the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

adaptive biorthogonal spline schemes for advection reaction equations in this paper based on eulerian lagrangian localized adjoint method ellam we use biorthogonal spline wavelets to develop numerical schemes for multidimensional advection reaction equations the derived schemes produce accurate numerical solutions even if large time steps are used these schemes are explicit but unconditionally stable they also have the ability to carry out adaptive compression while preserving the total mass numerical experiments including observations on their convergence rates are presented to show the strong potential of these methods

second order expressions for velocity moments in two and three dimensional statistically anisotropic media second order expressions are derived for the mean and covariance of steady state seepage velocity under mean uniform flows in infinite two and three dimensional domains the order of approximation is defined in terms of the variance 2 of a statistically homogeneous and anisotropic natural log hydraulic conductivity field y with a gaussian spatial autocorrelation function results show that second order mean velocity either exceeds or is close to its first order counterpart depending on anisotropy head fluctuations of order larger than affect second order velocity moments to the same extent as do head fluctuations of order in virtually all cases hence neglecting the former renders the results nonasymptotic velocity variances are generally larger when approximated consistently to second than to first order the ratio between second and first order variance approximations is larger in three than in two dimensions larger for transverse than for longitudinal velocity and increases with 2 anisotropy has a significant effect on second order velocity variance second order effects have the greatest influence on longitudinal velocity variance at extreme anisotropy ratios and on transverse velocity variance in isotropic domains

a one time truncate and encode multiresolution stochastic framework in this work a novel adaptive strategy for stochastic problems inspired from the classical email protected s framework is presented the proposed algorithm allows building in a very general manner stochastic numerical schemes starting from a whatever type of deterministic schemes and handling a large class of problems from unsteady to discontinuous solutions its formulations permits to recover the same results concerning the interpolation theory of the classical multiresolution approach but with an extension to uncertainty quantification problems the present strategy permits to build numerical scheme with a higher accuracy with respect to other classical uncertainty quantification techniques but with a strong reduction of the numerical cost and memory requirements moreover the flexibility of the proposed approach allows to employ any kind of probability density function even discontinuous and time varying without introducing further complications in the algorithm the advantages of the present strategy are demonstrated by performing several numerical problems where different forms of uncertainty distributions are taken into account such as discontinuous and unsteady custom defined probability density functions in addition to algebraic and ordinary differential equations numerical results for the challenging 1d kraichnan orszag are reported in terms of accuracy and convergence finally a two degree of freedom aeroelastic model for a subsonic case is presented though quite simple the model allows recovering some physical key aspect on the fluid structure interaction thanks to the quasi steady aerodynamic approximation employed the injection of an uncertainty is chosen in order to obtain a complete parameterization of the mass matrix all the numerical results are compared with respect to classical monte carlo solution and with a non intrusive polynomial chaos method

a wavelet mra based adaptive semi lagrangian method for the relativistic vlasov maxwell system in this paper we present a new method for the numerical solution of the relativistic vlasov maxwell system on a phase space grid using an adaptive semi lagrangian method the adaptivity is performed through a wavelet multiresolution analysis which gives a powerful and natural refinement criterion based on the local measurement of the approximation error and regularity of the distribution function therefore the multiscale expansion of the distribution function allows to get a sparse representation of the data and thus save memory space and cpu time we apply this numerical scheme to reduced vlasov maxwell systems arising in laser plasma physics interaction of relativistically strong laser pulses with overdense plasma slabs is investigated these vlasov simulations revealed a rich variety of phenomena associated with the fast particle dynamics induced by electromagnetic waves as electron trapping particle acceleration and electron plasma wavebreaking however the wavelet based adaptive method that we developed here does not yield significant improvements compared to vlasov solvers on a uniform mesh due to the substantial overhead that the method introduces nonetheless they might be a first step towards more efficient adaptive solvers based on different ideas for the grid refinement or on a more efficient implementation here the vlasov simulations are performed in a two dimensional phase space where the development of thin filaments strongly amplified by relativistic effects requires an important increase of the total number of points of the phase space grid as they get finer as time goes on the adaptive method could be more useful in cases where these thin filaments that need to be resolved are a very small fraction of the hyper volume which arises in higher dimensions because of the surface to volume scaling and the essentially one dimensional structure of the filaments moreover the main way to improve the efficiency of the adaptive method is to increase the local character in phase space of the numerical scheme by considering multiscale reconstruction with more compact support and by replacing the semi lagrangian method with more local in space numerical scheme as compact finite difference schemes discontinuous galerkin method or finite element residual schemes which are well suited for parallel domain decomposition techniques

adaptive multiresolution approach for two dimensional pdes a multiresolution adaptive approach for the solution of two dimensional partial differential equations pdes is presented this methodology is a multidimensional extension of that presented in a previous work comput methods appl mech engrg 191 2002 3909 the method proposed is unconditionally stable by incorporating convection differencing schemes with the tvd property and the grid is dynamically adapted so that higher spatial resolution is automatically allocated to domain regions where strong gradients are observed the two desired properties of any pde solver stability and accuracy are therefore retained numerical results for four test problems are presented which serve to demonstrate the robustness and cost effectiveness of the method

grid structure impact in sparse point representation of derivatives in the sparse point representation spr method the principle is to retain the function data indicated by significant interpolatory wavelet coefficients which are defined as interpolation errors by means of an interpolating subdivision scheme typically a spr grid is coarse in smooth regions and refined close to irregularities furthermore the computation of partial derivatives of a function from the information of its spr content is performed in two steps the first one is a refinement procedure to extend the spr by the inclusion of new interpolated point values in a security zone then for points in the refined grid such derivatives are approximated by uniform finite differences using a step size proportional to each point local scale if required neighboring stencils are not present in the grid the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme using the cubic interpolation subdivision scheme we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the spr for this purpose we prove some results concerning the local behavior of such wavelet reconstruction operators which stand for spr grids having appropriate structures this statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at spr grid points consequently in addition to the refinement strategy our analysis indicates that some care must be taken concerning the grid structure in order to keep the truncation error under a certain accuracy limit illustrating results are presented for 2d maxwell s equation numerical solutions

adaptive multiresolution weno schemes for multi species kinematic flow models multi species kinematic flow models lead to strongly coupled nonlinear systems of first order spatially one dimensional conservation laws the number of unknowns the concentrations of the species may be arbitrarily high models of this class include a multi species generalization of the lighthill whitham richards traffic model and a model for the sedimentation of polydisperse suspensions their solutions typically involve kinematic shocks separating areas of constancy and should be approximated by high resolution schemes a fifth order weighted essentially non oscillatory weno scheme is combined with a multiresolution technique that adaptively generates a sparse point representation spr of the evolving numerical solution thus computational effort is concentrated on zones of strong variation near shocks numerical examples from the traffic and sedimentation models demonstrate the effectiveness of the resulting weno multiresolution weno mrs scheme

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

higher order effects on flow and transport in randomly heterogeneous porous media advective solute transport in nonuniform geologic media is generally nonlocal and non fickian neuman 1993 cushman and ginn 1993 in statistically homogeneous log conductivity fields under uniform mean flow the transport is expected to become asymptotically local and fickian at late time during the earlier preasymptotic regime macrodispersivity a measure of the rate at which a plume spreads is expected to vary with solute residence time a first order linear in the natural log hydraulic conductivity variance 2 analysis of this variation has been performed by dagan 1984 1987 1988 he found that when local dispersion is neglected the longitudinal macrodispersivity increases monotonically from zero toward a constant asymptote however the transverse macrodispersivity first increases from zero to a peak value then decreases monotonically toward zero the first order asymptotic analyses by winter 1982 gelhar and axness 1983 and winter et al 1984 also yield zero transverse macrodispersivity when local dispersion is disregarded

effective hydraulic conductivity of bounded strongly heterogeneous porous media we develop analytical expressions for the effective hydraulic conductivity ke of a three dimensional heterogeneous porous medium in the presence of randomly prescribed head and flux boundaries the log hydraulic conductivity y forms a gaussian statistically homogeneous and anisotropic random field with an exponential autocovariance by effective hydraulic conductivity of a finite volume in such a field we mean the ensemble mean expected value of all random equivalent conductivities that one could associate with a similar volume under uniform mean flow we start by deriving a first order approximation of an exact expression developed in 1993 by neuman and orr we then generalize this to strongly heterogeneous media by invoking the landau lifshitz conjecture upon evaluating our expressions we find that ke decreases rapidly from the arithmetic mean ka toward an asymptotic value as distance between the prescribed head boundaries increases from zero to about eight integral scales of y the more heterogeneous is the medium the larger is ke relative to its asymptote at any given separation distance our theory compares well with published results of spatially power averaged expressions and with a first order expression developed intuitively by kitanidis in 1990

an adaptive multiresolution method on dyadic grids we propose a modified adaptive multiresolution scheme for solving d dimensional hyperbolic conservation laws which is based on cell average discretization in dyadic grids adaptivity is obtained by interrupting the refinement at the locations where appropriate scale wavelet coefficients are sufficiently small one important aspect of such a multiresolution representation is that we can use the same binary tree data structure for domains of any dimension the tree structure allows us to succinctly represent the data and efficiently navigate through it dyadic grids also provide a more gradual refinement as compared with the traditional quad trees 2d or oct trees 3d that are commonly used for multiresolution analysis we show some examples of adaptive binary tree representations with significant savings in data storage when compared to quad tree based schemes as a test problem we also consider this modified adaptive multiresolution method using a dynamic binary tree data structure applied to a transport equation in 2d domain based on a second order finite volume discretization highlights we describe an adaptive multiresolution method for d dimensional pdes the method is based on cell average discretization on a dyadic grid the grid refinement is dynamically adapted to the current solution the same binary tree structure is used for domains of any dimension the method is tested on a transport problem in a two dimensional domain

bridge functions near the liquid vapor coexistence curve in binary lennard jones mixtures we have carried out extensive molecular dynamics simulation studies of binary lennard jones mixtures to calculate directly the bridge function at state points lying in a very narrow single fluid phase region between the vapor liquid and solid liquid coexistence lines lamm and hall fluid phase equilib 182 37 2001 194 197 197 2002 by varying the density close to the liquid vapor coexistence line significant deviations are observed at intermediate distances between the simulated bridge function and two widely used approximate closures in the integral equation theory of liquids viz the hybrid mean spherical approximation and the duh henderson closures the overall qualitative agreement remains the same with small variation in temperature that brings the system closer to either the liquid vapor or liquid solid coexistence curve we also report a comparison of the direct and indirect correlation functions obtained from our simulation studies as well as from the integral equation theory of liquids our results emphasize the need for developing new closures applicable to binary fluid mixtures over a wide range of thermodynamic parameters

critical end point behavior in a binary fluid mixture we consider the liquid gas phase boundary in a binary fluid mixture near its critical end point using general scaling arguments we show that the diameter of the liquid gas coexistence curve exhibits singular behaviour as the critical end point is approached this prediction is tested by means of extensive monte carlo simulations of a symmetrical lennard jones binary mixture within the grand canonical ensemble the simulation results show clear evidence for the proposed singularity as well as confirming a previously predicted singularity in the coexistence chemical potential fisher and upton phys rev lett 65 2402 1990 the results suggest that the observed singularities particularly that in the coexistence diameter should also be detectable experimentally

variational multiscale stabilized fem formulations for transport equations stochastic advection diffusion and incompressible stochastic navier stokes equations an extension of the deterministic variational multiscale vms approach with algebraic subgrid scale sgs modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic navier stokes equations the stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method ssfem generalized polynomial chaos and karhunen loeve expansion techniques are used for representation of uncertain quantities the proposed stabilized method is then applied to various standard advection diffusion and fluid flow examples with uncertainty in essential boundary conditions comparisons are drawn between the numerical solutions and monte carlo analytical solutions wherever possible

upscaling of permeability of anisotropic heterogeneous formations 1 the general framework in numerical simulations of flow through heterogeneous formations the domain is partitioned into numerical elements the solution requires assigning physical properties to each numerical block the process of transferring information from the scale of actual heterogeneity to that of the numerical elements is known as upscaling we consider steady one phase flow and the only property of interest is the permeability which is regarded as a random space function and the same is true for the upscaled conductivity in this paper we establish the necessary and sufficient conditions to be satisfied by upscaling the necessary conditions are expressed with the aid of the global response of the formation e g the total flux caused by a constant pressure head drop applied on the boundaries the requirement is that the expected value and the variance of the total flux are the same in the actual and upscaled domains these are supplemented by local conditions which stipulate that space averages of flow variables over the numerical blocks in the upscaled domain tend to those in the actual formation when the elements size tends to 0 the result of this analysis is that the upscaled permeability must be such as to ensure the equality in a statistical sense between the space averaged dissipation in the two media by assuming an unbounded domain average uniform flow and stationarity simplified relationships between the moments of the dissipation in the upscaled and actual media are derived it is shown that the effective permeability of the two media is the same the approach is illustrated for stratified formations for which exact composition rules are possible even in this simple case the upscaled permeability is a tensor characterized at second order by two mean components and three covariance functions

upscaling of permeability of anisotropic heterogeneous formations 3 applications the general results of indelman and dagan this issue a b are applied to some cases of interest in applications the exact solution of upscaling of stratified formations is used in order to assess the range of applicability of the first order analysis the explicit expressions of the two first moments of the upscaled permeability are derived for media of two dimensional isotropic and three dimensional axisymmetric actual log permeability of gaussian covariance and for parallelepipedic rectangular for two dimensional averaging elements the effects of the actual log permeability covariance and of the anisotropy of partition elements on the upscaled permeability statistics are analyzed the asymptotic behavior of the upscaled permeability for large and small blocks is discussed for general heterogeneity the strong dependence of the small element limit on the properties of actual heterogeneity and on the shape of the elements is illustrated by typical examples in contrast the homogenization of the upscaled media by large blocks depends primarily on the volume area for two dimensions of the partition element relative to the one based on the correlation scale of the log permeability

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

fully adaptive multiresolution finite volume schemes for conservation laws the use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by a harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation in this approach the solution is still represented at each time step on the finest grid resulting in an inherent limitation of the potential gain in memory space and computational time the present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes in which the solution is represented and computed in a dynamically evolved adaptive grid a crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages several solutions to this problem are proposed analyzed and compared in terms of accuracy and complexity

probabilistic characterization of transport in heterogeneous media abstract the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented thus the hydraulic processes are discretized using the spectral karhunen loeve expansion this expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes these amplitudes form a basis in the manifold associated with the random processes the concentrations over the whole domain are also random processes with unknown probabilistic structure these processes are represented using the polynomial chaos basis this is a basis in the functional space described by all second order random variables the deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator once the spatio temporal variation of the concentrations has been specified in terms of the polynomial chaos expansion individual realizations can be readily computed

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

effective permittivity of log normal isotropic random media a method of deriving the effective permittivity epsilon eff of log normal d dimensional isotropic media is developed the approach is based on applying the regularized green function technique for the laplace operator the feasibility of exact determining of the effective permittivity is analysed by deriving epsilon eff at third order in the log permittivity variance it is shown that the 3d effective permittivity is a functional of the spectrum of inhomogeneities and therefore cannot be expressed by a closed formula in particular the formula for the effective permittivity discussed previously by landau and lifshitz 1960 and matheron 1967 deviates from the exact one in the third order term

well balanced adaptive mesh refinement for shallow water flows well balanced shock capturing wbsc schemes constitute nowadays the state of the art in the numerical simulation of shallow water flows they allow to accurately represent discontinuous behavior known to occur due to the non linear hyperbolic nature of the shallow water system and at the same time numerically maintain stationary solutions in situations of practical interest these schemes often need to be combined with some kind of adaptivity in order to speed up computing times in this paper we discuss what ingredients need to be modified in a block structured amr technique in order to ensure that when combined with a wbsc scheme the so called water at rest stationary solutions are exactly preserved

nonlocal properties of nonuniform averaged flows in heterogeneous media the properties of nonuniform average potential flows in media of stationary random conductivity are studied the mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head the averaged darcy s law determines the effective conductivity as an integral operator of the convolution type relating the mean velocity to the mean head gradient in a nonlocal way in fourier domain the mean velocity is proportional to the mean head gradient the coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments it is shown that the effective conductivity cannot be defined uniquely for potential flows however this nonuniqueness does not affect the spatial distribution of the mean head and velocity analytical expressions of the effective conductivity tensor are derived for two and three dimensional flows and for exponential and gaussian correlations of isotropic conductivity the fundamental solution of the governing equations in effective media mean green function is calculated for the same cases two new asymptotic models of the averaged darcy s law are developed to be applicable to large and small scales of heterogeneity several asymptotic expansions of tue two dimensional and three dimensional mean green functions are derived for exponential and gaussian correlations

analysis of variance method for the equivalent conductivity of rectangular blocks we use an analysis of variance anova decomposition of the log hydraulic conductivity f x ln k x as the sum of an average value f plus main effects and interactions of various orders to evaluate the equivalent conductivity of rectangular blocks in d dimensional space some of the anova components are dealt with exactly the effect of the other components is approximated through calibration to the effective conductivity keff under ergodic conditions our analysis applies to both isotropic and anisotropic lognormal k fields we evaluate the theoretical findings through analytical and numerical comparisons with exact results and previously proposed approximations we find that the anova formula is nearly unbiased although its performance is generally inferior to numerical renormalization advantages of the anova formula are that it is less computationally demanding and explicitly shows the dependence of the block conductivity kb on the various components of fluctuation of f inside the block such dependence is more complex than previously assumed considering the anova formula to be unbiased we derive the bias of the geometric mean of the cardwell parsons bounds for three dimensional rectangular blocks

stochastic diagrammatic analysis of groundwater flow in heterogeneous porous media the diagrammatic approach is an alternative to standard analytical methods for solving stochastic differential equations governing groundwater flow with spatially variable hydraulic conductivity this approach uses diagrams instead of abstract symbols to visualize complex multifold integrals that appear in the perturbative expansion of the stochastic flow solution and reduces the original flow problem to a closed set of equations for the mean and the covariance functions diagrammatic analysis provides an improved formulation of the flow problem over conventional first order series approximations which are based on assumptions such as constant mean hydraulic gradient infinite flow domain and neglect of cross correlation terms this formulation includes simple schemes like finite order diagrammatic perturbations that account for mean gradient trends and boundary condition effects as well as more advanced schemes like diagrammatic porous media description operators which contain infinite order correlations in other words diagrammatic analysis covers not only the cases where low order diagrams lead to good approximations of flow but also those situations where low order perturbation is insufficient and a more sophisticated analysis is needed diagrams lead to a nonlocal equation for the mean hydraulic gradient in terms of which necessary conditions are formulated for the existence of an effective hydraulic conductivity three dimensional flow in an isotropic bounded domain with dirichlet boundary conditions is considered and an integral equation for the mean hydraulic head is derived by means of diagrams this formulation provides an explicit expression for the boundary effects within the three dimensional flow domain in addition to these theoretical results the numerical performance of the diagrammatic approach is tested and useful insight is obtained by means of one dimensional flow examples where the exact stochastic solutions are available

pumping tests in heterogeneous aquifers an analytical study of what can be obtained from their interpretation using jacob s method interpretation of pumping tests to estimate hydraulic parameter values is typically based on the assumption of aquifer homogeneity the applicability of the traditional methods of interpretation in real aquifers can be questioned since the evaluation of the drawdown curves observed at different locations in a single test may not result in one consistent set of hydraulic parameters thus most hydrogeologists tend to look at estimated transmissivities t as some average property of the medium while estimated storativities s are disregarded in some cases particularly when they are obtained from data measured at the pumping well an analytical study of drawdown under radially convergent flow toward a single point in heterogeneous aquifers shows that large time drawdown values form a straight line on a drawdown versus log time plot jacob s method consists of obtaining estimates for t and s from the slope and intercept of this line we find that even in a heterogeneous field these estimates provide valuable information about the aquifer estimated t values for different observation points tend to converge to a single value which corresponds to the effective t derived under parallel flow conditions estimated storativities however display higher variability but the geometric mean of the sest values can be used as an unbiased estimator of the actual s thus it appears that although jacob s method was originally derived for homogeneous media it can provide valuable information in real aquifers

finite analytic numerical method for two dimensional fluid flow in heterogeneous porous media starting from the investigation on the detailed flow pattern finite analytic numerical method is developed to solve the two dimensional fluid flows in heterogeneous porous media it is shown that only for some specific permeability distributions the pressure has the piecewise linear distribution where harmonic average scheme works very well in general case the pressure will have the power law behavior and its gradient will diverge as approaching the node joining the different permeability areas the nodal flow effects cause the flow fingering into the high permeability region it is a challenge problem to numerically describe the nodal fingering effects with the help of the specific properties of pressure and its gradient around the node a local analytical nodal solution is derived and then applied to construct a finite analytic numerical scheme numerical examples show that the detailed flow pattern can be reconstructed with the proposed numerical scheme under few grid refinements only with 2x2 or 3x3 subdivisions the proposed numerical scheme can provide rather accurate solutions the convergent speed of the numerical scheme is independent of the permeability heterogeneity in contrast the refinement ratio for the grid cell needs to be increased dramatically to get an accurate result when the traditional numerical method is used for strong heterogeneous cases

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

field theoretical analysis of singularities at critical end points abstract continuum models with critical end points are considered whose hamiltonian h depends on two densities and field theoretic methods are used to show the equivalence of the critical behavior on the critical line and at the critical end point and to give a systematic derivation of critical end point singularities like the thermal singularity t 2 of the spectator phase boundary and the coexistence singularities t 1 or t of the secondary density the appearance of a discontinuity eigenexponent associated with the critical end point is confirmed and the mechanism by which it arises in field theory is clarified

multiresolution schemes on triangles for scalar conservation laws this paper proposes a multiresolution procedure adapted to triangular cell averages to improve the performance of finite volume schemes by reducing flux evaluation cost using the approach introduced by a harten a specific coarse to fine prediction scheme is proposed that ensures the stability of the computations even when a large number of scales are involved numerical tests are presented that illustrate the computational gain as well as the order of accuracy of the scheme

a higher order approximation to effective conductivity in media of anisotropic random structure properties of the effective conductivity tensor keff are studied by deriving the second order terms in its expansion in the variance 2 of normally distributed log conductivity it is shown that for media of anisotropic structure the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance that is it depends on the shape of the correlation function and not only on anisotropy ratios variance 2 and space dimensions however the trace of keff is independent of the log conductivity autocovariance and for a given mean conductivity depends only on 2 the dependence of the effective conductivity on the correlation structure is illustrated for gaussian and exponential autocovariances of log conductivity and for two and three dimensional flows

wavelet based schemes for linear advection dispersion equation abstract in this paper two wavelet based adaptive solvers are developed for linear advection dispersion equation the localization properties and multilevel structure of the wavelets in the physical space are used for adaptive computational methods for solution of equation which exhibit both smooth and shock like behaviour the first framework is based on wavelet galerkin and the second is based on multiscale decomposition of finite element method coiflet wavelet filter is incorporated in both the methods the main advantage of both the adaptive methods is the elimination of spurious oscillations at very high peclet number

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

new resolution strategy for multiscale reaction waves using time operator splitting space adaptive multiresolution and dedicated high order implicit explicit time integrators we tackle the numerical simulation of reaction diffusion equations modeling multi scale reaction waves this type of problem induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts spatially very localized in this paper we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts the paper considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is restricted by neither the fastest scales in the source term nor by stability constraints of the diffusive steps but only by the physics of the phenomenon we aim thus at solving complete models including all time and space scales within a prescribed accuracy considering large simulation domains with conventional computing resources the efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for two dimensional spiral waves and three dimensional scroll waves as an illustration future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are discussed

upscaling of permeability of anisotropic heterogeneous formations 2 general structure and small perturbation analysis the general methodology developed in part 1 indelman and dagan this issue of this study is applied to the detailed analysis of upscaled permeability first the structure of the dissipation function for the general conductivity a tensor of stationary random components is examined with the aid of dimensional analysis it is shown that for arbitrary shapes of the numerical elements the upscaled permeability has also this general structure and the numbers of unknown parameters and equations match this result suggests that the upscaling problem has an unique solution in the particular case of scalar permeability of isotropic covariance of the actual formation it is shown that a similar upscaled permeability is possible only for spherical circular numerical elements otherwise the upscaled permeability has to be a tensor of anisotropic covariance if the actual formation has a scalar permeability of axisymmetric covariance upscaling preserves the last property only for axisymmetric partition elements i e for a sphere cylinder and ellipsoid explicit expressions for the first moments of the upscaled permeability mean covariance are derived at first order in the log permeability variance the detailed computations for a scalar permeability of axisymmetric covariance and for axisymmetric numerical elements lead to simple results the upscaled permeability expected values are components of an axisymmetric tensor whereas the fluctuations are determined by a scalar random space function of anisotropic covariance the general case of upscaling by a first order approximation is examined in appendix b these results will be applied to a few particular cases in part 3 indelman this issue

fourth order approximation of fluid flow through rough walled rock fracture 1 a stochastic approach for estimating the permeability between rough walled rock fracture surfaces is presented in the present analysis the reynolds lubrication equation which is adopted for analyzing fluid flow through a rough surface of rock fracture is solved as a stochastic partial differential equation using perturbative expansion method the results are compared with the numerical deterministic analysis for randomly generated aperture distributions the analytical results are in a good agreement with numerical results obtained by the monte carlo approach through the present analysis a fourth order accurate approximation to the effective permeability of a single joint is given in terms of the mean the standard deviation and the spatial correlation so that the effect of higher order terms is clarified more accurately than the classical second order approximation

the use of field theoretic methods for the study of flow in a heterogeneous porous medium the effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined it is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system it is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means this allows the calculation of effective permeability the mean pressure and the pressure variance the method as well as recovering familiar results gives a formal means of improving the approximation and approaching more complex systems

adaptive multiresolution semi lagrangian discontinuous galerkin methods for the vlasov equations we develop adaptive numerical schemes for the vlasov equation by combining discontinuous galerkin discretisation multiresolution analysis and semi lagrangian time integration we implement a tree based structure in order to achieve adaptivity both multi wavelets and discontinuous galerkin rely on a local polynomial basis the schemes are tested and validated using vlasov poisson equations for plasma physics and astrophysics

coherent vortex simulation of weakly compressible turbulent mixing layers using adaptive multiresolution methods an adaptive mulitresolution method based on a second order finite volume discretization is presented for solving the three dimensional compressible navier stokes equations in cartesian geometry the explicit time discretization is of second order and for flux evaluation a 2 4 mac cormack scheme is used coherent vortex simulations cvs are performed by decomposing the flow variables into coherent and incoherent contributions the coherent part is computed deterministically on a locally refined grid using the adaptive multiresolution method while the influence of the incoherent part is neglected to model turbulent dissipation the computational efficiency of this approach in terms of memory and cpu time compression is illustrated for turbulent mixing layers in the weakly compressible regime and for reynolds numbers based on the mixing layer thickness between 50 and 200 comparisons with direct numerical simulations allow to assess the precision and efficiency of cvs

adaptive multiresolution schemes for shock computations in this paper we present adaptive multiresolution schemes for the computation of discontinuous solutions of hyperbolic conservation laws starting with the given grid we consider the grid averages of the numerical solution for a hierarchy of nested grids which is obtained by diadic coarsening and we compute its equivalent multiresolution representation this representation of the numerical solution consists of the grid averages of the coarsest grid and the set of errors in predicting the grid averages of each level of resolution from the next coarser one these errors depend on the size of the grid and the local regularity of the solution at a jump discontinuity they remain the size of the jump independent of the level of resolution this observation enables us to identify the location of discontinuities in the numerical solution in a region of smoothness once the numerical solution is resolved to our satisfaction at a certain locality of some grid then the prediction errors there for this grid and all finer ones are small this enables us to obtain data compression by setting to zero the terms of the multiresolution representation that fall below a specified tolerance the numerical flux of the adaptive scheme is taken to be that of a standard centered scheme unless it corresponds to an identified discontinuity in which case it is taken to be the flux of an eno scheme we use the data compression of the numerical solution in order to reduce the number of numerical flux evaluations as follows we start with the computation of the exact numerical fluxes at the few grid points of the coarsest grid and then we proceed through diadic refinement to the given grid at each step of refinement we add values for the numerical flux at the new grid points which are the centers of the coarser cells wherever the solution is locally well resolved i e the corresponding prediction error is below the specified tolerance the costly exact value of the numerical flux function is replaced by an accurate enough approximate value which is obtained by an inexpensive interpolation from the values of the coarser grid

multiwavelet based grid adaptation with discontinuous galerkin schemes for shallow water equations we provide an adaptive strategy for solving shallow water equations with dynamic grid adaptation including a sparse representation of the bottom topography a challenge in computing approximate solutions to the shallow water equations including wetting and drying is to achieve the positivity of the water height and the well balancing of the approximate solution a key property of our adaptive strategy is that it guarantees that these properties are preserved during the refinement and coarsening steps in the adaptation process the underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids this provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients furthermore we use the multiresolution analysis of the underlying data as an additional indicator of whether the limiter has to be applied on a cell or not by this the number of cells where the limiter is applied is reduced without spoiling the accuracy of the solution by means of well known 1d and 2d benchmark problems we verify that multiwavelet based grid adaptation can significantly reduce the computational cost by sparsening the computational grids while retaining accuracy and keeping well balancing and positivity

assessment of the predictive capabilities of stochastic theories in a three dimensional laboratory test aquifer effective hydraulic conductivity and temporal moments of breakthrough curves 1 conservative and sorptive tracer experiments were conducted in a highly heterogeneous lnk2 1 79 and anisotropic h v 3 52 three dimensional test aquifer under well controlled laboratory conditions to evaluate the effective conductivity and temporal moments predicted by stochastic theories the spatial distribution of lnk in the test aquifer conformed to a statistically homogeneous system thus allowing evaluation of results from stochastic theories for stationary random fields effective hydraulic conductivity keff in the mean flow direction for the test aquifer was compared with different stochastic theoretical expressions breakthrough curves btcs at individual deep penetrating observation wells and averaged breakthrough curves at control planes were examined the mean arrival time and the travel time variance estimated using averaged btcs from many observation wells at the same control plane for bromide and lithium were slightly overestimated by stochastic theories yet most of the experimental data ranged within the confidence interval rendered by the uncertainty in the statistical properties of the test aquifer a highlight of our experimental approach is that the continuous sampling procedure also permitted the evaluation of higher order temporal moments that were analyzed to study the asymmetry and peakedness of btcs by means of the coefficients of skewness and kurtosis respectively small perturbation analytical solutions for the coefficient of skewness and kurtosis were provided it is seen that although stochastic theories based upon small perturbations provide reasonably good estimates of the coefficients of skewness and kurtosis associated with averaged btcs at control planes in general they can largely underestimate the peakedness and tailing of btcs observed at individual deep penetrating observation wells

multiresolution algorithms for the numerical solution of hyperbolic conservation laws given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner to do so we consider the grid averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest and introduce an equivalent multiresolution representation the multiresolution representation of the numerical solution consists of its grid averages for the coarsest grid and the set of errors in predicting the grid averages of each level of resolution in this hierarchy from those of the next coarser one once the numerical solution is resolved to our satisfaction in a certain locality of some grid then the prediction errors there are small for this particular grid and all finer ones this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance therefore instead of computing the time evolution of the numerical solution on the given grid we compute the time evolution of its compressed multiresolution representation algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid at each step of refinement we add the values of the numerical flux at the center of the coarser cells the information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well resolved when this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid the computational efficiency of this multiresolution algorithm is proportional to the rate of data compression for a prescribed level of tolerance that can be achieved for the numerical solution of the given scheme

critical endpoint behavior in an asymmetric ising model application of wang landau sampling to calculate the density of states using the wang landau sampling method with a two dimensional random walk we determine the density of states for an asymmetric ising model with two and three body interactions on a triangular lattice in the presence of an external field with an accurate density of states we were able to map out the phase diagram accurately and perform quantitative finite size analyses at and away from the critical endpoint we observe a clear divergence of the curvature of the spectator phase boundary and of the magnetization coexistence diameter derivative at the critical endpoint and the exponents for both divergences agree well with previous theoretical predictions

an adaptive multiresolution scheme with local time stepping for evolutionary pdes we present a fully adaptive numerical scheme for evolutionary pdes in cartesian geometry based on a second order finite volume discretization a multiresolution strategy allows local grid refinement while controlling the approximation error in space for time discretization we use an explicit runge kutta scheme of second order with a scale dependent time step on the finest scale the size of the time step is imposed by the stability condition of the explicit scheme on larger scales the time step can be increased without violating the stability requirement of the explicit scheme the implementation uses a dynamic tree data structure numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid fully adaptive three dimensional computations for reaction diffusion equations illustrate the additional speed up of the local time stepping for a thermo diffusive flame instability

perturbation schemes for flow in random media we utilize a many body style formalism to compute the long range effective diffusivity kappa e for dispersal of a passive scalar field in a random irrotational flow with molecular diffusivity by comparing the theoretical results with the outcome of computer simulations we show that a renormalization group approach that includes an appropriate vertex renormalization is superior to low order perturbation theory and to the hartree fock approximation and gives a rather accurate method of computing kappa e for a range of parameter values

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

multiresolution schemes for the numerical solution of 2 d conservation laws i a generalization of harten s multiresolution algorithms to two dimensional 2 d hyperbolic conservation laws is presented given a cartesian grid and a discretized function on it the method computes the local scale components of the function by recursive diadic coarsening of the grid since the function s regularity can be described in terms of its scale or multiresolution analysis the numerical solution of conservation laws becomes more efficient by eliminating flux computations wherever the solution is smooth instead in those locations the divergence of the solution is interpolated from the next coarser grid level first the basic 2 d essentially nonoscillatory eno scheme is presented then the 2 d multiresolution analysis is developed and finally the subsequent scheme is tested numerically the computational results confirm that the efficiency of the numerical scheme can be considerably improved in two dimensions as well

effective hydraulic conductivity of a randomly heterogeneous porous medium 1 the effective hydraulic conductivity of a randomly heterogeneous isotropic porous medium has been calculated by means of a new perturbative technique which differs greatly from that commonly used in the approach developed we construct the perturbation series for a seepage velocity and then utilize darcy s law for calculating pressure on the basis of the analysis of high order approximations it is shown that in the general case the effective conductivity in the large scale limit does depend on the form of the correlation function thus the widespread landau lifshitz matheron formula which operates on the conjecture that the first two terms of the perturbation series are those for the taylor series expansion of an exponential function proves to be invalid the calculations were carried out in a space of arbitrary dimension the dependence of the effective conductivity on the variance has been obtained up to the third order approximation inclusive in terms of a conductivity logarithm for convenience and simplification of series analysis the feynman diagrammatic technique was developed and utilized in a one dimensional 1d case our result in the large scale limit when a heterogeneity scale is small enough with respect to the characteristic scale of groundwater flow gives an exact formula for the effective conductivity

a semi lagrangian amr scheme for 2d transport problems in conservation form in this paper we construct a semi lagrangian sl adaptive mesh refinement amr solver for 1d and 2d transport problems in conservation form first we describe the a la harten amr framework the adaptation process selects a hierarchical set of grids with different resolutions depending on the features of the integrand function using as criteria the point value prediction via interpolation from coarser meshes and the appearance of large gradients we integrate in time by reconstructing at the feet of the characteristics through the point value weighted essentially non oscillatory pv weno interpolator we propose then an extension to the 2d setting by making the time integration dimension by dimension thanks to a strang splitting we discuss the quality of the results and the speedup with respect to a fixed mesh fm strategy through the following benchmark tests in 1d constant and variable coefficient advections in 2d the 1d vlasov poisson system 2d in the phase space for the case of constant coefficient advections and for the case of variable coefficient advections the deformation flow and the guiding center model

renormalization group analysis of macrodispersion in a directed random flow we apply field theoretic methods to the calculation of the effective diffusivity macrodispersion coefficient in a random flow we show how this approach can be utilized to calculate a perturbation series for the effective diffusivity of conservative tracers in an incompressible velocity field with nonvanishing mean the first order i e one loop approximation of this series coincides with classical results derived by gelhar and axness and dagan a renormalization group rng approach is utilized and the results are compared to the classical first order perturbation theory for a moderate variability of the permeability the renormalized theory predicts only small corrections to the longitudinal dispersivity however the transverse dispersivity can be larger than that predicted by the first order perturbation theory by several orders of magnitude we compare these values to the outcome of monte carlo simulations and find that the rng predictions are in much better though not perfect accordance with the results of simulations moreover the results are in good quantitative agreement with reported observations from the borden site field experiment

convergence of an adaptive semi lagrangian scheme for the vlasov poisson system an adaptive semi lagrangian scheme for solving the cauchy problem associated to the periodic 1 1 dimensional vlasov poisson system in the two dimensional phase space is proposed and analyzed a key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow the accuracy of the scheme is monitored by a prescribed tolerance parameter e which represents the local interpolation error at each time step in the l metric the numerical solutions are proved to converge in l towards the exact ones as e and t tend to zero provided the initial data is lipschitz and has a finite total curvature or in other words that it belongs to w 1 infty cap w 2 1 the rate of convergence is mathcal o delta t 2 varepsilon delta t which should be compared to the results of besse who recently established in siam j numer anal 42 1 350 382 2004 similar rates for a uniform semi lagrangian scheme but requiring that the initial data are in mathcal c 2 several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

scales of fluctuation and the propagation of uncertainty in random porous media the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the heterogeneity of the medium is modeled as a superposition of scales of heterogeneity that are statistically uncorrelated the stochastic process making up a hydraulic property is thus represented as a linear combination of normalized deterministic shapes multiplied by a set of uncorrelated random variables the karhunen loeve expansion is used to formalize this representation the uncertainty contained in each of these scales is then propagated through the system using the governing differential equation as a constraint the solution process will thus consist of the original scales used to represent the porous medium as well as various interaction between these scales the objective of this paper is to develop a procedure for quantifying the uncertainty in the field variables using the scales of fluctuation of the porous medium as the measuring stick

adaptive multiresolution discontinuous galerkin schemes for conservation laws a multiresolution based adaptation concept is proposed that aims at accelerating discontinuous galerkin schemes applied to nonlinear hyperbolic conservation laws opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement instead of this a multiresolution analysis is performed on a hierarchy of nested grids for the data given on a uniformly refined mesh this provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients a central mathematical problem addressed in this work is then to show that the adaptive solution is of the same accuracy as the reference solution on a uniformly refined mesh numerical comparisons demonstrate the efficiency of the concept and provide reliable estimates of the actual error in the numerical solution

a multiresolution finite volume scheme for two dimensional hyperbolic conservation laws in this paper high resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two dimensional hyperbolic conservation laws the method is conservative moreover it is stable which is proven numerically in this paper the computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed using this proposed scheme we compute several two dimensional model problems and a compressive rate ranging from about 5 10 is observed in all simulations

high resolution monte carlo simulation of flow and conservative transport in heterogeneous porous media 1 methodology and flow results in this the first of two papers concerned with the use of numerical simulation to examine flow and transport parameters in heterogeneous porous media via monte carlo methods various aspects of the modelling effort are examined in particular the need to save on core memory causes one to use only specific realizations that have certain initial characteristics in effect these transport simulations are conditioned by these characteristics also the need to independently estimate length scales for the generated fields is discussed the statistical uniformity of the flow field is investigated by plotting the variance of the seepage velocity for vector components in the x y and z directions finally specific features of the velocity field itself are illuminated in this first paper in particular these data give one the opportunity to investigate the effective hydraulic conductivity in a flow field which is approximately statistically uniform comparisons are made with first and second order perturbation analyses the mean cloud velocity is examined to ascertain whether it is identical to the mean seepage velocity of the model finally the variance in the cloud centroid velocity is examined for the effect of source size and differing strengths of local transverse dispersion

on second order correction to the effective conductivity of two dimensional anisotropic media it has been shown recently that the effective conductivity keff of anisotropic media depends on the shape of correlations indelman and abramovich 1994 this property results from the analysis of the second order expression of keff in the log conductivity variance 2 which is derived in terms of quadratures and calculated by numerical integration for two and three dimensional flows and for exponential and gaussian covariances the present paper is a continuation of this study and aims at deriving an analytical closed form expression of keff at order 4 for the case of two dimensional flow in lognormal anisotropic media of gaussian log conductivity correlation

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization

a numerical strategy to discretize and solve the poisson equation on dynamically adapted multiresolution grids for time dependent streamer discharge simulations we develop a numerical strategy to solve multi dimensional poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts the method is an extension of the multiresolution finite volume scheme used to solve hyperbolic and parabolic time dependent pdes such an approach guarantees a numerical solution of the poisson equation within a user defined accuracy tolerance most adaptive meshing approaches in the literature solve elliptic pdes level wise and hence at uniform resolution throughout the set of adapted grids here we introduce a numerical procedure to represent the elliptic operators on the adapted grid strongly coupling inter grid relations that guarantee the conservation and accuracy properties of multiresolution finite volume schemes the discrete poisson equation is solved at once over the entire computational domain as a completely separate process the accuracy and numerical performance of the method are assessed in the context of streamer discharge simulations

effective medium approximation and exact formulae for electrokinetic phenomena in porous media electrokinetic phenomena in porous media are studied by application of the effective medium theory and the theory of duality transformation we deduce new exact relations and analytical formulae for the effective constants of the macroscopic tensor we also prove that the effective tensors obtained by these approaches coincide for 2d problems the obtained results for the electrokinetic processes are closely related to similar results derived for piezoelectric composites because of a common mathematical background