameters of models of the covariance and the drift functions of the pertinent rsfs the fitted models were used to evaluate the mean and covariance functions of the hydraulic conductivity and water rentention functions for given water saturations covariance functions of log saturated conductivity log ks and the shape parameters of the gardner russo and van genuchten models exhibited statistical anisotropy characterized by aspect ratios that vary between 3 to 4 correlation scales of log ks were larger than those of the shape parameters consequently the product of the variance of log unsaturated conductivity and its correlation scale remained essentially invariant for a considerable range of water saturation the implications of these results regarding stochastic modeling of transport in heterogeneous porous formations and possible applications of the results of this study are discussed briefly

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

accuracy of operator splitting for advection dispersion reaction problems an operator splitting approach is often used for the numerical solution of advection dispersion reaction problems operationally this approach advances the solution over a single time step in two stages one involving the solution of the nonreactive advection dispersion equation and the other the solution of the reaction equations the first stage is usually solved with a finite difference finite element or related technique while the second stage is normally solved with an ordinary differential equation integrator the only generally published guidelines on numerical accuracy suggest that the discretization errors associated with each stage must be small in order to achieve high accuracy of the overall solution however in this note we demonstrate that there is an inherent mass balance error present in the operator splitting algorithm for problems involving continuous mass influx boundary conditions the mass balance error does not exist for instantaneous mass input problems these conclusions are based upon analysis of a simple first order decay problem for which each stage of the calculation can be performed analytically i e without discretization error for this linear decay problem we find that the product of the first order decay coefficient k times t must be less than approximately 0 1 in order for the mass balance error to be less than 5 we also present a variant of the normal operator splitting algorithm in which the order of solving the advection dispersion and reaction operators is reversed at each time step this modification reduces the mass balance error by more than a factor of 10 for a wide range of k t values

piecewise linearized and linearized θ methods for ordinary and partial differential equations lnitial and boundary value problems appear frequently in many branches of physics in this paper several numerical methods based on linearization techniques for solving these prob lems are reviewed first piecewise linearized methods and linearized 0 methods are considered for the solution of initial value problems in ordinary differential equations second piecewise linearized techniques for two point boundary value problems in ordinary differential equations are developed and used in conjunction with a shooting method in order to overcome the lack of convergence asso ciated with shooting piecewise linearized methods which provide piecewise analytical solutions and yield nonstandard finite difference schemes axe presented third methods of lines in either space or time for the solution of one dimensional convection reaction diffusion problems that transform the original problem into an initial or boundary value one are reviewed methods of lines in time that result in boundary value problems at each time step can be solved by means of the techniques described here whereas methods of lines in space that yield initial value problems and employ ei ther piecewise linearized techniques or linearized 0 methods in time are also developed finally for multidimensional problems approximate factorization methods are first used to transform the multi dimensional problem into a sequence of one dimensional ones which are then solved by means of the linearized and piecewise linearized methods presented here 2003 elsevier science ltd all rights reserved

a multiscale finite element linearization scheme for the unsaturated flow problems in heterogeneous porous media 1 numerical solution of unsaturated water flow in heterogeneous porous media is difficult not only because of the spatial variability in the parameters used to characterize the relevant physical properties of the natural porous media but also because of the mathematical problems that arise in dealing with the nonlinearities in this study a multiscale finite element linearization scheme is presented for effectively simulating unsaturated flow in heterogeneous porous media spanning over many scales the central goal of this method is to obtain the large scale solution of richards equation with heterogeneous coefficients accurately and efficiently on a coarse grid without resolving all the small scale details the underlying idea is to employ the slodicka linear relaxation approximation scheme to address the nonlinear characteristics of the equation and to use the multiscale finite element base functions to account for the spatial variability in the equation coefficients we describe the principle for constructing such a method and give an algorithm for implementing it numerical experiments are carried out for the unsaturated flow equation with both periodic and randomly generated lognormal hydraulic conductivity to demonstrate the efficiency and accuracy of the proposed method

n th order operator splitting schemes and nonreversible systems this paper is concerned with partitioned nth order accurate split operator schemes built using m distinct solution operators for semidiscrete equations of the form frac d dt u j f j 1 u k cdots f j m u k which arise among others from constant coefficient parabolic partial differential equations we prove that for every n geq 3 and m geqslant 2 each solution operator must be applied for at least one backward fractional time step during each complete time step this result has important consequences for applications to the complex ginzburg landau equation with periodic boundary conditions and other partial differential equations with both reversible and nonreversible components furthermore for the special case of n 3 and m 2 we analytically determine all possible schemes

an improvement of fractional step methods for the incompressible navier stokes equations abstract a numerical method for computing three dimensional unsteady incompressible flows is presented the method is a predictor corrector technique combined with a fractional step method each time step is advanced in three sub steps the novel feature of the present scheme is that the poisson equation for the pressure is solved only at the final sub step resulting in substantial savings in computing time it is shown that the method allows a larger cfl number and reduces the computing cost without loss of accuracy by satisfying the continuity equation only at the last sub step numerical solutions for the decaying vortices and flow over a backward facing step are obtained and compared with analytical and other numerical results

rational approximants and numerical methods for initial value problems abstract we survey three instances of an interaction between numerical analysis of initial value problems and rational approximation theory rational functions can be used to design numerical algorithms for differential equations assist in their implementation and analyse their behaviour and our three examples have been designed to highlight all these aspects they are a order stars and their application to the derivation of bounds on the order of stable methods for linear hyperbolic differential equations b dimensional splitting and its implementation to solve efficiently nonlinear parabolic differential equations c estimates and bounds on the truncation error when the matrix exponential is approximated by rational functions

split step spectral schemes for nonlinear dirac systems abstract the paper considers split step spectral schemes for the numerical integration of nonlinear dirac systems in 1 1 dimensions proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split step finite difference algorithms

accuracy of product formula algorithms abstract to obtain accurate solutions to the time dependent schrodinger equation a product formula algorithm has been proposed in literature which is both explicit and unconditionally stable in this paper the accuracy of this algorithm is considered for a few basic problems in mathematical physics viz the convection equation and the diffusion equation in addition to the schrodinger equation from a theoretical analysis which is confirmed by numerical experiments it is concluded that the product formula method can indeed produce accurate solutions but only for small time steps so that the unconditional stability is not of very much use a comparison is also made with standard finite difference methods such as leap frog and crank nicholson

optimal time splitting for two and three dimensional navier stokes equations with mixed derivatives a new explicit time splitting algorithm has been developed for finite difference modelling of the full two and three dimensional time dependent compressible viscous navier stokes equations of fluid mechanics the scheme is optimal in the sense that the split operators achieve their maximum allowable time step i e the corresponding courant number the algorithm allows a conservation form formulation stability is proven analytically and verified numerically in proving stability it was shown that all nine matrix coefficients of the navier stokes equations are simultaneously symmetrizable by a similarity transformation two such transformations and their resulting symmetric matrix coefficients are presented explicitly

numerical homogenization of the richards equation for unsaturated water flow through heterogeneous soils homogenized equations and the corresponding effective constitutive relations are generally necessary for numerically modeling large scale unsaturated flow processes in soils recently based on the kirchhoff transformation and the two scale convergence theory a homogenization method for the richards equation with the mualem van genuchten model has been proposed with a constant model parameter relating to the inverse of the air entry pressure and the soil pore size distribution the homogenized model is computationally efficient and convenient to use because of its explicit expression in this study we generalize this method allowing to be a spatially distributed random field and proposing a homogenized richards equation in the mixed form h under the condition that the effective hydraulic conductivity tensor is diagonal this generalization eliminates the limitation of a constant in practical applications the proposed homogenized model is meaningful in most situations because the flow problems are influenced mainly by the diagonal terms of conductivity and the off diagonal terms are often neglected two dimensional numerical tests are conducted in soil profiles with different degrees of spatial heterogeneity structure to illustrate that the homogenized model can capture the fine scale flow behaviors on coarse grids effectively homogenization for the richards equation with other two commonly used constitutive relations the brooks corey model and the gardner russo model is also illustrated in this study