use of iterative refinement in the solution of sparse linear systems it is well known that if gaussian elimination with iterative refinement ir is used in the solution of systems of linear algebraic equations ax b whose matrices are dense then the accuracy of the results will usually be greater than the accuracy obtained by the use of gaussian elimination without iterative refinement ds however both more storage about 100 because a copy of matrix a is needed and more computing time some extra time is needed to perform the iterative process must be used with ir normally when the matrix is sparse the accuracy of the solution computed by some sparse matrix technique and ir will still be greater in this paper it is verified by many numerical experiments that the use of sparse matrix techniques with ir may also result in a reduction of both the computing time and the storage requirements this will never happen when ir is applied for dense matrices two parameters a drop tolerance t geqq 0 and a stability factor u 1 are introduced in the effo

a mathematical formulation for reactive transport that eliminates mineral concentrations we present a concise and general mathematical formulation for reactive transport in groundwater for general applications by means of linear algebraic manipulations of the stoichiometric coefficients of the chemical reactions we are able to reduce the number of unknowns of the equations to be solved to the number of degrees of freedom according to thermodynamic rules we present six formulations that differ from each other by number and type of unknowns and discuss their advantages and disadvantages with respect to the two most important numerical solution methods the sequential iteration approach sia and the direct substitution approach dsa our conclusion is that the proposed reduction of the number of variables is of special interest for the dsa we have applied one of these formulations to an example of the flushing of saline water by fresh water

a global strategy for solving reactive transport equations reactive transport models are complex nonlinear partial differential algebraic equations pdae coupling the transport engine with the geochemical operator we propose an efficient and robust global numerical method based on a method of lines and differential algebraic equations dae solvers combined with a newton method using a powerful sparse linear solver numerical experiments show the performances of the method we also propose a unified framework to describe classical methods such as sequential non iterative approach sequential iterative approach direct substitution approach and compare them with our global method

analysis of operator splitting in the nonasymptotic regime for nonlinear reaction diffusion equations application to the dynamics of premixed flames in this paper we mathematically characterize through a lie formalism the local errors induced by operator splitting when solving nonlinear reaction diffusion equations especially in the nonasymptotic regime the nonasymptotic regime is often attained in practice when the splitting time step is much larger than some of the scales associated with either source terms or the diffusion operator when large gradients are present in a series of previous works a reduction of the asymptotic orders for a range of large splitting time steps related to very short time scales in the nonlinear source term has been studied as well as that associated with large gradients but for linearized equations this study provides a key theoretical step forward since it characterizes the numerical behavior of splitting errors within a more general nonlinear framework for which new error estimates can be derived by coupling lie formalism and regularizing effects of the heat equation the validity of these theoretical results is t

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

enhancing random heterogeneity representation by mixing the kriging method with the zonation structure 1 this research develops a zonation kriging zk method and extends the geostatistical theory to investigate the spatially distributed random field and obtain an optimal nonsmooth conditional estimate in the zk method the zonal structure honors the parameter measurements such that each sample location represents a distinct zone while the kriging method also honors the same data set and results in a smooth distribution the zk method integrates the conditional estimates of the zonal structure and the kriged field over a set of weighting coefficients on the sample data the extended geostatistical theory using the zk method is applied to the intrinsic field stationary field and nonstationary field the study shows that the conditional variance using the zk method is bounded between the kriging variance and variogram when estimating hydraulic conductivity by inversion of a groundwater model the optimal values of weighting coefficients are determined via predictive probability of the bayesian decision theory the numerical examples show that the zk method has less dependence on the semivariogram model the zk method outperforms the kriging method in both smooth and nonsmooth fields the zk method shows high efficiency to better the conditional estimate of hydraulic conductivity distribution with low conditional uncertainty and good agreement to the groundwater head observations

computationally efficient algorithms for parameter estimation and uncertainty propagation in numerical models of groundwater flow finite difference and finite element methods are frequently used to study aquifer flow however additional analysis is required when model parameters and hence predicted heads are uncertain computational algorithms are presented for steady and transient models in which aquifer storage coefficients transmissivities distributed inputs and boundary values may all be simultaneously uncertain innovative aspects of these algorithms include a new form of generalized boundary condition a concise discrete derivation of the adjoint problem for transient models with variable time steps an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation and a new efficient first order second moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values the techniques are presented in matrix form and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations details of matrix structures are provided for a two dimensional linear triangular finite element model

robustness of the multiple realization method for stochastic hydraulic aquifer management hydraulic containment of a groundwater plume is modeled as a pump and treat system of extraction wells in a confined aquifer with uncertain transmissivity the transmissivity is modeled as a random field with a specified covariance structure conditioned at measurement locations the response matrix method is used to form a linear program to obtain the optimal pumping strategy high reliability is achieved by employing a multiple realization or stacking method which requires the solution to simultaneously satisfy a number of realizations of the random field empirical tests indicate that this reliability is a monotonically increasing function of the number of stacked realizations and appears to be relatively insensitive to moderate changes in the variance correlation length and covariance structure of the stochastic transmissivity and to changes in the boundary conditions and measurement data two simple statistical models based on bayesian analysis and order statistics are formulated leading to predicted reliabilities of n 1 n 2 and n n 1 respectively where n is the number of stacked realizations these simple curves are shown to compare quite well with the empirical findings these results indicate that it may be possible to determine the stack size necessary to achieve a prespecified reliability the robustness of the reliability curve indicates that it may be useful as a preliminary design tool for aquifer management

nonstationary stochastic analysis in well capture zone design using first order taylor s series approximation 1 nonstationarity of flow fields due to pumping wells and its impact on advective transport is of particular interest in well capture zone design and wellhead protection however techniques based on monte carlo methods to characterize the associated capture zone uncertainty are time consuming and cumbersome this paper introduces an alternative approach the mean and covariance of system state variables i e head pore water velocity and particle trajectory are approximated using a first order taylor s series with sensitivity coefficients estimated from the adjoint operator for a system of discrete equations the approach allows nonstationarity due to several sources e g transmissivity pumping boundary conditions to be treated by employing numerical solution methods it is able to handle irregular geometry varying boundary conditions complicated sink source terms and different covariance functions all of which are important factors for real world applications a comparison of results for the taylor s series approximation with those from monte carlo analysis showed in general good agreement for most of the tested particles particle trajectory variance calculated using taylor s series approximation is then used to predict well capture zone probabilities under the assumption of normality of the mass transport s state variables verification of this assumption showed that not all particle trajectories depending on their starting location are normally or log normally distributed however the risk of using the first order method to delineate the confidence interval of a well capture zone is minimal since it marginally overestimates the 2 5 probability contour furthermore this should be balanced against its greater computation efficiency over the monte carlo approach

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

a comparison of four inverse approaches to groundwater flow and transport parameter identification four different methods of parameterizing spatially varying log transmissivities in an inverse approach are compared with respect to prediction accuracy of simulated flow and transport transport parameter estimation is included by two stage feedback optimization in stage one the log transmissivities are estimated by fitting both head and concentration data given initial values of the source concentration and the dispersivities in stage two the source concentration and the dispersivities are estimated by fitting the concentration data with the updated transport parameters final estimates of the log transmissivities are obtained by repeating the optimization of stage one the formulated objective functions are minimized using levenberg marquardt s algorithm the models are applied to synthetic two dimensional transport problems in steady state flow regimes the true log transmissivity fields are generated by the turning bands method thereby incorporating spatial variability the test cases differ in the input variances of the generated fields and with respect to the amount and accuracy of observed transmissivity data

stability of the richardson extrapolation combined with some implicit runge kutta methods the implementation of the richardson extrapolation in combination with different numerical methods for solving systems of ordinary differential equations odes is relatively simple but the important requirement for stability of the computational process may cause serious difficulties for example the commonly used by scientists and engineers trapezoidal rule has good stability properties but its combination with the richardson extrapolation is unstable therefore it is necessary to study in advance and very carefully the stability of the new numerical methods arising when the scientists and the engineers use this computational device in combination with different algorithms for solving systems of odes we are presenting a systematic investigation of the implementation of richardson extrapolation for two implicit runge kutta methods three numerical examples including an atmospheric chemical scheme used successfully in several extensive environmental studies and described mathematically by a very stiff and badly scaled nonlinear system of odes are presented to illustrate the advantages of the presented approach the numerical results show that not only are the computations stable but also the achieved accuracy is higher when the richardson extrapolation is additionally applied it will be possible to derive similar stability and accuracy results for other implicit runge kutta methods

asymptotic expansions of the global error of fixed stepsize methods in his fundamental paper on general fixed stepsize methods skeel 6 studied convergence properties but left the existence of asymptotic expansions as an open problem in this paper we give a complete answer to this question for the special cases of one step and linear multistep methods our proof is shorter than the published ones asymptotic expansions are the theoretical base for extrapolation methods

studying absolute stability properties of the richardson extrapolation combined with explicit runge kutta methods abstract explicit runge kutta methods are considered it is assumed that the number of stages m m 1 2 3 4 is equal to the order p of the selected method the impact of the application of the richardson extrapolation on the absolute stability properties is studied the richardson extrapolation was used until now only in an attempt to increase the accuracy of the numerical approximations or in order to keep the computational errors under some prescribed in advance level another issue the absolute stability of the richardson extrapolation in connection with several numerical methods is the major topic of this study it is shown that not only are the combinations of the richardson extrapolation with explicit runge kutta methods more accurate than the underlying numerical methods but also their absolute stability regions are larger this means that larger time stepsizes can be used during the integration when richardson extrapolation is used the validity of the theoretical results is confirmed by numerical experiments with three carefully chosen examples it is pointed out that the application of richardson extrapolation together with explicit runge kutta methods might be useful when some large scale mathematical models described by systems of partial differential equations are handled numerically

identification of large scale spatial trends in hydrologic data it is often useful to distinguish different scales of variability in hydrologic properties such as hydraulic conductivity in the simplest two scale case large scale variations can be viewed as a trend while small scale fluctuations about this trend can be viewed as a random residual this paper describes a method for estimating spatial trends from scattered field measurements the basic concept is to treat both the trend and the residual as stationary random functions these functions are distinguished by their spatial spectral or covariance properties which may be estimated from available data or simply hypothesized we present two versions of a general algorithm for estimating spatial trends 1 a discrete version which is useful in practical applications where data are limited and irregularly spaced and 2 a continuous version which can be used to study the effects of using incorrect spectral parameters applications of the discrete algorithm to both synthetically generated data and field measurements yield satisfactory trend estimates an analysis based on the continuous algorithm shows that the estimation error lower bound for these applications depends on two dimensionless ratios that is the scale disparity ratio of the trend and residual correlation scales and the signal to noise ratio ratio of the trend and residual variances these ratios may be used to evaluate the feasibility of trend estimation before field samples are actually collected

richardson extrapolated sequential splitting and its application during numerical time integration the accuracy of the numerical solution obtained with a given step size often proves unsatisfactory in this case one usually reduces the step size and repeats the computation while the results obtained for the coarser grid are not used however we can also combine the two solutions and obtain a better result this idea is based on the richardson extrapolation a general technique for increasing the order of an approximation method this technique also allows us to estimate the absolute error of the underlying method in this paper we apply richardson extrapolation to the sequential splitting and investigate the performance of the resulting scheme on several test examples

richardson extrapolation based sensitivity analysis in the multi objective optimization of analog circuits the feasible solutions provided by a multi objective evolutionary algorithm moea in the optimal sizing of analog integrated circuits ics can be very sensitive to process variations therefore to select the optimal sizes of metal oxide semiconductor field effect transistors mosfets but with low sensitivities we propose to perform multi parameter sensitivity analysis however since moeas generate feasible solutions without an explicit equation then we show the application of richardson extrapolation to approximate the partial derivatives associated to the sensitivities of the performances of an amplifier with respect to the sizes of every mosfet the proposed multi parameter sensitivity analysis is verified through the optimization of a recycled folded cascode rfc operational transconductance amplifier ota we show the behavior of the multi parameter sensitivity approach versus generations the final results show that the optimal sizes selected after executing the sensitivity approach guarantee the lowest sensitivities values while improving the performances of the rfc ota

an optimized runge kutta method for the solution of orbital problems we present a new explicit runge kutta method with algebraic order four minimum error of the fifth algebraic order the limit of the error is zero when the step size tends to zero infinite order of dispersion and eighth order of dissipation the efficiency of the newly constructed method is shown through the numerical results of a wide range of methods when these are applied to well known periodic orbital problems

a stochastic inverse solution for transient groundwater flow parameter identification and reliability analysis in this paper a stochastic approach is developed for solving the inverse problem of parameter identification for transient groundwater flow adjoint state equations are derived for the stochastic partial differential equation spde relating transient head and log hydraulic conductivity perturbations the derived equations can be used to calculate the covariance matrix of head observations and the cross covariance matrix between head observations and log hydraulic conductivity measurements for each observation time as well as the covariance and cross covariance matrices between different observation times the number of simulation runs required for obtaining these matrices is equal to the number of head observations the reliability of model prediction is evaluated through the variance estimate method using adjoint sensitivity analysis and the cokriging estimate the reliability evaluation of model prediction can be used as an aid in the optimization of experimental design for sampling strategies a numerical example is given to illustrate the stochastic inverse procedure for a general transient flow problem the example shows that using head observations of all observation times simultaneously produces much better results than using them sequentially commonly known as the quasi steady approach the example also shows that it is possible to obtain a reliable stochastic inverse solution for meeting a given prediction objective based on a limited number of observations

a new class of diagonally implicit runge kutta methods with zero dissipation and minimized dispersion error abstract in this work we propose a new class of a stable diagonally implicit four stage runge kutta r k methods with minimal dissipation and optimally low dispersion error these schemes obtained by minimizing both amplification and phase error enjoy fourth order of accuracy and are suitable for stiff systems emphasis here is to outline an algorithm that can be used to develop diagonally implicit r k methods of diverse stages having low dissipation low dispersion virtues while retaining to a large extent inherent stability and high accuracy this algorithm is subsequently applied to propose two three and four stage diagonally implicit r k schemes one and two dimensional linear and non linear propagation problems are numerically tackled at a relatively higher cfl number and a comprehensive comparison is carried out with other stable diagonally implicit schemes available in the literature to exhibit benefits of optimization

runge kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics in this paper a new runge kutta method with minimal dispersion and dissipation error is developed the chebyshev pseudospectral method is utilized using spatial discretization and a new fourth order six stage runge kutta scheme is used for time advancing the proposed scheme is more efficient than the existing ones for acoustic computations

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

minimum storage runge kutta schemes for computational acoustics abstract in this paper an implementation of some runge kutta schemes that requires 2n storage where n is the number of degrees of freedom of the system is proposed it is shown that some runge kutta schemes for wave propagation proposed by hu hussaini and manthey 1 can be written by using our implementation thus we provide an alternative to the 2n storage implementation proposed by stanescu and habashi 2 which follows the approach of williamson 3 carpenter and kennedy 4 and others in addition new runge kutta schemes with small dissipation and dispersion errors that can be written also by using our implementation are given finally the results of some numerical experiments are presented to compare the behaviour of these methods

a dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations in this work we have derived an optimized upwind compact difference scheme for achieving excellent spatial resolution the derived numerical scheme adds numerical diffusion which is strictly restricted to a high wavenumber region in order to control numerical instabilities as well as to achieve de aliasing ability more importantly the derived numerical scheme has excellent dispersion relation preserving drp property the applicability of the proposed scheme in simulating real flow problems has been demonstrated by solving flow inside a lid driven cavity a transitional flow past ag24 aerofoil and a two dimensional decaying turbulence flow

a comparison of several methods for the solution of the inverse problem in two dimensional steady state groundwater flow modeling two geostatistical approaches for the estimation of hydraulic conductivity and hydraulic head from hydraulic conductivity and hydraulic head measurements are developed for two dimensional steady flow with sinks for both approaches the field of the logarithm of hydraulic conductivity log conductivity is represented as a random field with mean 1 2x 3y where x and y denote cartesian coordinates variance 2 and covariance 2 exp r 2 exp r2 or 2 rr where and r are constants and r is separation distance the first approach uses linearization of the discretized flow equations to allow the construction of the joint covariance matrix of the hydraulic head and log conductivity measurements as functions of the parameters 2 3 2 and or it then uses maximum likelihood estimation to obtain these parameters and also a parameter associated with log conductivity measurement error having found values for the parameters it then uses kriging to form predictors for log conductivity and hydraulic head from measured values of hydraulic conductivity and hydraulic head the second approach uses kriging to form a parameter dependent predictor for log conductivity from measured hydraulic conductivity and then uses this predicted log conductivity placed into the discretized flow equations to compute hydraulic head the parameters are determined by the minimization of the sum of the squares of the difference between the measured and computed hydraulic heads a third approach simply allows the hydraulic conductivity field to be a step function with a different value for hydraulic conductivity assigned to each of several chosen regions in the two dimensional aquifer the assigned hydraulic conductivities are determined by the minimization of the sum of the squares of the difference between the measured and computed hydraulic heads the three approaches are tested for hydraulic head prediction accuracy on two generated test problems one of which is statistically generated and also on a field problem the third approach despite its simplicity performs as well or better than the other approaches

efficient implementation of stable richardson extrapolation algorithms richardson extrapolation is a powerful computational tool which can successfully be used in the efforts to improve the accuracy of the approximate solutions of systems of ordinary differential equations odes obtained by different numerical methods including here combined numerical methods consisting of appropriately chosen splitting procedures and classical numerical methods some stability results related to two implementations of the richardson extrapolation active richardson extrapolation and passive richardson extrapolation are formulated and proved in this paper an advanced atmospheric chemistry scheme which is commonly used in many well known operational environmental models is applied in a long sequence of experiments in order to demonstrate the fact that a it is indeed possible to improve the accuracy of the numerical results when the richardson extrapolation is used also when very difficult badly scaled and stiff non linear systems of odes are to be treated b the computations can become unstable when the combination of the trapezoidal rule and the active richardson extrapolation is used c the application of the active richardson extrapolation with the backward euler formula is leading to a stable computational process d experiments with different algorithms for solving linear systems of algebraic equations are very useful in the efforts to select the most suitable approach for the particular problems solved and e the computational cost of the richardson extrapolation is much less than that of the underlying numerical method when a prescribed accuracy has to be achieved

a stochastic approach to the problem of upscaling of conductivity in disordered media theory and unconditional numerical simulations a general problem in groundwater modeling is how to assign conductivity values to blocks of a numerical model in this study a method is proposed to upscale the conductivity measurements observed at a given scale to block conductivity values for arbitrary block size assuming a multilognormal distribution for the point conductivities the block conductivity appears as a random function whose distribution conditional to point measurements can be derived this conditional distribution allows filling the flow simulator blocks with equiprobable realizations of block conductivities locally conditioned to actual data these realizations could also be conditioned to conductivity values measured at any scale if the size of the measurement support is known the analytical results are compared to results obtained through a monte carlo analysis and a good agreement was found even for variances of ln t larger than 1

the convergence of diagonally implicit runge kutta methods combined with richardson extrapolation runge kutta methods are widely used in the solution of systems of ordinary differential equations richardson extrapolation is an efficient tool for enhancing the accuracy of time integration schemes in this paper we investigate the convergence of the combination of any of the diagonally implicit including also the explicit runge kutta methods with active richardson extrapolation and show that the numerical solution obtained converges under rather natural conditions

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

sdirk methods for stiff odes with oscillating solutions new sdirk methods specially adapted to the numerical solution of stiff systems of odes which are assumed to possess oscillating solutions are obtained our interest is centered on the dispersion phase errors and the dissipation numerical damping of the dominant components in the numerical oscillations when these methods are applied to a homogeneous linear test model two a stable methods with algebraic order 3 and higher order of dissipation are obtained among the members of a family of methods proposed by van der houwen and sommeijer 1989 some numerical results are presented to show the efficiency of the new methods when they are compared with other methods presented in van der houwen and sommeijer 1989

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

operator splitting for chemical reaction systems with fast chemistry abstract splitting methods are frequently used for large scale chemical reaction systems the main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem however a splitting error is introduced the most popular splitting schemes are the first order lie trotter splitting and the second order strang splitting however in case of stiff differential equations the strang splitting suffers from order reduction and both schemes have order one for stiff differential equations therefore the step size restrictions due to the low order can result in a prohibitive small step size hence splitting schemes with order larger than one are necessary for stiff differential equations the richardson extrapolation of the lie trotter splitting is a second order scheme in this paper we examine differential equations with a fast chemical source term and a slow transport term thus all stiffness of the system is related to the chemical source term we show that the richardson extrapolation does not suffer from order reduction in this case furthermore we perform a stability analysis of the extrapolated splitting scheme for a linear test problem thereby the operators of the test problem do not commute

one step and extrapolation methods for differential algebraic systems the paper analyzes one step methods for differential algebraic equations dae in terms of convergence order in view of extrapolation methods certain perturbed asymptotic expansions are shown to hold for the special dae extrapolation solver based on the semi implicit euler discretization the perturbed order pattern of the extrapolation tableau is derived in detail the theoretical results lead to modifications of the known code the efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion

a distributed parameter approach for evaluating the accuracy of groundwater model predictions 2 application to groundwater flow the distributed parameter theory presented by mclaughlin and wood this issue is used to evaluate the accuracy of a groundwater flow model the special case investigated assumes that the model s prediction errors are due primarily to data limitations in this case approximate prediction error moments mean and covariance are obtained by solving two sets of coupled partial differential equations which have the same basic structure as the original flow equation solutions can be obtained with spectral green s function or numerical techniques depending on the assumptions made two examples are solved using a finite element approach the first two dimensional example confirms steady state infinite domain results obtained with spectral methods the second three dimensional example investigates the influence of spatial variability sampling strategy and suboptimal estimation on model prediction accuracy

finite difference schemes for long time integration a general method for constructing finite difference schemes for longtime integration problems is presented it is demonstrated for discretizations of first and second space derivatives however the approach is not limited to these cases the schemes are constructed so as to minimize the global truncation error taking into account the initial data the resulting second order compact schemes can be used for integration times fourfold or more longer than previously studied schemes with similar computational complexity a similar approach was used to obtain improved integration schemes

a geostatistical approach to the inverse problem in groundwater modeling steady state and one dimensional simulations the problem of estimating hydrogeologic parameters in particular permeability from input output measurements is reexamined in a geostatistical framework the field of the unknown parameters is represented as a random field and the estimation procedure consists of two main steps first the structure of the parameter field is identified i e mathematical representations of the variogram and the trend are selected and their parameters are established by using all available information including measurements of hydraulic head and permeability second linear estimation theory is applied to provide minimum variance and unbiased point estimates of hydrogeologic parameters kriging structure identification is achieved iteratively in three substeps structure selection maximum likelihood estimation and model validation and diagnostic checking the methodology was extensively tested through simulations on a simple one dimensional case the results are remarkably stable and well behaved the estimated field is smooth while small scale variability is statistically described as the quality of measurements improves the procedure reproduces more features of the original field the results are also shown to be rather insensitive to deviations from assumptions about the geostatistical structure of the field

application of richardson extrapolation for multi dimensional advection equations abstract a crank nicolson type scheme which is of order two with respect to all independent variables is used in the numerical solution of multi dimensional advection equations normally the order of accuracy of any numerical scheme can be increased by one when the well known richardson extrapolation is used it is proved that in this particular case the order of accuracy of the combined numerical method the method consisting of the crank nicolson scheme and the richardson extrapolation is not three but four

integral expressions of the derivatives of the small angle scattering correlation function the integral expressions of the derivatives of the small angle scattering correlation function are obtained in terms of the parametric equations of the interphase surfaces for each derivative its integral expression is the surface average of a loop integral of a vectorial field that is the only quantity depending on the derivative order the field relevant to the nth derivative is obtained by letting a suitable differential operator act n 2 times on the field relevant to the second derivative both the differential operator and the last vectorial field are given in terms of the interface parametric equations starting from these expressions the limiting values of the derivatives of the correlation function as r 0 are studied by a procedure similar to the one used by kirste and porod kolloid z 184 1 1962 and wu and schmidt j appl crystallogr 4 224 1971 in this way the conjecture put forth by wu and schmidt that when the interfaces are sufficiently smooth all even order derivatives

a theoretical study of the estimation of the correlation scale in spatially variable fields 2 nonstationary fields a nonstationary field may be viewed as a process composed of two components a stochastic component characterized by a relatively small scale high frequency variation hfv and a deterministic drift component characterized by a relatively large scale low frequency variation lfv in this case the estimation of the variogram or the covariance function and its associated correlation scale requires the prior estimation of the drift in this study the effects of the type of drift function the size of the integral scale and the variances associated with the two components on the estimation of the correlation scale were analyzed using a theoretically generated master population of points representing the nonstationary field a set of nonstationary fields was generated by superimposing different drift functions with a given variance l2 on stationary fields with a given integral scale and variance h2 results showed that the shape of the estimated variograms and their associated estimated correlation scales the fractal dimensions and the estimates of the covariance function provided by the parametric estimation procedures were affected by the type of the imposed drift function by the integral scale of the stochastic component and by the ratio between the variances of the two components both the maximum likelihood ml and the restricted maximum likelihood rml estimation procedures may fail to correctly identify and separate the two components when their respective variances are of the same order of magnitude and when the correlation scale or spatial extent of the stochastic component is small compared with the typical size of the field domain in these situations the fractal dimension analysis also may not indicate the dominance of the lfv over the hfv even when l2 2 h2 the model validation tests also may fail to indicate inconsistency between the estimates of the parameters of the structural model and the data when l2 h2 in general the ml and the rml estimation procedures overestimated the integral scale j and the variance c o of the underlying process whereas the iterative generalized least squares and the ordinary least squares procedures coupled with the minimum mean square deviation msd criterion for variogram model fitting underestimated the value of the integral scale however results of model validation tests suggested that these estimates were accurate and consistent with the data and almost indistinguishable in terms of the values of the akaike information criterion consequently different sets of and 0 values may adequately describe a given realization of the nonstationary field to the same level of precision careful examination of the estimated variograms over the entire set of lag classes application of fractal dimension analysis and a validation test of the fitted variogram models are necessary steps for drift detection quantification of the drift however may not be achieved in certain situations

the correlation functions of plane polygons the correlation function of a bounded plane figure of arbitrary shape and its derivatives are studied using their integral expressions from these follows that the correlation function and its first derivative are continuous functions while the second and the higher order derivatives can show finite and or algebraic singularities with exponents equal to n 1 2 and n z in the presence of some well defined geometrical features of the figure boundary the limit values of the first second and third derivatives at the origin are obtained they are similar to those of the three dimensional case in the case of a plane polygon of arbitrary shape the correlation function has a closed analytical form essentially equal to the sum of the values taken by two analytically known functions at appropriate sets of points two simple cases are explicitly worked out for illustration

effects of kriging and inverse modeling on conditional simulation of the avra valley aquifer in southern arizona the avra valley aquifer in southern arizona is modeled stochastically at three levels of uncertainty the highest level of uncertainty occurs when log transmissivity estimates are based on measured values of this parameter but without regard to the geographic location of each measurement point the resulting steady state hydraulic heads in the aquifer computed by unconditional simulation with the aid of a multivariate normal random number generator coupled with a finite element model have a relatively large variance this variance can be reduced by conditioning the log transmissivity estimates on the spatial arrangement of the data by means of kriging when conditional simulation of the aquifer is performed at this intermediate level of uncertainty with the aid of the same technique as in the unconditional case the variance of the predicted head values is reduced by a factor of 3 2 the lowest level of uncertainty is achieved when the log transmissivity estimates are further conditioned on data relating to the flow regime such as flow rates and water levels in observation wells by means of a statistical inverse procedure conditional simulation at this level of uncertainty is a novel concept it results in a hydraulic head prediction variance that is 14 3 times lower than the corresponding variance based on kriged log transmissivities the net effect of conditioning by means of kriging and inverse modeling is to reduce the prediction variance by a factor of 46 0 a similar study performed by binsarti 1980 on the cortaro aquifer in southern arizona showed insignificant variance reduction due to kriging but a factor of four reduction after inverse modeling these results indicate that the conditioning effect of the inverse method may be much greater than that of kriging the fact that conditioning can reduce the hydraulic head prediction variance by as much as a factor of 46 implies that one should be cautious when dealing with the results of stochastic aquifer models in which conditioning effects are disregarded

high order low dissipation low dispersion diagonally implicit runge kutta schemes high order low dissipation low dispersion diagonally implicit runge kutta schemes are analyzed and introduced based on the optimization of amplification and phase errors for wave propagation various optimized schemes can be obtained the new scheme shows negligible dissipation it is illustrated mathematically and numerically that the new scheme preserves fourth order accuracy while the recently developed diagonally implicit runge kutta scheme does not the numerical applications contain the advection equation with and without a stiff nonlinear source term and an oscillatory test the new scheme is a stable as desired for the solution of stiff problems

the behaviour of the local error in splitting methods applied to stiff problems splitting methods are frequently used in solving stiff differential equations and it is common to split the system of equations into a stiff and a nonstiff part the classical theory for the local order of consistency is valid only for stepsizes which are smaller than what one would typically prefer to use in the integration error control and stepsize selection devices based on classical local order theory may lead to unstable error behaviour and inefficient stepsize sequences here the behaviour of the local error in the strang and godunov splitting methods is explained by using two different tools lie series and singular perturbation theory the two approaches provide an understanding of the phenomena from different points of view but both are consistent with what is observed in numerical experiments

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

exponential operator splitting time integration for spectral methods pseudospectral spatial discretization by orthogonal polynomials and strang splitting method for time integration are applied to second order linear evolutionary pdes before such a numerical integration can be used the original pde is transformed into a suitable form trigonometric jacobi and some of their special cases generalized laguerre and hermite polynomials are considered a double representation of a function by coefficients of a polynomial expansion and by values at the nodes associated with a suitable quadrature formula is used for numerical implementation so that it is possible to avoid calculations of matrix exponentials

runge kutta interpolants with minimal phase lag abstract we develop the basic theory for the construction of explicit scaled runge kutta methods srk for the integration of first order differential equations having an oscillatory solution based on this theory and on the theory of runge kutta methods with minimal phase lag we construct an embedded runge kutta 10 12 method and an explicit scaled runge kutta method numerical results indicate the efficiency of the new algorithms

a multiresolution method for distributed parameter estimation a multiresolution method for distributed parameter estimation or inverse problems is studied numerically the identification of the coefficient of an elliptic equation in one dimension is considered as our model problem first multiscale bases are used to analyze the degree of ill posedness of the inverse problem second based on some numerical results it is shown that the method of scale by scale multiresolution yields robust and fast convergence finally it is shown how the method gives a natural regularization approach which is complementary to tikhonov s regularization

asymptotic error expansions for stiff equations the implicit euler scheme this paper discusses the existence of an asymptotic expansion for the global error of the implicit euler scheme applied to stiff nonlinear systems of ordinary differential equations it is shown that in strongly stiff situations a full asymptotic expansion exists at all gridpoints for the mildly stiff case it is shown that the full order of the remainder term which inevitably breaks down at the first gridpoints after a stepsize change reappears at the subsequent gridpoints our analysis is based on singular perturbation techniques

analysis of operator splitting for advection diffusion reaction problems from air pollution modelling abstract operator or time splitting is often used in the numerical solution of initial boundary value problems for differential equations it is for example standard practice in computational air pollution modelling where we encounter systems of three dimensional time dependent partial differential equations of the advection diffusion reaction type for such systems little attention has been devoted to the analysis of splitting and to the question why splitting can work so well from the theoretical point of view the success of splitting is primarily determined by the splitting error this paper presents an analysis of operator splitting aimed at providing insight into the splitting error using the lie operator formalism a general expression is derived for a three term strang splitting in the pure initial value case for a class of advection diffusion reaction problems the splitting error is analyzed in greater detail a special case is discussed in which the splitting error can be reduced also some attention is paid to the use of operator splitting in initial boundary value problems

an analysis of operator splitting techniques in the stiff case operator splitting methods are commonly used in many applications we focus here on the case where the evolution equations to be simulated are stiff we will more particularly consider the case of two operators a stiff one and a nonstiff one this occurs in numerous application fields e g combustion air pollution and reactive flows the classical analysis of the splitting error may then fail since the chosen splitting timestep t is in practice much larger than the fastest time scales the asymptotic expansion t 0 is therefore no longer valid we show here that singular perturbation theory provides an interesting framework for the study of splitting error some new results concerning the order of local errors are derived the main result deals with the choice of the sequential order for the operators the stiff operator must always be last in the splitting scheme

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

numerical integration of damped maxwell equations we study the numerical time integration of maxwell s equations from electromagnetism following the method of lines approach we start from a general semidiscrete maxwell system for which a number of time integration methods are considered these methods have in common an explicit treatment of the curl terms central in our investigation is the question how to efficiently raise the temporal convergence order beyond the standard order of two in particular in the presence of an explicitly or implicitly treated damping term which models conduction

operator splitting for nonlinear reaction diffusion systems with an entropic structure singular perturbation and order reduction in this paper we perform the numerical analysis of operator splitting techniques for nonlinear reaction diffusion systems with an entropic structure in the presence of fast scales in the reaction term we consider both linear diagonal and quasi linear non diagonal diffusion the entropic structure implies the well posedness and stability of the system as well as a tikhonov normal form for the nonlinear reaction term 23 it allows to perform a singular perturbation analysis and to obtain a reduced and well posed system of equations on a partial equilibrium manifold as well as an asymptotic expansion of the solution we then conduct an error analysis in this particular framework where the time scale associated to the fast part of the reaction term is much shorter that the splitting time step t thus leading to the failure of the usual splitting analysis techniques we define the conditions on diffusion and reaction for the order of the local error associated with the time splitting to be reduced or to be preserved in the presence of fast scales all the results obtained theoretically on local error estimates are then illustrated on a numerical test case where the global error clearly reproduces the scenarios foreseen at the local level we finally investigate the discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis

first order analysis of uncertainty in numerical models of groundwater flow part 1 mathematical development in part 1 of this paper first and second order analysis of uncertainty is applied to numerical models of groundwater flow the models are cast in state space form with boundary conditions and inputs that are functionally dependent but statistically independent of time using a compact matrix calculus notation first and second order taylor series expansions of the model equations are derived and used to estimate the mean and variance covariance properties of piezometric head predictions given corresponding statistics for aquifer parameters material properties initial conditions boundary conditions and inputs the mathematical results demonstrate that the prediction uncertainty is a function of the magnitude of the parameter uncertainty and sensitivity of the predictions to the parameters furthermore the first order estimate of the piezometric head is identical to the dete ministic result part 2 of this paper to be presented later will illustrate these and other results through numerous applications of the methodology

stability of operator splitting methods for systems with indefinite operators advection diffusion reaction systems this brief paper presents an a stability result for operator splitting type time integration methods applied to advection diffusion reaction equations with possibly indefinite source terms these results extend our earlier work on diffusion reaction systems d l ropp j n shadid stability of operator splitting methods for systems with indefinite operators reaction diffusion systems j comput phys 203 2 2005 449 466 the a stability result presents sufficient conditions that control both low and high wave number instabilities a corollary shows that if l stable methods are used for the diffusion term the high wave number instability will be controlled more easily numerical results are presented that verify second order convergence for the operator splitting methods and demonstrate control of instabilities on a chemotaxis problem by use of an l stable diffusion integrator

incorporation of prior information on parameters into nonlinear regression groundwater flow models 2 applications this paper investigates factors influencing the degree of improvement in estimates of parameters of a nonlinear regression groundwater flow model by incorporating prior information of unknown reliability consideration of expected behavior of the regression solutions and results of a hypothetical modeling problem lead to several general conclusions first if the parameters are properly scaled linearized expressions for the mean square error mse in parameter estimates of a nonlinear model will often behave very nearly as if the model were linear second by using prior information the mse in properly scaled parameters can be reduced greatly over the mse of ordinary least squares estimates of parameters third plots of estimated mse and the estimated standard deviation of mse versus an auxiliary parameter the ridge parameter specifying the degree of influence of the prior information on regression results can help determine the potential for improvement of parameter estimates fourth proposed criteria can be used to make appropriate choices for the ridge parameter and another parameter expressing degree of overall bias in the prior information results of a case study of truckee meadows reno sparks area washoe county nevada conform closely to the results of the hypothetical problem in the truckee meadows case incorporation of prior information did not greatly change the parameter estimates from those obtained by ordinary least squares however the analysis showed that both sets of estimates are more reliable than suggested by the standard errors from ordinary least squares

optimal high order diagonally implicit runge kutta schemes for nonlinear diffusive systems on atmospheric boundary layer abstract nonlinear diffusion equations are extensively applicable in diverse fields of science and engineering numerical stability is a common concern in this class of equations in the present study a three stage third order diagonally implicit runge kutta dirk scheme is introduced by optimizing the error and linear stability analysis for a commonly used nonlinear diffusive system in atmospheric boundary layer the proposed scheme is stable for a wide range of time steps and able to resolve different diffusive systems with diagnostic turbulence closures or prognostic ones with a diagnostic length scale with enhanced accuracy and stability compared to current schemes it maintains a stability which makes it appropriate for the solution of stiff problems the procedure implemented in this study is quite general and can be used in other diffusive systems as well

stability of the richardson extrapolation applied together with the θ method consider a system of ordinary differential equations odes dy dt f t y where a t a b with b a b y is a vector containing s components and c y a is given the q method is applied to solve approximately the system of odes on a set of prescribed grid points if n is the number of time steps that are to be carried out then this numerical method can be defined using the following set of relationships y n y n 1 h 1 q f t n 1 y n 1 h qf t n y n q 0 5 1 0 n 1 2 n h b a n t n t n 1 h t 0 nh t 0 a t n b as a rule the accuracy of approximations y n n 1 2 n can be improved by applying the richardson extrapolation under the assumption that the stability of the computational process is preserved therefore it is natural to require that the combined numerical method richardson extrapolation the q method is in some sense stable it is proved in this paper that the combined method is strongly a stable when q 2 3 1 0 it is furthermore shown that some theorems proved in a previous paper by the same authors farago et al 2009 1 are simple corollaries of the main result obtained in the present work the usefulness of the main result in the solution of many problems arising in different scientific and engineering areas is demonstrated by performing a series of tests with an extremely badly scaled and very stiff atmospheric chemistry scheme which is actually used in several well known large scale air pollution models

an efficient and rapid numerical quadrature to generate element matrices for quadrilateral and hexahedral elements in functionally graded materials fgms abstract this research paper introduces a robust efficient and stabilized midpoint quadrature to generate element stiffness matrices for quadrilateral and hexahedral elements in finite element method fem this new quadrature is established based on passive richardson extrapolation which has the inherent ability to stabilize and improve the accuracy of the extrapolated results the new quadrature includes a simple artificial stabilizing function with gauss one point quadrature 1gp as an anti hourglass effect for the first approximation the 1gp sampling point to be considered at origin 0 0 is done to evaluate element matrices thermal conductivity matrices capacitance matrices etc in the 4 square element 2 2 x 2 2 in the plane as a stabilizing function sampling points are considered to be either at the center of the four element edges or center of the four quadrants in the reference square element with appropriate constant weighting functions for hexahedral elements also the element stiffness coefficients are calculated only in the center of the element and center of all the octants of the reference cubic element 2 2 x 2 2 x 2 2 the proposed method deals mostly with integer variables and sampling at 0 0 0 whereas gauss quadrature uses lengthy float variables corresponding to irrational numbers in terms of decimals in the stiffness matrix calculations thus consuming time a number of standard example problems are demonstrated to assess the efficiency of the proposed method the results of all the numerical examples are found to be in good agreement with the exact results and conventional gauss quadrature results

stochastic groundwater quality management role of spatial variability and conditioning 1 in this paper a methodology for the stochastic management of groundwater quality problems is presented a stochastic algorithm to solve the coupled flow and mass transport inverse problem is combined with a stochastic management approach to design a reliable pump and treat scheme three main differences exist between this research and previously published research in the field of groundwater management the first is the method of parameterization where we have explicitly considered spatial variability of flow and mass transport parameters in our method second we have used adjoint and or sensitivity equations to calculate first and second derivatives of state variables with respect to hydraulic parameters finally the third point is the nonlinear multiobjective formulation we adopt which enables us to explore the whole trade off curve between reliability and cost of remediation a number of key results are listed here the stochastic inverse algorithm used here the representers method makes it possible to estimate heterogeneity as reflected by the measurements of parameters and state variables meanwhile using the adjoint operator to accurately estimate sensitivity of state variables with respect to the parameters has facilitated the use of the head and concentration measurements in the inverse algorithm moreover stochastic management which is conditioned on head and concentration measurements simultaneously are shown to provide more realistic and reliable remediation schemes finally with the aid of the multiobjective formulation of the management problem and analysis of its results it is possible to identify a significant remediation scheme such a scheme has the advantage of producing maximum possible reliability with the least possible increase in the total cost of remediation

exponential rosenbrock type methods we introduce a new class of exponential integrators for the numerical integration of large scale systems of stiff differential equations these so called rosenbrock type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the jacobian in contrast to standard integrators the methods are fully explicit and do not require the numerical solution of linear systems we analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in banach spaces in particular we derive an abstract stability and convergence result for variable step sizes this analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods we present a third order method with two stages and a fourth order method with three stages respectively the application of the required matrix functions to vectors are computed by krylov subspace approximations we briefly discuss these implementation issues and we give numerical examples that demonstrate the efficiency of the new integrators

asymptotic analysis of two reduction methods for systems of chemical reactions abstract this paper concerns two methods for reducing large systems of chemical kinetics equations namely the method of intrinsic low dimensional manifolds ildms due to maas and pope combust flame 88 1992 239 and an iterative method due to fraser j chem phys 88 1988 4732 and further developed by roussel and fraser j chem phys 93 1990 1072 both methods exploit the separation of fast and slow reaction time scales to find low dimensional manifolds in the space of species concentrations where the long term dynamics are played out the asymptotic expansions of these manifolds e 0 where e measures the ratio of the reaction time scales are compared with the asymptotic expansion of m e the slow manifold given by geometric singular perturbation theory it is shown that the expansions of the ildm and m e agree up to and including terms of o e the former has an error at o e 2 that is proportional to the local curvature of m 0 the error vanishes if and only if the curvature is zero everywhere the iterative method generates term by term the asymptotic expansion of m e starting from m 0 the ith application of the algorithm yields the correct expansion coefficient at o e i while leaving the lower order coefficients invariant thus after l applications the expansion is accurate up to and including the terms of o e l the analytical results are illustrated on a planar system from enzyme kinetics michaelis menten henri and a model planar system due to davis and skodje

the universal structure of the groundwater flow equations most groundwater flow models are not sufficiently detailed to allow an explicit representation of all dynamical scales instead most models are constructed at such a coarse scale of resolution that unresolved subgrid variability often exists it is therefore important to understand the interaction between unresolved dynamics and explicitly resolved dynamics the concept of universality is central to this interaction and is here related to the problem of rescaling models of groundwater flow we examine when it is possible to construct an accurate model for the explicitly resolved large scales without an explicit description of the subgrid scale dynamics we demonstrate how unresolved subgrid scale dynamics interact with resolved scale dynamics we show that a universally valid resolved scale model can be constructed if the resolved dynamics are sufficiently independent of the details of the subgrid scale dynamics in that event a resolved scale model is composed of a universal structure and accompanying model parameters the model parameters represent the effect of unresolved dynamics upon resolved dynamics we examine two possible model structures for groundwater flow we show theoretically and numerically that a local darcy law is a universally valid resolved scale model if the resolved and subgrid scales of the hydraulic conductivity are separated in scale by a so called spectral gap if however the hydraulic conductivity possesses many scales of variability then a more general nonlocal darcy law is a more appropriate model structure when the nonlocal darcy s law is more appropriate numerical experiments suggest that errors using a local darcy s law with effective parameters are most significant at the smallest resolving scale of the model and are minimal at scales between 8 and 16 times the resolving scale

relation between the chord length distribution of an infinitely long cylinder and that of its base chord length distributions are defined for planar and spatial geometric figures there exist connections between the planar and the spatial cases the chord length distribution densities clds of a cylinder b and its base s the cylinder s orthogonal cross section are interrelated by a simple surveyable integral transformation from this transformation it was concluded that the odd moments of the cld f x of s define the leading asymptotic terms of the cld a r of b the power series of a r at r 0 can be traced back to the derivatives of f x in the origin as a general conclusion there exist different geometric bodies b in r3 with the same cld a r an exact recognition of b via its cld is not possible clds do not characterize spatial sets this result is of importance in materials science in order to avoid wrong interpretations in structure research the new cld integral transformation has been illustrated in connection with the transitions triangle triangular rod and rectangle rectangular rod

optimal time advancing dispersion relation preserving schemes in this paper we examine the constrained optimization of explicit runge kutta rk schemes coupled with central spatial discretization schemes to solve the one dimensional convection equation the constraints are defined with respect to the correct error propagation equation which goes beyond the traditional von neumann analysis developed in sengupta et al t k sengupta a dipankar p sagaut error dynamics beyond von neumann analysis j comput phys 226 2007 1211 1218 the efficiency of these optimal schemes is demonstrated for the one dimensional convection problem and also by solving the navier stokes equations for a two dimensional lid driven cavity ldc problem for the ldc problem results for re 1000 are compared with results using spectral methods in botella and peyret o botella r peyret benchmark spectral results on the lid driven cavity flow comput fluids 27 1998 421 433 to calibrate the method in solving the steady state problem we also report the results of the same flow at re 10 000 and compare them with some recent results to establish the correctness and accuracy of the scheme for solving unsteady flow problems finally we also compare our results for a wave packet propagation problem with another method developed for computational aeroacoustics

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

explicit exponential runge kutta methods of high order for parabolic problems exponential runge kutta methods constitute efficient integrators for semilinear stiff problems so far however explicit exponential runge kutta methods are available in the literature up to order 4 only the aim of this paper is to construct a fifth order method for this purpose we make use of a novel approach to derive the stiff order conditions for high order exponential methods this allows us to obtain the conditions for a method of order 5 in an elegant way after stating the conditions we first show that there does not exist an explicit exponential runge kutta method of order 5 with less than or equal to 6 stages then we construct a fifth order method with 8 stages and prove its convergence for semilinear parabolic problems finally a numerical example is given that illustrates our convergence bound

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100