analytical solution for one dimensional drainage burgers and simplified forms solutions are developed for burgers equation and for the surface water content subject to one dimensional drainage these are found as limiting cases of richards equation using diffusivity and conductivity functions from fujita as extended by broadbridge and white the solution from burgers equation can generally be expressed as a function of only a dimensionless time depth and initial reduced water content for the more strongly nonlinear case initially the surface water content decreases in proportion to the square root of time for large times the surface water content approaches the residual water content inversely with the square root of time

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

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

analytical decomposition of the nonlinear unsaturated flow equation an analytical procedure for the solution of the nonlinear unsaturated flow equation is proposed the horizontal and vertical flow equations subject to constant boundary conditions and continuous wetting are approximated using analytical decomposition series the results are verified with the philip 1955 and the parlange 1971 classical solutions and with experimental data a convergence theorem with proof and discussion on the numerical accuracy of decomposition series are included applications to the modeling of infiltration subject to hysteresis in the soil water functional relationships soil heterogeneity and time variable point rainfall are also discussed other potential applications in water management models under more realistic conditions are possible the methodology offers advantages over the common small perturbation solutions the possibility to study nonstationarity in the random quantities statistical separability from the physics of the problem i e independence between the system parameters and the input quantities rather than forced through closure approximations and the construction of an analytical series solution that converges uniformly to the true nonlinear solution finally because of stability and computational economy with respect to linearized numerical solutions an analytical solution appears promising

high accuracy wave simulation revised derivation numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation the nearly analytic integration discrete naid method for solving the two dimensional acoustic wave equation has been fully mathematically revised analyzed and tested the naid method is an alternative numerical modeling method for generating synthetic seismograms the acoustic wave equation is first transformed into a system of first order ordinary differential equations odes with respect to time variable t and then directly integrated at a small time interval of tn tn 1 to obtain semi discrete ordinary differential equations the integral kernel is expanded into a truncated taylor series to which the integration operator is explicitly applied the high order temporal derivatives involved in the integral kernel are replaced by high order spatial derivatives which then are approximately calculated as a weighted linear combination of the displacement the particle velocity and their spatial gradients in this article we investigate the theoretical properties of the revised naid method including the discrete error and the stability criteria numerical results for constant and layered velocity models show that comparing to the lax wendroff correction lwc scheme and the staggered grid finite difference method the naid method can effectively suppress the numerical dispersion and source noises caused by the discretization of the acoustic wave equation with too coarse spatial grids or when models have strong velocity contrasts between adjacent layers the proposed naid method has been applied in computing the acoustic wavefields for two heterogeneous models the corner edge model and the marmousi model promising numerical results illustrate that the naid method provides an encouraging tool for large scale and complex wave simulation and inversion problems based on the acoustic equation

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

lie symmetry classification analysis for nonlinear coupled diffusion the group properties of the 1 1 dimensional matrix diffusion equation of the form with respect to point symmetries are given it is shown that in particular cases the group properties of this equation are similar to those of the corresponding scalar equation namely the lie algebra is extended for power and exponential functions with additional extensions when powers of and 2 exist however a specific form of is shown to exist for cases admitting both infinite and finite groups the result obtained is used to construct new group invariant solutions for impulsive boundary inputs using two similarity variables a reduction is applied to the coupled diffusion of temperature and volumetric moisture content in porous media under periodic boundary conditions a perturbation method is employed to obtain an explicit solution for the case when can be expressed exponentially in terms of the similarity variables

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

new approximate analytical technique to solve richards equation for arbitrary surface boundary conditions a general approximation for the solution to the one dimensional richards equation is presented it applies to arbitrary soil properties and boundary conditions but only uniform initial conditions at current stage the result is very accurate within 2 when the diffusivity is constant suggesting that the present general formulation is reliable since the approximation becomes increasingly accurate as the soil water diffusivity approaches a delta function

experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model 1 an optimal experimental design algorithm is developed to select locations for a network of observation wells that provide maximum information about unknown groundwater pumping in a confined anisotropic aquifer the design uses a maximal information criterion that chooses among competing designs the design that maximizes the sum of squared sensitivities while conforming to specified design constraints the formulated optimization problem is non convex and contains integer variables necessitating a combinatorial search given a realistic large scale model the size of the combinatorial search required can make the problem difficult if not impossible to solve using traditional mathematical programming techniques genetic algorithms gas can be used to perform the global search however because a ga requires a large number of calls to a groundwater model the formulated optimization problem still may be infeasible to solve as a result proper orthogonal decomposition pod is applied to the groundwater model to reduce its dimensionality then the information matrix in the full model space can be searched without solving the full model results from a small scale test case show identical optimal solutions among the ga integer programming and exhaustive search methods this demonstrates the ga s ability to determine the optimal solution in addition the results show that a ga with pod model reduction is several orders of magnitude faster in finding the optimal solution than a ga using the full model the proposed experimental design algorithm is applied to a realistic two dimensional large scale groundwater problem the ga converged to a solution for this large scale problem

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

an efficient finite element procedure for analyzing three phase porous media based on the relaxed picard method summary effective simulation of the solid liquid gas coupling effect in unsaturated porous media is of great significance in many diverse areas because of the strongly nonlinear characteristics of the fully coupled formulations for the three phase porous media an effective numerical solution scheme such as the finite element method with an efficient iterative algorithm has to be employed in this paper an efficient finite element procedure based on the adaptive relaxed picard method is developed for analyzing the coupled solid liquid gas interactions in porous media the coupled model and the finite element analysis procedure are implemented into a computer code poroush2m and the proposed procedure is validated through comparing the numerical simulations with the experimental benchmarks it is shown that the adaptive relaxed picard method has salient advantage over the traditional one with respect to both the efficiency and the robustness especially for the case of relatively large time step sizes compared with the newton raphson scheme the picard method successfully avoids the unphysical spurious unloading phenomenon under the plastic deformation condition although the latter shows a better convergence rate the proposed procedure provides an important reference for analyzing the fully coupled problems related to the multi phase multi field coupling in porous media copyright 2014 john wiley sons ltd

three dimensional unsaturated flow modeling using cellular automata 1 the parabolic partial differential equation describing fluid flow in partially saturated porous media richards equation is highly nonlinear due to pressure head dependencies in the specific soil moisture capacity and relative hydraulic conductivity terms in order to solve richards equation several numerical techniques have been developed which starting from the discretization of the partial differential equation produced even more accurate models leading to complex and computationally expensive simulations for large scale systems a three dimensional unsaturated flow modeling developed through a simulation environment based on cellular automata ca is described in this paper the proposed model represents an extension of the original computational paradigm of cellular automata because it uses a macroscopic ca approach where local laws with a clear physical meaning govern interactions among automata this ca structure aimed at simulating a large scale system is based on functionalities capable of increasing its computational capacity both in terms of working environment and in terms of the optimal number of processors available for parallel computing the model has been validated with reference multidimensional solutions taken from benchmarks in literature showing a good agreement even in the cases where nonlinearity is very marked furthermore some analyses have been carried out considering quantization techniques aimed at transforming the ca model into an asynchronous structure the use of these techniques in a three dimensional benchmark allowed a considerable reduction in the number of local interactions among adjacent automata without changing the efficiency of the model especially when simulations are characterized by scarce mass exchanges finally from a computational point of view the higher efficiency values were achieved running the model on a parallel architecture obtaining a high speedup very close to the optimal with the maximum number of processors available

a nearly analytic discrete method for solving the acoustic wave equations in the frequency domain abstractwe have developed a nearly analytic discrete nad method to discretize frequency domain acoustic wave equations with an absorbing boundary condition we evaluate in detail the discrete process of wave equations to derive a linear system the sparse structure and eigenproperties of its coefficient matrix also called the impedance matrix were analyzed to reveal the intrinsic difficulty in solving the linear system efficiently to accelerate the forward modeling process in the frequency domain we introduce a class of inexact rotated block triangular preconditioners incorporated with krylov subspace methods to solve this linear system and test their numerical behaviors by comparing with other two commonly used preconditioners to this end we perform wavefield simulation by the nad method and another two conventional numerical schemes in various media numerical dispersion analysis and waveform comparison are also implemented for these numerical schemes our results show the superiority of our pro

implementation of the jacobian free newton krylov method for solving the first order ice sheet momentum balance we have implemented the jacobian free newton krylov jfnk method for solving the first order ice sheet momentum equation in order to improve the numerical performance of the glimmer community ice sheet model glimmer cism the land ice component of the community earth system model cesm our jfnk implementation is based on significant re use of existing code for example our physics based preconditioner uses the original picard linear solver in glimmer cism for several test cases spanning a range of geometries and boundary conditions our jfnk implementation is 1 8 3 6 times more efficient than the standard picard solver in glimmer cism importantly this computational gain of jfnk over the picard solver increases when refining the grid global convergence of the jfnk solver has been significantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact newton method while a diverse set of test cases show that our jfnk implementation is usually robust for some problems it may fail to converge with increasing resolution as does the picard solver globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of picard and jfnk and the use of homotopy methods

application and improvement of a recent approximate analytical solution of richards equation a method for developing wetting front profiles for the one dimensional richards equation is given the method is obtained by simplifying and extending a recent approximate solution and brings together features from several different studies difficulties associated with profile development are discussed by applying the method to several examples providing a deeper understanding of the solutions of richards equation for simplicity we illustrate the improvements for soil water diffusivities and conductivities which have a power law dependence on the water content although they should apply in general and the appropriate generalization is given in the conclusion

techniques for making finite elements competitve in modeling flow in variably saturated porous media a galerkin finite element formulation is developed for the numerical simulation of water flow in variably saturated soil systems included in this formulation is a solution strategy based on picard and newton raphson algorithms both algorithms are designed especially to cope with severely nonlinear field problems the two algorithms are formulated for both rectangular and triangular elements the element matrices are evaluated in a simple and efficient manner using a technique referred to as the influence coefficient technique this technique avoids numerical integration and leads to a substantial saving of computational cost four examples are presented to demonstrate the effectiveness of the present finite element approach these examples show that the nonlinear solution schemes are capable of accomodating cases involving large variations in the saturated hydraulic conductivity as well as highly nonlinear soil moisture characteristics a comparative study of the picard and the newton raphson algorithms is also provided the study indicates that despite the higher cost per iteration of the newton raphson scheme it usually requires a substantially smaller number of iterations than the picard scheme in some instances where convergence difficulties are experienced with the latter scheme it is desirable to use the newton raphson scheme in order to obtain a cost effective solution to the problem

time to ponding comparison of analytic quasi analytic and approximate predictions an analytic expression for time to ponding is introduced using the nonlinear model of broadbridge and white 1987 the hydraulic properties of this model can encompass properties ranging from those of a highly nonlinear green ampt like soil to those satisfying the weakly nonlinear burgers equation because of its versatility this analytic solution is used as a benchmark against which extant analytic quasi analytic and approximate expressions are compared time to ponding is parameterized here in terms of the readily measured field properties sorptivity and hydraulic conductivity in the limit of green ampt like properties the analytic solution reduces exactly to the parlange and smith 1976 approximation a similar functional dependence of time to ponding on rainfall rate is found from quasi analytic approximations based on this a modified approximation is suggested which should give time to ponding for most soils to within 10 some existing approximations are found to have unacceptable deviations from the analytic solution and their continued use appears unwarranted finally we address the field problem of predicting time to ponding at any antecedent water content given sorptivity measured at only one initial water content

high order finite differences and the pseudospectral method on staggered grids finite difference approximations for the first derivative valid halfway between equidistant gridpoints are in general much more accurate than the corresponding approximations which are valid at gridpoints the pseudospectral fourier method can be viewed as the limit of finite difference approximations when the order of accuracy tends to infinity a fundamentally different and more accurate pseudospectral method is obtained from the halfway approximations this study derives these and related methods and discusses their accuracies

exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption several authors have now used the backlund transformation to linearize richard s equation and so obtain analytical solutions for infiltration and redistribution processes for the linearizable class of diffusivities and hydraulic conductivities after transformation a convection diffusion equation is obtained to date all the analytical results making use of the backlund transform have been based on a specified constant flux that is assumed to exist at the surface boundary we generalize the previous work by removing this restriction and obtain a general solution for an arbitrary temporal variation in surface flux the solution contains a function that is specified by a volterra integral equation in practice the volterra equation must be evaluated numerically this task is accomplished using a relatively straightforward algorithm the analytical results for infiltration can be applied to the process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are equivalent under another mapping our new analytical results therefore also apply to this class of solute transport processes the general form of the adsorption isotherm for which exact results can be obtained is derived this isotherm appears capable of representing a relatively large family of physically relevant cases although this topic remains open for future investigation

the mast edge centred lumped scheme for the flow simulation in variably saturated heterogeneous porous media a novel methodology is proposed for the solution of the flow equation in a variably saturated heterogeneous porous medium the computational domain is descretized using triangular meshes and the governing pdes are discretized using a lumped in the edge centres numerical technique the dependent unknown variable of the problem is the piezometric head a fractional time step methodology is applied for the solution of the original system solving consecutively a prediction and a correction problem a scalar potential of the flow field exists and in the prediction step a marching in space and time mast formulation is applied for the sequential solution of the ordinary differential equation of the cells ordered according to their potential value computed at the beginning of the time step in the correction step the solution of a large linear system with order equal to the number of edges is required a semi analytical procedure is also proposed for the solution of the prediction step the computational performance the order of convergence and the mass balance error have been estimated in several tests and compared with the results of other literature models

constant rate rainfall infiltration a versatile nonlinear model 1 analytic solution analytic solutions are presented for a nonlinear diffusion convection model describing constant rate rainfall infiltration in uniform soils and other porous materials the model is based on the darcy buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity d and hydraulic conductivity k which depend on a single free parameter c and readily measured soil hydraulic properties these d and k yield physically reasonable analytic moisture characteristics the relation between this model and other models which give analytic solutions is explored as c the model reduces to the weakly nonlinear burgers equation which has been applied in certain field situations at the other end of the range as c 1 the model approaches a green ampt like model a wide range of realistic soil hydraulic properties is encompassed by varying the c parameter the general features of the analytic solutions are illustrated for selected c values gradual and steep wetting profiles develop during rainfall aspects seen in the laboratory and field in addition the time dependent surface water content and surface water pressure potential are presented explicitly a simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times

parameter estimation in flow through partially saturated porous materials a class of numerical simulators were developed and critically evaluated to be incorporated as the solver of a forward problem in the framework of an inverse modeling strategy the strategy couples a mass lumped galerkin linear finite element solution of the mixed form richards equation with an experimental time space series and the osborne more revised form of the levenberg marquardt algorithm to retrieve hydraulic parameters of a partially saturated porous medium the numerical simulator shows excellent agreement with a reference solution obtained on a dense grid and infinitesimal time step in terms of fluid pressure head fluid content and fluid volumetric flux density and perfectly conserves the global mass an adaptive algorithm was implemented to estimate sensitivity matrix in the inverse algorithm a multi criterion stopping rule was developed and successfully implemented to end the inverse code at the solution the result of the optimization was compared with a large scale in situ soil moisture space time series measured during the course of a drainage experiment and excellent agreements were found analysis of the parameter response surfaces and hyper space plots closeness of the gradient of the penalty function at minimum to zero and positive definiteness of the approximation for the hessian at the solution eigs h 0 indicate that the obtained solution is a strong local minimum a state of the art sensitivity analysis carried out to quantify sensitivity of the state variable with respect to uncertainty and changes in different model parameters

finite element formulation of unilateral boundary conditions for unsaturated flow in porous continua this paper presents the numerical resolution of unilateral boundary conditions able to effectively model several problems of unsaturated flow as those involving rainfall infiltration and seepage faces besides the penalty technique we also consider the novel regularization of these conditions by means of the more effective augmented lagrangian method the performance of the so obtained finite element method is carefully investigated in terms of accuracy and ill conditioning effects including comparisons with analytical solutions and a complete identification of the analogies with the problem of frictionless contact in this way we provide a priori estimates of optimal and admissible ranges for the penalty coefficient as functions of permeability and spatial discretization the proposed method and the estimated coefficient ranges are validated in further numerical examples involving the propagation of a wetting front due to rainfall and the partial saturation of an aged concrete dam these applications show that the proposed regularizations do not induce any detrimental effect on solution accuracy and on convergence rate of the employed newton raphson method hence the present approach should be preferred to the commonly used iterative switching between the imposed flow and the imposed pressure conditions which often leads to spurious oscillations and convergence failures

accuracy of galerkin finite elements for groundwater flow simulations in two and three dimensional triangulations summary in standard nite element simulations of groundwater 4ow the correspondence between hydraulic head gradients and groundwater 4uxes is represented by the sti ness matrix in two dimensional problems the use of linear triangular elements on delaunay triangulations guarantees a sti ness matrix of type m this implies that the local numerical 4uxes are physically consistent with darcy s law this condition is fundame ntal to avoid theoccurre nceof local maxima or minima and is of crucial importancewhe n the calculated 4ow eld is used in contaminant transport simulations or pathline evaluation in three spatial dimensions the linear galerkin approach on tetrahedra does not lead to m matrices even on delaunay meshes by interpretation of the galerkin approach as a subdomain collocation scheme we develop a new approach osc orthogonal subdomain collocation that is shown to produce m matrices in three dimensional delaunay triangulations in case of heterogeneous and anisotropic coeacients extra mesh properties required for m sti ness matrices will also be discussed copyright 2001 john wiley sons ltd

an interblock conductivity scheme for finite difference models of steady unsaturated flow in heterogeneous media with the increased ability to model subsurface heterogeneity provided by stochastic simulation methods there is a corresponding increase in the need for numerical flow models to accomodate the spatial variability of hydrodynamic parameters these trends make it necessary to reexamine methods for calculating finite difference interblock conductivities originally developed for use in homogeneous media this study describes a new method for calculating interblock conductivities better suited to the numerical modeling of unsaturated flow in heterogeneous media an expression for interblock conductivity is derived by equating the finite difference flux between adjacent blocks with the exact analytical flux for one dimensional steady unsaturated flow assuming an exponential model for relative conductivity this expression appears as a function of the saturated conductivities alpha parameters and matric potentials of the adjacent blocks and the potential at the interblock boundary since this last quantity is not directly featured in the discretized flow equation it must be evaluated iteratively the interblock conductivity calculation scheme is implemented within a block centered seven point finite difference model of steady unsaturated flow where nonlinearity is resolved using a newton raphson iterative approach in a series of numerical experiments this scheme is compared with the more familiar geometric averaging method the proposed interblock conductivity scheme is shown to provide perfect agreement between numerical and analytical solutions for the one dimensional case of steady infiltration into a perfectly layered medium in problems of two dimensional unsaturated flow in heterogeneous media comparisons with the geometric averaging method show that the proposed scheme allows more rapid convergence of newton raphson iterations and consideration of greater parameter spatial variability before the onset of divergence in these iterations

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

on the construction of a functional solution method for the infiltration in porous media problems abstract in this paper we consider a transient vertical one dimensional flow problem of water in unsaturated porous media modelled by the non linear richards equation constitutive relations of van genuchten will be employed and pade approximants are used to represent the hydraulic capacity and conduction in a simplified fashion from the proposed methodology a construction of a functional solution is presented with the objective to define an initialisation of a recursive scheme in the spirit of adomian decomposition this solution is optimised and evaluated using the governing equation for a self consistency test the results are presented for some soil types and its related soil parameters that are reported in the literature

clean two and three dimensional analytical solutions of richards equation for testing numerical solvers 1 this technical note derives clean analytical solutions of richards equation for three dimensional unsaturated groundwater flow clean means that the boundary conditions and steady state solutions are closed form expressions and the transient solutions have relatively simple additional fourier series terms two dimensional versions of these solutions are also given the primary purpose for the solutions is to test linear and nonlinear solvers in finite difference volume element computer programs for accuracy and scalability using architectures ranging from pcs to parallel high performance computers this derivation starts from the quasi linear assumption of relative hydraulic conductivity varying exponentially with pressure head and the separate approximation that relative hydraulic conductivity varies linearly with moisture content this allows a transformation to be used to create a linear partial differential equation separation of variables and fourier series are then used to obtain the final solution physically reasonable material properties are also used a total of four solutions are given in this technical note steady state and transient solutions for two different boundary conditions of the sample problem

analytical model of infiltration under constant concentration boundary conditions 1 known integrable models for 1d flow in unsaturated soil have a rescaled soil water diffusivity that is either constant or proportional to c c 1 c 2 where is the degree of saturation and c 1 is constant with a wider more realistic range of hydraulic conductivity functions than has been used in this context before a formal series solution is developed for infiltration subject to constant concentration boundary conditions a readily programmed iteration algorithm applicable for any value of c is used to construct many coefficients of the infiltration series without requiring any numerical integration in particular for either c 1 small or 1 c small several infiltration series coefficients are constructed as formal power series in c 1 or in 1 c for which we construct a number of terms explicitly in the limit as the diffusivity approaches a delta function the infiltration coefficients are obtained in simpler closed form all but the sorptivity depend on the form of the conductivity function

mixed finite elements and newton type linearizations for the solution of richards equation we present the development of a two dimensional mixed hybrid finite element mhfe model for the solution of the non linear equation of variably saturated flow in groundwater on unstructured triangular meshes by this approach the darcy velocity is approximated using lowest order raviart thomas rt0 elements and is exactly mass conserving hybridization is used to overcome the ill conditioning of the mixed system the scheme is globally first order in space nevertheless numerical results employing non uniform meshes show second order accuracy of the pressure head and normal fluxes on specific grid points the non linear systems of algebraic equations resulting from the mhfe discretization are solved using picard or newton iterations realistic sample tests show that the mhfe newton approach achieves fast convergence in many situations in particular when a good initial guess is provided by either the picard scheme or relaxation techniques copyright 1999 john wiley sons ltd

series solution for richards equation under concentration boundary conditions and uniform initial conditions a series solution which expresses the moisture profiles and surface fluxes during infiltration and exfiltration is derived the solution applies for conditions of fixed surface moisture content and uniform initial moisture content within an unbounded homogeneous nonhysteretic soil column the series derivation is similar to j r philip s but with a transformed time variable the series converges for infiltration problems involving soils which exhibit steep increase in capillary pressure near saturation under these conditions the solution captures the transition from early square root of time behavior to the longtime traveling wave solution without the use of approximate joining techniques the soil water retention models for which convergence occurs include the widely applied brooks corey and van genuchten models with minor restrictions on the shape parameters of the latter other cases examined include infiltration and exfiltration assuming power law linear quadratic and constant dependence of diffusivity and conductivity on moisture content for these cases the convergence limits were finite the solution is particularly well suited for infiltration into sandy soils for which the transition between short time and longtime behavior occurs relatively early

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

innovative modelling of 3d unsaturated flow in porous media by coupling independent models for vertical and lateral flows unsaturated groundwater flows are mathematically represented by the richards equation hitherto in hydrology solutions of this equation mainly serve as an alimentation of the source term for the surface runoff modelling therefore the complete resolution of the 3d model looks surplus to requirements and the infiltration is dealt either thanks to 1d vertical modelling of the richards equation or through derivate models like e g the green ampt infiltration model or the horton law thus ignoring eventual horizontal transfers nowadays the request for more detailed information is real and the physics of groundwater unsaturated flow needs to be represented more reliably this information could be furnished by the resolution of the complete 3d model but although numerically mastered and well documented it is very costly for large scale both in time and space real applications climate change adaptation of watersheds the authors propose an original solution decoupling the 3d equations into 1d vertical equations and a 2d depth integrated horizontal equation the aim is to consider independent vertical columns of infiltration coupled with lateral transfer of mass through the boundary conditions on this basis they postulate that the mass transfers in the three dimensions are correctly represented this way problematic like the supply of the aquifers the re emergence of groundwater to surface water or especially the capability of memorization of past rainy events could be reliably depicted the two coupled models are solved on a unique numerical frame a cell centred finite volume method is used to solve the parabolic partial differential equations the spatial derivatives are approximate by a second order central difference scheme while the time splitting follows an implicit backward euler scheme coupled with picard iteration the method has been tested and its reliability assessed on different theoretical two dimensional cross sectional test cases representing infiltration phenomena

implementation of the jacobian free newton krylov method for solving the for solving the first order ice sheet momentum balance we have implemented the jacobian free newton krylov jfnk method for solving the rst order ice sheet momentum equation in order to improve the numerical performance of the community ice sheet model cism the land ice component of the community earth system model cesm our jfnk implementation is based on signicant re use of existing code for example our physics based preconditioner uses the original picard linear solver in cism for several test cases spanning a range of geometries and boundary conditions our jfnk implementation is 1 84 3 62 times more efficient than the standard picard solver in cism importantly this computational gain of jfnk over the picard solver increases when rening the grid global convergence of the jfnk solver has been signicantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact newton method while a diverse set of test cases show that our jfnk implementation is usually robust for some problems it may fail to converge with increasing resolution as does the picard solver globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of picard and jfnk and the use of homotopy methods

locally conservative stabilized finite element methods for variably saturated flow standard galerkin finite element methods for variably saturated groundwater flow have several deficiencies for instance local oscillations can appear around sharp infiltration fronts without the use of mass lumping and velocity fields obtained from differentiation of pressure fields are discontinuous at element boundaries here we consider conforming finite element discretizations based on a multiscale formulation along with recently developed local postprocessing schemes the resulting approach maintains the basic flexibility and appeal of traditional finite element methods while controlling nonphysical oscillations and producing element wise mass conservative velocity fields accuracy and efficiency of the proposed schemes are evaluated through a series of steady state and transient variably saturated groundwater flow problems in homogeneous as well as heterogeneous domains the schemes are formulated for a generic nonlinear advection diffusion equation and are thus applicable to many other flow models

numerical simulation of interface waves by high order spectral modeling techniques few problems in elastodynamics have a closed form analytical solution the others can be investigated with semianalytical methods but in general one is not sure whether these methods give reliable solutions the same happens with numerical techniques for instance finite difference methods solve in principle any complex problem including those with arbitrary inhomogeneities and boundary conditions however there is no way to verify the quantitative correctness of the solutions the major problems are stability with respect to material properties numerical dispersion and the treatment of boundary conditions in practice these problems may produce inaccurate solutions in this paper the study of complex problems with two different numerical grid techniques in order to cross check the solutions is proposed interface waves in particular are emphasized since they pose the major difficulties due to the need to implement boundary conditions the first method is based on global differential operators where the solution is expanded in terms of the fourier basis and chebyshev polynomials while the second is the spectral element method an extension of the finite element method that uses chebyshev polynomials as interpolating functions both methods have spectral accuracy up to approximately the nyquist wave number of the grid moreover both methods implement the boundary conditions in a natural way particularly the spectral element algorithm we first solve lamb s problem and compare numerical and analytical solutions then the problem of dispersed rayleigh waves and finally the two quarter space problem we show that the modeling algorithms correctly reproduce the analytical solutions and yield a perfect matching when these solutions do not exist the combined modeling techniques provide a powerful tool for solving complex problems in elastodynamics

the quasilinear approximation for unsaturated porous media flow this paper gives a review of the quasilinear approximation an exponential relationship between hydraulic conductivity and moisture potential and the uses this has been put to since the time of its creation one of the major attractions of this approximation is that under steady regimes the highly nonlinear richards equation governing unsaturated flow can be linearized this simplification has enabled analytic and semianalytic solutions to be found for many unsaturated flow problems these are reviewed and the current research areas in which the quasilinear approximation is being used are discussed

infiltration models for soil profiles bounded by a water table 1 approximate solutions of richards equation are developed for rational forms of the soil hydraulic conductivity and moisture retention functions using a perturbation expansion method a novel approach is devised in the perturbation technique whereby a priori introduced parameters are estimated in an optimal context to ensure the accuracy of the perturbation solution at all times and for all flow conditions the perturbation model describes the one dimensional nonlinear nonsteady infiltration for arbitrary initial conditions under either a time dependent surface flux or a constant head boundary condition into either a semi infinite or a finite medium bounded from below by a shallow water table the optimal perturbation approach proved to be an effective method by which closed form linear solutions can be adapted to nonlinear soils through proper scaling factors the results show that the first order linear term of the optimal perturbation solution is sufficiently accurate for practical applications especially for shallow water table conditions they offer the means to analyze the effect of nonuniform initial conditions and the influence of the water table on the infiltration process practical results in terms of well defined soil water parameters and the antecedent moisture content include algebraic expressions for the surface moisture content variation under variable rainfall conditions time to ponding expressions and infiltration equations for deep and shallow water table conditions and a prediction formula for the water table recharge rate

sensitivity analysis for unobserved confounding of direct and indirect effects using uncertainty intervals to estimate direct and indirect effects of an exposure on an outcome from observed data strong assumptions about unconfoundedness are required since these assumptions cannot be tested using the o

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

estimation of controlled direct effects when regression models adjust for mediators on the causal path from exposure to outcome the regression coefficient of exposure is commonly viewed as a measure of the direct exposure effect this interpretation can be misleading even with a randomly assigned exposure this is because adjustment for post exposure measurements introduces bias whenever their association with the outcome is confounded by more than just the exposure by the same token adjustment for such confounders stays problematic when these are themselves affected by the exposure robins accommodated this by introducing linear structural nested direct effect models with direct effect parameters that can be estimated by using inverse probability weighting by a conditional distribution of the mediator the resulting estimators are consistent but inefficient and can be extremely unstable when the mediator is absolutely continuous we develop direct effect estimators which are not only more efficient but also consistent under a less demanding model for a conditional expectation of the outcome we find that the one estimator which avoids inverse probability weighting altogether performs best this estimator is intuitive computationally straightforward and as demonstrated by simulation competes extremely well with ordinary least squares estimators in settings where standard regression is valid

elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport finite difference acoustic wave modeling and reverse time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well however conventional finite difference methods for solving the acoustic or elastic wave equation suffer from numerical dispersion when too few samples per wavelength are used the flux corrected transport fct algorithm adapted from hydrodynamics reduces the numerical dispersion in finite difference wavefield continuation the flux correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion incorporating the flux correction procedure in conventional finite difference modeling or reverse time migration can provide finite difference solutions with no numerical dispersion even for impulsive sources the fct correction which can be applied to finite difference approximations of any order in space and time is an efficient alternative to use for finite difference approximations of increasing order through demonstrations of modeling and migration on both synthetic and field data we show the benefits of the fct algorithm as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse

burgers equation a solution for soil water flow in a finite length burgers equation can be considered a special case of the one dimensional richards equation for vertical flow for which the soil water diffusivity is constant and the unsaturated hydraulic conductivity is proportional to the square of the reduced water content here a fourier series solution for burgers 1948 equation for a finite region is presented the solution is useful to modelers seeking a simplified representation of vertical water flow through the vadose zone and as an analytical solution for benchmarking which maintains many of the nonlinear characteristics of solutions to richards equation two example problems are presented in the first infiltration into a dry soil is modeled showing that burgers equation gives developing wetting front profiles similar to those observed for richards equation in the second example one step outflow from an 8 cm core is modeled comparisons between the analytical and finite difference solutions for burgers equation are presented to help verify the analytical solutions

efficient numerical methods for infiltration using richards equation two efficient finite difference methods for solving richards equation in one dimension are presented and their use in a range of soils and conditions is investigated large time steps are possible when the mass conserving mixed form of richards equation is combined with an implicit iterative scheme while a hyperbolic sine transform for the matric potential allows large spatial increments even in dry inhomogeneous soil infiltration in a range of soils can be simulated in a few seconds on a personal computer with errors of only a few percent in the amount and distribution of soil water one of the methods adds points to the space grid as an infiltration or redistribution front advances thus gaining considerably in efficiency over the other fixed grid method for infiltration problems in 17 s computing this advancing front method simulated infiltration redistribution and drainage for 50 days in an inhomogeneous soil with nonuniform initial conditions only 16 space and 21 time steps were needed for the simulation which included early ponding with the development and dissipation of a perched water table

an efficient fourth order low dispersive finite difference scheme for a 2 d acoustic wave equation in this paper we propose an efficient fourth order compact finite difference scheme with low numerical dispersion to solve the two dimensional acoustic wave equation combined with the alternating direction implicit adi technique and pade approximation the standard second order finite difference scheme can be improved to fourth order and solved as a sequence of one dimensional problems with high computational efficiency however such compact higher order methods suffer from high numerical dispersion to suppress numerical dispersion the compact and non compact stages are interlinked to produce a hybrid scheme in which the compact stage is based on pade approximation in both y and temporal dimensions while the non compact stage is based on pade approximation in y dimension only stability analysis shows that the new scheme is conditionally stable and superior to some existing methods in terms of the courant friedrichs lewy cfl condition the dispersion analysis shows that the new scheme has lower numerical dispersion in comparison to the existing compact adi scheme and the higher order locally one dimensional lod scheme three numerical examples are solved to demonstrate the accuracy and efficiency of the new method

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

finite volume integrated surface subsurface flow modeling on nonorthogonal grids in this paper we present an innovative finite volume surface subsurface integrated flow model on nonorthogonal grids the shallow water equation with diffusion wave approximation is used to formulate the surface flow system while the richards equation is used to formulate the saturated unsaturated subsurface flow system these two flow systems are discretized using a finite volume method and are then coupled by enforcing the continuity of pressure and flux at the surface subsurface interface which does not require unphysical parameters such as the interface permeability and thickness the numerical instability caused by enforcing the continuity of pressure and flux at the interface is resolved using a cell centered finite volume discretization the coupled systems are solved simultaneously by the newton iterative method a battery of benchmark analyses and laboratory experiments verify the proposed model s superior performance relative to existing models two numerical experiments over irregular terrain show that the nonorthogonal grids and diffusive wave approximation used in the proposed model accurately represent the interaction between surface and subsurface flows for irregular topographies in particular they capture the significant topographical effects on runoff discharges especially where gentle slopes are involved

a comparison of the jacobian free newton krylov method and the evp model for solving the sea ice momentum equation with a viscous plastic formulation a serial algorithm study numerical convergence properties of a recently developed jacobian free newton krylov jfnk solver are compared to the ones of the widely used evp model when solving the sea ice momentum equation with a viscous plastic vp formulation to do so very accurate reference solutions are produced with an independent picard solver with an advective time step of 10s and a tight nonlinear convergence criterion on 10 20 40 and 80 km grids approximate solutions with the jfnk and evp solvers are obtained for advective time steps of 10 20 and 30min because of an artificial elastic term the evp model permits an explicit time stepping scheme with a relatively large subcycling time step the elastic waves excited during the subcycling are intended to damp out and almost entirely disappear such that the approximate solution should be close to the vp solution results show that residual elastic waves cause the evp approximate solution to have notable differences with the reference solution and that these differences get more important as the grid is refined compared to the reference solution additional shear lines and zones of strong convergence divergence are seen in the evp approximate solution the approximate solution obtained with the jfnk solver is very close to the reference solution for all spatial resolutions tested

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

numerical evaluation of iterative and noniterative methods for the solution of the nonlinear richards equation several noniterative procedures for solving the nonlinear richards equation are introduced and compared to the conventional newton and picard iteration methods noniterative strategies for the numerical solution of transient nonlinear equations arise from explicit or linear time discretizations or they can be obtained by linearizing an implicit differencing scheme we present two first order accurate linearization methods a second order accurate two level implicit factored scheme and a second order accurate three level lees method the accuracy and computational efficiency of these four schemes and of the newton and picard methods are evaluated for a series of test problems simulating one dimensional flow processes in unsaturated porous media the results indicate that first order accurate schemes are inefficient compared to second order accurate methods that second order accurate noniterative schemes can be quite competitive with the iterative newton and picard methods and that the newton scheme is no less efficient than the picard method and for strongly nonlinear problems can outperform the picard scheme the two second order accurate noniterative schemes appear to be attractive alternatives to the iterative methods although there are concerns regarding the stability behavior of the three level scheme which need to be resolved we conclude that of the four noniterative strategies presented the implicit factored scheme is the most promising and we suggest improved formulations of the method

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

analytical solution to transient richards equation with realistic water profiles for vertical infiltration and parameter estimation a general analytical model for one dimensional transient vertical infiltration is presented the model is based on a combination of the brooks and corey soil water retention function and a generalized hydraulic conductivity function this leads to power law diffusivity and convective term for which the exponents are functions of the inverse of the pore size distribution index accordingly the proposed analytical solution covers many existing realistic models in the literature the general form of the analytical solution is simple and it expresses implicitly the depth as function of water content and time it can be used to model infiltration through semi infinite dry soils with prescribed water content or flux boundary conditions some mathematical expressions of practical importance are also derived the general form solution is useful for comparison between models validation of numerical solutions and for better understanding the effect of some hydraulic parameters based on the analytical expression a complete inverse procedure which allows the estimation of the hydraulic parameters from water content measurements is presented

extension of the heaslet alksne technique to arbitrary soil water diffusivities note 28 2793 2798 reference eflum article 1992 002 record created on 2005 09 08 modified on 2017 02 23

non integrability of non linear diffusion convection equations in two spatial dimensions it is generally accepted that integrable partial differential time evolution equations possess lie backlund symmetries of arbitrarily high finite order it has already been established that in 1 1 dimensions the full class of nonlinear fokker planck convection diffusion equations having this property consists of the known integrable examples these are closely related either to the burgers equations or to the fokas yortsos rosen equation both of which have been applied to unsaturated flow in porous media here the author shows that higher order lie backlund symmetries do not exist for any examples of the nonlinear fokker planck equation in 1 2 one time and two space dimensions

semiparametric theory for causal mediation analysis efficiency bounds multiple robustness and sensitivity analysis whilst estimation of the marginal total causal effect of a point exposure on an outcome is arguably the most common objective of experimental and observational studies in the health and social sciences in recent years investigators have also become increasingly interested in mediation analysis specifically upon evaluating the total effect of the exposure investigators routinely wish to make inferences about the direct or indirect pathways of the effect of the exposure not through or through a mediator variable that occurs subsequently to the exposure and prior to the outcome although powerful semiparametric methodologies have been developed to analyze observational studies that produce double robust and highly efficient estimates of the marginal total causal effect similar methods for mediation analysis are currently lacking thus this paper develops a general semiparametric framework for obtaining inferences about so called marginal natural direct and indirect causal effects while appropriately accounting for a large number of pre exposure confounding factors for the exposure and the mediator variables our analytic framework is particularly appealing because it gives new insights on issues of efficiency and robustness in the context of mediation analysis in particular we propose new multiply robust locally efficient estimators of the marginal natural indirect and direct causal effects and develop a novel double robust sensitivity analysis framework for the assumption of ignorability of the mediator variable

the green ampt limit with reference to infiltration coefficients 1 recent progress with an analytic nonlinear model has provided the exact infiltration coefficients for realistic soil behaviors with nonsingular hydraulic functions as well as their exact delta function diffusivity limits after some correction and reinterpretation of the approximate analytical method the exactly solvable model validates some previously obtained approximate infiltration functions the green ampt infiltration function follows from a delta function diffusivity limit with a hydraulic conductivity that may be among other possibilities a linear function of water content just as a linear conductivity function is an overestimate for a realistic soil the second philip infiltration coefficient s1 in the green ampt infiltration function is too large due to conductivity being overestimated better agreement with experiment halving the value of s1 is obtained from the analytic nonlinear model with a limiting delta function diffusivity and a matching gardner exponential hydraulic conductivity function in general infiltration behavior is determined by the limiting forms of the diffusivity and conductivity relative to one another at the saturated water content or alternatively the relationship between the conductivity and soil moisture potential a new infiltration model demonstrates the possible range of s1 for physically valid limiting conductivity functions we show that in the delta function diffusivity limit the solution behaves as if the potential at the wet front were time dependent decreasing in magnitude from an initial value at the traditional green ampt level

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

linearized unsteady multidimensional infiltration the parameters in the linearized multidimensional unsteady equation for unsaturated flow in homogeneous soils can be chosen so that small time flow rates are given correctly with the quasi linear steady solution as the large time limit matched in this way the linearized equation offers insight into the time course of multidimensional infiltration the laplace transform reduces the equation and its governing conditions into forms similar to those for quasi linear steady flows the problems of unsteady infiltration from buried spherical and cylindrical cavities are solved in many cases the solutions are approximately reducible to the product of the steady solution and a function only of time and radial coordinate theorems show that approximate product solutions apply not only to these configurations but generally to multidimensional cavities of arbitrary shape multidimensional infiltration from finite supply surfaces involves two characteristic lengths and two characteristic times one set deriving from the capillarity gravity interaction and the other from the capillarity geometry interaction we explore the influence of the two characteristic times on the time scales of approach to steady moisture distribution near the supply surface and to steady discharge the analysis provides explanation of the relatively rapid approach to the steady state observed in three dimensional infiltration from small supply surfaces

a perturbative view on the subsurface water pressure response at hillslope scale 1 this work deals with a rearrangement of richards equation to obtain a better understanding of the components of the subsurface water and pressure flows and their interactions in a hillslope the basic idea proposed in this paper is the normalization of the spatial and temporal coordinates according to the physical knowledge of the processes and the geometry of a hillslope the pressure head and all the other quantities contained in richards equation are rescaled to obtain a dimensionless equation the pressure head is also split into an equilibrium component and a transient component and expanded in a perturbation series subsequently richards equation itself is conveniently split into two equations connected by a source sink term results of these manipulations show that for the first order approximation the water and pressure flow is slope normal while slope parallel effects only occur at successive orders integrating the perturbed equation obtained for the long term pressure head component results in the well known boussinesq equation for the water table level further integrations then lead to the hillslope storage boussinesq model and finally to a generalization of the o loughlin formula for the slope parallel long term subsurface flow the introduced source sink term is estimated at the first order of approximation to be proportional to the temporal variation of water pressure head at the bottom of the integration domain the treatment of the equations clarifies some of the mechanisms acting in water pressure redistribution at hillslope scale even without solving the equation

a new family of finite difference schemes to solve the heterogeneous acoustic wave equation equivalent staggered grid scheme esg is a new family of schemes based on the finite difference method fdm the method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic fdm approaches the main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations ssg but with the same memory requirements of simple grid schemes hence it results in a reduction of memory usage by 33 in 2d and 50 in 3d problems compared to the memory usage of ssg the first order system of equations in terms of pressure and velocity is not used here instead the formulation is based on applying new central difference operators to the secondorder acoustic wave equation in terms of pressure obtaining the same level of accuracy and stability as the ssg schemes the equivalence between the esg and ssg is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed