models for disc infiltrometers disc infiltrometers are popular devices for determining in situ hydraulic properties of unsaturated soils this paper compares alternative solutions of richards equation for both steady state and time dependent cases the steady state solutions were in general agreement using alternative hydraulic conductivity functions of the same capillary length scale small time solutions for the nonlinear cases were consistent with linear diffusion from a disc source using an average diffusivity value which is simply related to the capillary length this offers a refinement over the one dimensional solution for short times in that the geometric effect of the circular source is included simulations for two examples indicate that the approach to the steady state solution may take considerably longer than what is commonly reported in the literature for field applications

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

theory of external two state markov noise in the presence of internal fluctuations a theory is presented to take into account internal fluctuations in the study of stochastically driven systems internal fluctuations are modeled by a master equation in which external noise is introduced external noise is modeled by a two state markov process a unified theory of internal and external fluctuations is described in terms of an effective integrodifferential master equation or its equivalent generating function representation two examples for which exact analytical results can be obtained are presented a discussion of the white noise limit of the theory is also given

unbounded solutions of the equation asymptotic behavior of solutions of first order differential equation with deviating arguments in the email protected t email protected email protected i t y email protected i y email protected i is discussed for t a criterion for representing solutions in exponential form is proved inequalities for solution estimation are given sufficient conditions for the existence of unbounded solutions are derived a relevant illustrative example is given as well known results are discussed and compared

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

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

the continuous time resolvent matrix for non markovian chains abstract a unified theory of discrete and continuous time resolvent matrix method is described in terms of the waiting time density we obtain a differential evolution equation for the marginal probability distribution after summing over the internal states of a markovian chain the example of the continuous time lorentz gas model is presented as an application of this theory

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

a simplified newton iteration method with linear finite elements for transient unsaturated flow a simplified newton iteration method with galerkin linear finite elements is developed for the solution of the nonlinear transient unsaturated flow equation in mixed form the use of the galerkin linear finite element method in the spatial discretization produces a recurrence equation in which the interaode hydraulic conductivity is represented by an integrated average expression which generates the exact darcy flow flux through each element under the steady horizontal flow condition the numerical computation of the integrated average hydraulic conductivity using simpson s 1 3 rule leads to an accurate solution even in the presence of a sharp front in the time discretization using a backward euler scheme the standard newton iteration method is simplified by taking account of the integrated average expression of the hydraulic conductivity and the diffusion dominated nature of the flow process the resulting two level scheme is of the same efficiency on a per iteration basis as the modified picard scheme but converges at a faster rate

second order accurate monotone finite volume scheme for richards equation in this work we perform a theoretical and numerical analysis of richards equation for certain types of nonlinearities we provide explicit analytical solutions these solutions are used to show that conventional unconditionally monotone finite volume schemes have only first order accuracy we derive necessary and sufficient conditions for the monotonicity of finite volume discretizations and use these conditions to construct a monotone finite volume discretization accurate to second order

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

estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils simulations of transient unsaturated vertical flow in different initially dry homogeneous soils using richards equation suggest that for a relatively coarse numerical grid there is no universal weighting scheme to estimate interblock conductivities capable of giving an accurate resolution of the resulting steep wetting fronts except for the scheme associated with the kirchhoff transformation kt the principal drawback of kt is that it can hardly be applied to heterogeneous soils a new asymptotic weighting aw scheme based on the asymptotic behavior of the hydraulic conductivity function in the vicinity of residual saturation is proposed numerical tests demonstrate that the aw scheme describes wetting fronts in homogeneous soils nearly as accurately as a scheme based on kt applicability and robustness of the aw scheme for simulating water infiltration into heterogeneous soils having substantial vertical variation of hydraulic properties are also demonstrated extension of the proposed scheme to higher dimensions is straightforward

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

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

asymptotic convergence criteria of solutions of delayed functional differential equations abstract the convergence of all solutions of the scalar equation x f t x t is studied under the main assumption that every constant is its solution the criterion and the sufficient conditions for convergence are proved for the considered classes of equations it is among others proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution as applications criteria for some classes of linear equations as well as for nonlinear equations are obtained some comparisons with already known results are given too

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

convergence of the solutions of the equation y t β t y t δ y t τ in the critical case we study the asymptotic behavior of the solutions of the first order differential equation containing two delays y t t y t y t with t0 r 0 the convergence of all solutions is characterized by the existence of a strictly increasing bounded solution a critical case is found for the coefficient function for coefficients below the critical function a strictly increasing and bounded solution is constructed and thus the convergence of all solutions is shown relations with known results are discussed too

asymptotic representation of solutions of equationẏ t β t y t y t τ t abstract in this paper the equation t t y t y t t is considered where c i 1 r i 1 t 1 r 0 t t is an increasing function on i 1 t t i 1 0 c i 1 r results about asymptotic structure and behavior of the solutions of this equation are proved a connection with the equation t c t x t t and the asymptotic representation of its solutions is obtained

mass balance errors when solving the convective form of the transport equation in transient flow problems 1 the original divergence form of the solute transport equation expresses the mass balance of solutes in groundwater the alternative convective form can be obtained by subtracting the flow equation multiplied by concentration from the original form many investigators prefer the latter because it facilitates the use of lagrangian or mixed eulerian lagrangian numerical methods for solution however it performs very poorly in transient contaminant transport problems with large temporal variations in water content we propose an alternative to overcome this difficulty it consists of evaluating the water content at time level k 1 if concentration in the convective term is evaluated at time level k for this reason we have called it reverse time weighting whereas we call an evaluation of the water content at time level k straight time weighting we show that reverse time weighting improves significantly over straight time weighting not only in terms of mass balance errors but also in terms of concentration errors

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

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

unbounded increasing solutions of a system of difference equations with delays we consider a homogeneous system of difference equations with deviating arguments in the form y n k 1 q k n y n p k y n r k where n n 0 n 0 z p k r k are integers r k p k 0 q is a positive integer y y 1 y s t y n 0 r n 0 r 1 r s is an unknown discrete vector function s 1 is an integer r max r 1 r q y n y n 1 y n and k n ij k n i j 1 s are real matrices such that ij k n 0 n 0 1 0 and k 1 q j 1 s ij k n 0 for each admissible i and all n n 0 the behavior of solutions of this system is discussed for n the existence of unbounded increasing solutions in an exponential form is proved and estimates of solutions are given the scalar case is discussed as well

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

solving the nonlinear richards equation model with adaptive domain decomposition modeling the transport processes in a vadose zone plays an important role in predicting the reactions of soil biotopes to anthropogenic activity e g modeling contaminant transport the effect of the soil water regime on changes in soil structure and composition etc water flow is governed by the richards equation while the constitutive laws are typically supplied by the van genuchten model which can be understood as a pore size distribution function certain materials with dominantly uniform pore sizes e g coarse grained materials can exhibit ranges of constitutive function values within several orders of magnitude possibly beyond the length of real numbers that computers can handle thus a numerical approximation of the richards equation often requires the solution of systems of equations that cannot be solved on computer arithmetics an appropriate domain decomposition into subdomains that cover only a limited range of constitutive function values and that will change adaptively reflecting the time progress of the model will enable an effective and reliable solution of this problem parts of this problem have already been treated in our work kuraz et al 2014 this paper focuses on improving the performance of a nonlinear solver by locating the areas with abrupt changes in the solution

exponential solutions of equation ẏ t β t y t δ y t τ abstract asymptotic behaviour of solutions of first order differential equation with two deviating arguments of the form y t t y t y t is discussed for t a criterion for representing solutions in exponential form is proved as consequences inequalities for such solutions are given connections with known results are discussed and a sufficient condition for existence of unbounded solutions generalizing previous ones is derived an illustrative example is considered too

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

the courant and peclet number criteria for the numerical solution of the richards equation recent studies indicate that inaccuracies exist in the numerical solutions of the richards equation a need exists to examine such inaccuracies and to correlate them formally to the soil hydraulic properties and the size of the spatial and temporal mesh in the absence of criteria for mesh design mass conservation and the rate of convergence are usually used as bases for acceptance of the solution it is well documented however that neither can guarantee an accurate solution this paper proposes the courant and peclet numbers as criteria for estimating spatial and temporal increments the study defines upper limits of these numbers for acceptable accuracy of the numerical solutions for cases that are characterized by decreasing pressure head gradients although the initial size of the time step may be extremely small efficient solutions are possible through a careful choice of an expansion factor for such a step

convergence of bounded solutions for a class of systems of delay differential equations abstract in this paper the convergence of a class of systems of delay differential equations is considered we show that every bounded solution of such systems tends to an equilibrium under certain hypotheses our results extend some corresponding ones already known

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

a theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations iii the markovian non markovian mix abstract the proof of a theorem that allows one to construct deterministic evolution equations from a set with two subsets containing two types of discrete stochastic evolution equation is developed one subset evolves markovianly and the other non markovianly as an illustrative example the deterministic evolution equations of quantum electrodynamics are derived from two sets of markovian and non markovian stochastic evolution equations of different type after an average over realization using the theorem this example shows that deterministic differential equations that contain both first order and second order time derivatives can be derived after a taylor series expansion of the dynamical variables it is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations two explicit examples the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations

asymptotic convergence of the solutions of a discrete system with delays abstract a system of s discrete equations y n n y n j y n k is considered where k and j are integers k j 0 n is a real s s square matrix defined for n n 0 k n 0 z with non negative elements ij n i j 1 s such that j 1 s ij n 0 y y 1 y 2 y s t n 0 k n 0 k 1 r s and y n y n 1 y n for n n 0 a method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically convergent under some conditions i e for every solution y n defined for all sufficiently large n there exists a finite limit lim n y n moreover it is proved that the asymptotic convergence of all solutions is equivalent to the existence of one asymptotically convergent solution with increasing coordinates some discussion related to the so called critical case known for scalar equations is given as well

a linearization of a backward euler scheme for a class of degenerate nonlinear advection diffusion equations abstract in this paper we propose a linearization of a backward scheme for a class of degenerate nonlinear advection diffusion equations modeling a two phase flow through a porous medium this is done using a taylor expansion of the fractional flow function and of the inverse function of an antiderivative of the diffusion coefficient the coercivity of the matrix obtained in the linearization is shown for a sufficiently small time step

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

non markovian stochastic evolution equations abstract non markovian continuum stochastic and deterministic equations are derived from a set of discrete stochastic and deterministic evolution equations examples are given of discrete evolution equations whose updating rules depend on two or more previous time steps among them the continuum stochastic evolution equation of the newton second law the stochastic evolution equation of a wave equation the stochastic evolution equation for the scalar meson field etc are obtained as special cases extension to systems of evolution equations and other extensions are considered and examples are given the concept of isomorphism and almost isomorphism are introduced in order to compare the coefficients of the continuum evolution equations of two different smoothing procedures that arise from two different approaches usually these discrepancies arising from two sources on the one hand the use of different representations of the generalized functions appearing in the models and on the other hand the different approaches used to describe the models these new concept allows to overcome controversies that were appearing during decades in the literature

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

a linear backward euler scheme for the saturation equation regularity results and consistency we consider a linearization of a numerical scheme for the saturation equation or porous medium equation email protected email protected email protected f s email protected email protected k s s 0 through first order expansions of the fractional function f and the inverse of the function k s email protected 0 sk t email protected after a regularization of the porous medium equation we establish a regularity result for the continuous galerkin method and a regularity result for the linearized scheme analogous to the corresponding nonlinear scheme we then show that the linearized scheme is consistent with the nonlinear scheme analyzed in a previous work

marginal distribution of non markovian stochastic processes with internal states abstract we obtain the differential equation for the dynamical evolution of the marginal probability distribution of non markovian processes with internal states 1 2 m the memory kernel for the generalized master equation of the marginal processes shows explicitly the dependence on the m initial conditions p k t 0 t p k t 0 m 1 t p k t 0 we apply this approach to study the composite non markovian 2 2 case

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

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

asymptotic convergence of the solutions of a discrete equation with several delays abstract a discrete equation y n i 1 s i n y n j i y n k i with integer delays k i and j i k i j i 0 is considered it is assumed that i z n 0 k 0 n 0 z k max k 1 k 2 k s z p p p 1 p z i 1 s i n 0 s z 1 is a fixed integer and n z n 0 criteria of asymptotic convergence of solutions are expressed in terms of inequalities for the functions i i 1 s some general properties of solutions are derived as well it is e g proved that for the asymptotical convergence of all solutions the existence of a strictly monotone and asymptotically convergent solution is sufficient a crucial role in the analysis of convergence is played by an auxiliary inequality derived from the form of a given discrete equation

a finite element collocation method for variably saturated flow in two space dimensions this paper introduces a finite element collocation technique for solving the equation governing two dimensional flow in a variably saturated porous medium the scheme uses a mass conserving formulation of richards equation as the basis for the finite difference time stepping method collocation in tensor product spaces of hermite cubics yields a computationally efficient finite element approximation of the spatial derivatives a newton like iteration gives a temporally stable implicit scheme the paper examines two sample problems including an initial boundary value problem involving subsurface irrigation

determining soil hydraulic properties by parameter estimation on the selection of a model for the hydraulic properties parameter estimation procedures involving solution of the inverse flow problem pertinent to a transient experiment constitute a powerful method of determining the soil hydraulic properties one of the problems associated with these procedures is the selection of an appropriate model to describe the soil hydraulic properties three different models were considered those of van genuchten vg and brooks and corey bc model and the exponential model of gardner for the hydraulic conductivity soil water pressure relationships coupled with a new expression for the soil water content soil water pressure relationship gr model for a given model the soil hydraulic properties were determined from simulated and measured outflow experiments supplemented with water content at a soil water pressure head of 15 000 cm h2o using the parameter estimation procedure of kool et al 1985a and data from two soils a hypothetical sandy loam soil assumed to be described by the vg model and a silt loam soil model validation tests were performed and the most appropriate model was selected from the candidate models by discrimination tests using the akaike information criterion aic in the case of the hypothetical sandy loam the vg model was found to be the most accurate and most consistent with the data the performance of both the bc and the gr models in terms of the aic was less good but very similar to each other in the case of the silt loam soil the vg model with additional parameter m which accounts for the correlation between pores and for the flow path tortuosity m 2 02 rather than a constant m 0 5 was found to be most accurate and most consistent with the data the performance of the bc and the gr models was similar when m was considered as an unknown parameter instead of a constant

convergence of solutions of a class of delay differential systems abstract this paper presents a new and concise approach for investigating the convergence of the solutions of a class of delay differential systems it is shown that each solution of the delayed system is convergent to a constant vector the result obtained extends the existing ones in the literature

numerical approximations of darcian flow through unsaturated soil the choice of appropriate intrablock approximations for the unsaturated hydraulic conductivity is a critical step for the numerical solution of richards equation forms commonly used include arithmetic geometric and harmonic averaging the choice is most often based on convenience and custom this study presents a framework for comparing any scheme for weighting the unsaturated hydraulic conductivity against the correct darcian flow across a block for steady conditions such comparisons are carried out for a number of commonly used averages and errors are shown to be alarmingly high in some cases a scheme for approximately matching the correct darcian flux is illustrated by two infiltration examples in the first the arithmetic and weighted schemes are superior to the geometric average when compared to a quasi analytical solution for the second example the weighted solution is slightly better than either the arithmetic or geometric averaging schemes these examples also illustrate that an acceptable mass balance should not be used as a sole criterion for success as it can probably be attained with most schemes the conclusions are believed relevant for algorithms which are pressure head based water content based or other choices and for multidimensional as well as one dimensional flow problems

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

non markovian diffusion like equation for transport processes with anisotropic scattering abstract in order to describe particles diffusing in a medium with anisotropic scattering we have derived a non markovian pseudo diffusion equation starting from the solution of the one dimensional persistent random walk model the differential equation has a space time convolution with a well defined kernel the analytic solution of this non markovian diffusion equation in fourier space gives the conditional probability essentially a renormalized gaussian distribution which is a function of the order parameter p q p and q being the forward and backward scattering probabilities respectively the velocity autocorrelation function has also been obtained analytically showing the dependence with the order parameter

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

a theorem allowing to derive deterministic evolution equations from stochastic evolution equations ii the non markovian extension abstract deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of non markovian stochastic evolution equations after an average over realization using a theorem examples are given show that deterministic differential equations that contain derivatives with respect to time higher than or equal to two can be derived after a taylor series expansion of the dynamical variables it is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations that increase in number after increasing the number of previous time steps in the updating rules that define a given model two explicit examples the first containing updating rules that depend on two previous time steps and the second on three are worked in some detail in order to show some features of the linear transformation that allow one to obtain the deterministic differential equations

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

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

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

exact generating function for the two dimensional lattice lorentz gas abstract we find the generating function for the two dimensional lattice lorentz gas by the resolvent matrix method introducing a direct procedure to calculate its elements the asymptotic behaviour of properties as moments of the distribution and the probability of returning to the origin is analyzed in the discrete time formulation and extended to continuous time we also discuss briefly a disordered model this result can be applied to a variety of physical 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

adaptive kronrod patterson integration of non linear finite element matrices efficient simulation of unsaturated moisture flow in porous media is of great importance in many engineering fields the highly non linear character of unsaturated flow typically gives sharp moving moisture fronts during wetting and drying of materials with strong local moisture permeability and capacity variations as result it is shown that these strong variations conflict with the common preference for low order numerical integration in finite element simulations of unsaturated moisture flow inaccurate numerical integration leads to errors that are often far more important than errors from inappropriate discretization in response this article develops adaptive integration based on nested kronrod patterson gauss integration schemes basically the integration order is adapted to the locally observed grade of non linearity adaptive integration is developed based on a standard infiltration problem and it is demonstrated that serious reductions in the numbers of required integration points and discretization nodes can be obtained thus significantly increasing computational efficiency the multi dimensional applicability is exemplified with two dimensional wetting and drying applications while developed for finite element unsaturated moisture transfer simulation adaptive integration is similarly applicable for other non linear problems and other discretization methods and whereas perhaps outperformed by mesh adaptive techniques adaptive integration requires much less implementation and computation both techniques can moreover be easily combined copyright 2009 john wiley sons ltd

stochastic evolution equations within the context of both the hamiltonian and lagrangian formalisms abstract it is proved that it is possible to obtain continuum deterministic and stochastic evolution equations from a set of discrete stochastic rules after an average over realizations and over near neighbors or coarse graining on the dynamical variables respectively examples are given that allow us to find the hamilton evolution equation for the dynamical variables and the euler evolution equation for the lagrangian of the system with additive noises

estimation of internodal permeabilities for numerical simulation of unsaturated flows 1 a scheme for the calculation of vertical interblock permeabilities for unsaturated flow in soil was developed and illustrated for one and two dimensional unsaturated flows in the proposed approach interblock permeabilities were calculated from an average weight of contiguous grid point permeabilities a large set of exact database weight values was fitted to a highly nonlinear empirical correlation equation as a function of the two contiguous block permeabilities the dimensionless grid spacing and a soil dependent parameter the present scheme was applied to the popular van genuchten and brooks and corey hydraulic functions for relative permeabilities ranging from 10 8 to 1 dimensionless grid spacing ranging from 0 01 to 1 and the soil parameter n of value between 1 05 and 5 the present approach is more accurate than other simple averaging schemes at the same time the computational effort for the present approach is comparable to that in which the arithmetic and geometric means are used while enabling to estimate fluxes even with coarse grids

a topological theorem and correlations within the context of stochastic evolution abstract a topological theorem that proves that any d dimensional lattice is equivalent to a one dimensional one allows to write the evolution equations as a function of only one spatial coordinate stochastic and continuum deterministic evolution equations are derived from a set of discrete stochastic evolution equations the evolution equations of the dynamical variables and correlations are obtained for processes that evolve non markovianly some relatively simple examples are given in order to illustrate the procedures

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

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

convergence in asymptotically autonomous functional differential equations in this paper we consider linear and nonlinear perturbations of a linear autonomous functional differential equation which has infinitely many equilibria we give sufficient conditions under which the solutions of the perturbed equation tend to the equilibria of the unperturbed equation at infinity as a consequence we obtain sufficient conditions for systems of delay differential equations to have asymptotic equilibrium

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

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