local projection stabilization for the oseen problem and its interpretation as a variational multiscale method we propose to apply the recently introduced local projection stabilization to the numerical computation of the oseen equation at high reynolds number the discretization is done by nested finite element spaces using a priori error estimation techniques we prove the convergence of the method the a priori estimates are independent of the local peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth moreover we show how this method may be cast in the framework of variational multiscale methods we indicate what modeling assumptions must be made to use the method for large eddy simulations

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

simulating mesoscopic reaction diffusion systems using the gillespie algorithm we examine an application of the gillespie algorithm to simulating spatially inhomogeneous reaction diffusion systems in mesoscopic volumes such as cells and microchambers the method involves discretizing the chamber into elements and modeling the diffusion of chemical species by the movement of molecules between neighboring elements these transitions are expressed in the form of a set of reactions which are added to the chemical system the derivation of the rates of these diffusion reactions is by comparison with a finite volume discretization of the heat equation on an unevenly spaced grid the diffusion coefficient of each species is allowed to be inhomogeneous in space including discontinuities the resulting system is solved by the gillespie algorithm using the fast direct method we show that in an appropriate limit the method reproduces exact solutions of the heat equation for a purely diffusive system and the nonlinear reaction rate equation describing the cubic autocatalytic reaction

an lp lq minimization method with cross validation for the restoration of impulse noise contaminated images abstract discrete ill posed problems arise in many areas of science and engineering their solutions if they exist are very sensitive to perturbations in the data regularization aims to reduce this sensitivity many regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term recently the use of a p norm to measure the fidelity term and a q norm to measure the regularization term has received considerable attention the relative importance of these terms is determined by a regularization parameter when the perturbation in the available data is made up of impulse noise and a sparse solution is desired it often is beneficial to let 0 p q 1 then the p and q norms are not norms the choice of a suitable regularization parameter is crucial for the quality of the computed solution it therefore is important to develop methods for determining this parameter automatically without user interaction however the latter has so far not received much attention when the data is contaminated by impulse noise this paper discusses two approaches based on cross validation for determining the regularization parameter in this situation computed examples that illustrate the performance of these approaches when applied to the restoration of impulse noise contaminated images are presented

a reduced order model based preconditioner for the efficient solution of transient diffusion equations this paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive definite linear systems arising from the numerical discretization of transient parabolic and selfadjoint partial differential equations the preconditioners are obtained by nesting appropriate projections of reduced order models into the classical iteration of the preconditioned conjugate gradient pcg the main idea is to employ the reduced order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases this approach is particularly appealing for transient systems where the full model solution has to be computed at each time step in these cases the natural reduced space is the one generated by full model solutions at previous time steps when increasing the size of the projection space the proposed methodology highly reduces the system conditioning number and the number of pcg iterations at every time step the cost of the application of the preconditioner linearly increases with the size of the projection basis and a trade off must be found to effectively reduce the pcg computational cost the quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models c 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

effective pore scale dispersion upscaling with a correlated continuous time random walk approach 1 we investigate the upscaling of dispersion from a pore scale analysis of lagrangian velocities a key challenge in the upscaling procedure is to relate the temporal evolution of spreading to the pore scale velocity field properties we test the hypothesis that one can represent lagrangian velocities at the pore scale as a markov process in space the resulting effective transport model is a continuous time random walk ctrw characterized by a correlated random time increment here denoted as correlated ctrw we consider a simplified sinusoidal wavy channel model as well as a more complex heterogeneous pore space for both systems the predictions of the correlated ctrw model with parameters defined from the velocity field properties both distribution and correlation are found to be in good agreement with results from direct pore scale simulations over preasymptotic and asymptotic times in this framework the nontrivial dependence of dispersion on the pore boundary fluctuations is shown to be related to the competition between distribution and correlation effects in particular explicit inclusion of spatial velocity correlation in the effective ctrw model is found to be important to represent incomplete mixing in the pore throats

catchments as simple dynamical systems a case study on methods and data requirements for parameter identification in many rainfall runoff models at least some calibration of model parameters has to take place especially for ungauged or poorly gauged basins this can be problematic because there is little or no data available for calibration a possible solution to overcome the problems caused by data scarcity is to set up a measurement campaign for a limited time period in this study we determine the minimum amount of data required to determine robust parameter values for a simple model with two parameters the model is constructed such that the parameters can be determined not only with automatic calibration but also by recession analysis and a priori from boussinesq theory the model has been applied to a research catchment in switzerland for automatic calibration and recession analysis one season 5 months is found to be sufficient to give robust parameters for simulation of high flows over the full observation period for automatic calibration this should be the season with the highest precipitation for recession analysis the season with least evapotranspiration the boussinesq equation is able to give good parameter estimates for modeling high flows but detailed in situ knowledge of the catchment is required automatic calibration outperforms recession analysis and boussinesq theory by far when it comes to parameter estimation with a focus on prediction of low flows it was shown that a single set of parameters cannot simultaneously describe high and low flows with a reasonable accuracy suggesting that more than two parameters are needed to characterize subsurface properties

reaction diffusion master equation diffusion limited reactions and singular potentials to model biochemical systems in which both noise in the chemical reaction process and spatial movement of molecules is important both the reaction diffusion master equation rdme and smoluchowski diffusion limited reaction sdlr partial differential equation pde models have been used in previous work we showed that the solution to the rdme may be interpreted as an asymptotic approximation in the reaction radius to the solution of the sdlr pde s a isaacson siam j appl math 70 77 2009 the approximation was shown to be divergent in the limit that the lattice spacing in the rdme approached zero in this work we expand upon these results for the special case of the two molecule annihilation reaction text a text b ensuremath rightarrow ensuremath emptyset we first introduce a third stochastic reaction diffusion pde model that incorporates a pseudopotential based bimolecular reaction mechanism the solution to the pseudopotential model is then shown to be an asymptotic approximation to the solution of the sdlr pde for small reaction radii we next illustrate how the rdme may be obtained by a formal discretization of the pseudopotential model motivating why the rdme is itself an asymptotic approximation of the sdlr pde finally we give a more detailed numerical analysis of the difference between solutions to the rdme and sdlr pde models as a function of both the reaction radius and the lattice spacing in the rdme

a particle tracking method for advective transport in fractures with diffusion into finite matrix blocks a particle tracking method has been developed to calculate tracer transport in fractures with diffusion into finite rock matrix blocks the method is an extension of the work of yamashita and kimura 1990 which is only applicable to diffusion into an infinite matrix the new method has been verified against a number of analytic or semianalytic solutions for transport in a homogeneous fracture medium with matrix diffusion the method is applied to the calculation of tracer breakthrough curves for a hypothetical tracer injection withdrawal experiment in a heterogeneous fracture zone with variable hydraulic properties and finite matrix blocks

particle tracking and the diffusion reaction equation 1 particle tracking algorithms are very useful methods to model conservative transport in surface and subsurface hydrological systems recently a novel ad hoc particle based method was proposed to account for multicomponent reactive transport by benson and meerschaert 2008 this one dimensional particle method has been shown to match theoretical predictions but to date there has been no rigorous demonstration that the particle method actually matches the governing equations for chemical transport we generalize this particle method to two dimensional and three dimensional systems and rigorously demonstrate that this particle method converges to the diffusion reaction equation at the limit of infinitely small time step we also investigate the numerical error associated with the method

on the stable solution of transient convection diffusion equations a transient convection diffusion equation is considered which particularly arises in optimization problems for the asymmetric flow field flow fractionation af4 process a time dependent generalization of the monotone edge averaged finite element eafe scheme is used to obtain a stable discretization in the convection dominated regime new error estimates are proved for this scheme and a comparison with the popular supg method is presented numerical experiments demonstrate that the eafe method is more suitable for problems where boundary layers are formed we consider transient convection diffusion equation with dominating convection the model has an application in optimal control of the asymmetric flow field flow fractionation process for a stable discretization we propose a monotone edge averaged finite element eafe scheme eafe is generalized to a time dependent case and a new error estimate is proved we present numerical results for comparison with a popular supg method

reactive random walk particle tracking and its equivalence with the advection diffusion reaction equation we acknowledge the support of the european research council erc through the project mhetscale 617511 and the spanish ministry of economy industry and competitiveness and european regional development fund through the project mecmat cgl2016 80022 r the numerical data can be reproduced following the numerical algorithms and parameter sets given in the paper alternatively the data can be obtained upon request from the corresponding author

relationship between the reaction diffusion master equation and particle tracking models the reaction diffusion master equation rdme is a model for chemical systems in which both noise in the chemical reaction process and the diffusion of molecules are important it extends the chemical master equation for well mixed chemical reactions by discretizing space into a collection of voxels in this work we show how the rdme may be rewritten as an equivalent particle tracking model following the motion and interaction of individual molecules on a lattice this new representation can be interpreted as a discrete version of the spatially continuous probability distribution function stochastic reaction diffusion model studied by doi we show how this new representation can be mapped to a quantum field theory complementing the existing work by peliti mapping the rdme and doi mapping his spatially continuous model to quantum field theories the formal continuum limit as the voxel size approaches zero of the particle tracking representation is studied to consider the question of whether the rdme approximates any spatially continuous model

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

impact of velocity correlation and distribution on transport in fractured media field evidence and theoretical model flow and transport through fractured geologic media often leads to anomalous non fickian transport behavior the origin of which remains a matter of debate whether it arises from variability in fracture permeability velocity distribution connectedness in the flow paths through fractures velocity correlation or interaction between fractures and matrix here we show that this uncertainty of distribution versus correlation controlled transport can be resolved by combining convergent and push pull tracer tests because flow reversibility is strongly dependent on velocity correlation whereas late time scaling of breakthrough curves is mainly controlled by velocity distribution we build on this insight and propose a lagrangian statistical model that takes the form of a continuous time random walk ctrw with correlated particle velocities in this framework velocity distribution and velocity correlation are quantified by a markov process of particle transition times that is characterized by a distribution function and a transition probability our transport model accurately captures the anomalous behavior in the breakthrough curves for both push pull and convergent flow geometries with the same set of parameters thus the proposed correlated ctrw modeling approach provides a simple yet powerful framework for characterizing the impact of velocity distribution and correlation on transport in fractured media

multigrid methods for convection diffusion problems discretized by a monotone scheme abstract we study multigrid mg methods for the solution of systems of linear algebraic equations obtained from a stable discretization of convection diffusion problems by an exponential fitting scheme the latter ensures the stability of the simplest possible coarse grid operators obtained from galerkin projections based on graph matching linear and nonlinear mg preconditioners are defined in the framework of algebraic multilevel iteration the option of using polynomial smoothers is investigated in context of nonsymmetric problems and a systematic performance comparison is presented for various algorithms on a representative set of two and three dimensional test problems

nonconforming streamline diffusion finite element methods for convection diffusion problems we analyze nonconforming finite element approximations of streamline diffusion type for solving convection diffusion problems both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same o h k 1 2 order of convergence in l 2 as in the conforming case for convection dominated problems a rigorous error analysis supported by numerical experiments is given

efficient random walk particle tracking algorithm for advective dispersive transport in media with discontinuous dispersion coefficients and water contents 1 random walk particle tracking rwpt is a well established and efficient alternative to grid based eulerian approaches when simulating the advection dispersion transport problem in highly heterogeneous porous media however rwpt methods suffer from a lack of accuracy when the dispersion tensor or the water content is spatially discontinuous we present improvements to the concept of a partially reflecting barrier used to account for these discontinuities 1 the nonlinear time splitting with that corrects for the systematic overestimation of the second dispersion displacement across an element interface when linear time splitting is used 2 the one sided reflection coefficient that correctly represents the effect of discontinuous dispersion coefficients and water content but eliminates redundant reflections of the two sided reflection coefficient and limits the error for discrete and 3 the transformation of the dispersive displacement across the element interface for complex multidimensional transport problems the proposed improvements are verified numerically by comparison with an analytical solution and a reference rwpt method the results indicate an increased efficiency and accuracy of the new rwpt algorithm because the new algorithm efficiently simulates both advection and dispersion dominated transport conditions it enhances the applicability of rwpt to scenarios in which both conditions occur as for example in the highly transient unsaturated zone the algorithm is easily implemented and it is shown that the computational benefit increases with increasing variability of the hydraulic parameter field

a particle number conserving lagrangian method for mixing driven reactive transport the purely lagrangian algorithm for chemical reactions introduced by benson and meerschaert 2008 suffers from a low concentration resolution problem we alleviate the problem by redefining the probabilistic collision reaction birth death stochastic process as a mass reduction operation theoretically this corresponds to replacing an on off particle with a large number of subparticles and tracking the number fraction the new particle reaction process maintains the original particle numbers but adjusts each particle s mass upon reaction several simulations show the veracity as well as the gains in low concentration resolution offered by the algorithm we also compare the results to those obtained by a traditional finite difference model with suitably defined initial condition demonstrating that the lagrangian models match these

connecting the dots semi analytical and random walk numerical solutions of the diffusion reaction equation with stochastic initial conditions we study a system with bimolecular irreversible kinetic reaction a b a where the underlying transport of reactants is governed by diffusion and the local reaction term is given by the law of mass action we consider the case where the initial concentrations are given in terms of an average and a white noise perturbation our goal is to solve the diffusion reaction equation which governs the system and we tackle it with both analytical and numerical approaches to obtain an analytical solution we develop the equations of moments and solve them approximately to obtain a numerical solution we develop a grid less monte carlo particle tracking approach where diffusion is modeled by a random walk of the particles and reaction is modeled by annihilation of particles the probability of annihilation is derived analytically from the particles co location probability we rigorously derive the relationship between the initial number of particles in the system and the amplitude of white noise represented by that number this enables us to compare the particle simulations and the approximate analytical solution and offer an explanation of the late time discrepancies

a conjugate gradient like method for p norm minimization in functional spaces we develop an iterative algorithm to recover the minimum p norm solution of the functional linear equation ax b where a mathcal x longrightarrow mathcal y is a continuous linear operator between the two banach spaces mathcal x l p 1 1 with x in mathcal x and b in mathcal y the algorithm is conceived within the same framework of the landweber method for functional linear equations in banach spaces proposed by schopfer et al inverse probl 22 311 329 2006 indeed the algorithm is based on using at the n th iteration a linear combination of the steepest current descent functional a j left b a x n right and the previous descent functional where j denotes a duality map of the banach space mathcal y in this regard the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in hilbert spaces we demonstrate that the proposed iterative algorithm converges strongly to the minimum p norm solution of the functional linear equation ax b and that it is also a regularization method by applying the discrepancy principle as stopping rule according to the geometrical properties of l p spaces numerical experiments show that the method is fast robust in terms of both restoration accuracy and stability promotes sparsity and reduces the over smoothness in reconstructing edges and abrupt intensity changes

evaluation of maximum likelihood parameter estimation techniques for conceptual rainfall runoff models influence of calibration data variability and length on model credibility the success of an automatic calibration procedure is highly dependent on the choice of the objective function and the nature quantity and quality of the data used the objective function should be selected on the basis of the stochastic properties of the errors present in the data and in the model also the data should be chosen so as to contain as much valuable information about the process as possible in this paper we compare the performance of two maximum likelihood estimators the amle which assumes the presence of first lag autocorrelated homogeneous variance errors and the hmle which assumes the presence of uncorrelated inhomogeneous variance errors to the commonly used simple least squares criterion sls the model calibrated was the soil moisture accounting model of the u s national weather service s river forecast system sma nwsrfs the results indicate that a properly chosen objective function can enhance the possibility of obtaining unique and conceptually realistic parameter estimates furthermore the sensitivity of the estimation results to various characteristics of the calibration data such as hydrologic variability and length are substantially reduced

a kernel based lagrangian method for imperfectly mixed chemical reactions abstract current lagrangian particle tracking algorithms used to simulate diffusion reaction equations must employ a certain number of particles to properly emulate the system dynamics particularly for imperfectly mixed systems the number of particles is tied to the statistics of the initial concentration fields of the system at hand systems with shorter range correlation and or smaller concentration variance require more particles potentially limiting the computational feasibility of the method for the well known problem of bimolecular reaction we show that using kernel based rather than dirac delta particles can significantly reduce the required number of particles we derive the fixed width of a gaussian kernel for a given reduced number of particles that analytically eliminates the error between kernel and dirac solutions at any specified time we also show how to solve for the fixed kernel size by minimizing the squared differences between solutions over any given time interval numerical results show that the width of the kernel should be kept below about 12 of the domain size and that the analytic equations used to derive kernel width suffer significantly from the neglect of higher order moments the simulations with a kernel width given by least squares minimization perform better than those made to match at one specific time a heuristic time variable kernel size based on the previous results performs on par with the least squares fixed kernel size

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part ii analysis for p1 and q1 finite elements abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers a review and a comparison of the most methods which have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers are given in part i v john p knobloch on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review comput methods appl mech engrg 196 2007 2197 2215 in the present paper the most promising of these sold methods are investigated in more detail for p 1 and q 1 finite elements in particular the dependence of the results on the mesh the data of the problems and parameters of the methods are studied analytically and numerically furthermore the numerical solution of the nonlinear discrete problems is discussed and the capability of adaptively refined grids for reducing spurious oscillations is examined our conclusion is that also for simple problems any of the sold methods generally provides solutions with non negligible spurious oscillations

incomplete mixing and reactions in laminar shear flow incomplete mixing of reactive solutes is well known to slow down reaction rates relative to what would be expected from assuming perfect mixing in purely diffusive systems for example it is known that small initial fluctuations in reactant concentrations can lead to reactant segregation which in the long run can reduce global reaction rates due to poor mixing in contrast nonuniform flows can enhance mixing between interacting solutes thus a natural question arises can nonuniform flows sufficiently enhance mixing to restrain incomplete mixing effects and if so under what conditions we address this question by considering a specific and simple case namely a laminar pure shear reactive flow two solution approaches are developed a lagrangian random walk method and a semianalytical solution the results consistently highlight that if shear effects in the system are not sufficiently strong incomplete mixing effects initially similar to purely diffusive systems will occur slowing down the overall reaction rate then at some later time dependent on the strength of the shear the system will return to behaving as if it were well mixed but represented by a reduced effective reaction rate

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

incorporating diffusion in complex geometries into stochastic chemical kinetics simulations a method is developed for incorporating diffusion of chemicals in complex geometries into stochastic chemical kinetics simulations systems are modeled using the reaction diffusion master equation with jump rates for diffusive motion between mesh cells calculated from the discretization weights of an embedded boundary method since diffusive jumps between cells are treated as first order reactions individual realizations of the stochastic process can be created by the gillespie method numerical convergence results for the underlying embedded boundary method and for the stochastic reaction diffusion method are presented in two dimensions a two dimensional model of transcription translation and nuclear membrane transport in eukaryotic cells is presented to demonstrate the feasibility of the method in studying cell wide biological processes

spatial markov processes for modeling lagrangian particle dynamics in heterogeneous porous media we investigate the representation of lagrangian velocities in heterogeneous porous media as markov processes we use numerical simulations to show that classical descriptions of particle velocities using markov processes in time fail because low velocities are much more strongly correlated in time than high velocities we demonstrate that lagrangian velocities describe a markov process at fixed distances along the particle trajectories i e a spatial markov process this remarkable property has significant implications for modeling effective transport in heterogeneous velocity fields i the spatial lagrangian velocity transition densities are sufficient to fully characterize these complex velocity field organizations ii classical effective transport descriptions that rely on markov processes in time for the particle velocities are not suited for describing transport in heterogeneous porous media and iii an alternative effective transport description derives from the markovian nature of the spatial velocity transitions it expresses particle movements as a random walk in space time characterized by a correlated random temporal increment and thus generalizes the continuous time random walk model to transport in correlated velocity fields

lagrangian statistical model for transport in highly heterogeneous velocity fields we define an effective lagrangian statistical model in phase space x t v for describing transport in highly heterogeneous velocity fields with complex spatial organizations the spatial markovian nature and temporal non markovian nature of lagrangian velocities leads to an effective transport description that turns out to be a correlated continuous time random walk this model correctly captures the lagrangian velocity correlation properties and is demonstrated to represent a forward model for predicting transport in highly heterogeneous porous media for different types of velocity organizations

time domain random walks for hydrodynamic transport in heterogeneous media we derive a general formulation of the time domain random walk tdrw approach to model the hydrodynamic transport of inert solutes in complex geometries and heterogeneous media we demonstrate its formal equivalence with the discretized advection dispersion equation and show that the tdrw is equivalent to a continuous time random walk ctrw characterized by space dependent transition times and transition probabilities the transition times are exponentially distributed we discuss the implementation of different concentration boundary conditions and initial conditions as well as the occurrence of numerical dispersion furthermore we propose an extension of the tdrw scheme to account for mobile immobile multirate mass transfer finally the proposed tdrw scheme is validated by comparison to analytical solutions for spatially homogeneous and heterogeneous transport scenarios

discontinuous galerkin methods for advection diffusion reaction problems we apply the weighted residual approach recently introduced in f brezzi et al comput methods appl mech engrg 195 2006 pp 3293 3310 to derive discontinuous galerkin formulations for advection diffusion reaction problems we devise the basic ingredients to ensure stability and optimal error estimates in suitable norms and propose two new methods

on finite element methods for 3d time dependent convection diffusion reaction equations with small diffusion the paper studies finite element methods for the simulation of time dependent convection diffusion reaction equations with small diffusion the supg method a sold method and two types of fem fct methods the methods are assessed in particular with respect to the size of the spurious oscillations in the computed solutions at a 3d example with nonhomogeneous dirichlet boundary conditions and homogeneous neumann boundary conditions

a first passage kinetic monte carlo method for reaction drift diffusion processes stochastic reaction diffusion models are now a popular tool for studying physical systems in which both the explicit diffusion of molecules and noise in the chemical reaction process play important roles the smoluchowski diffusion limited reaction model sdlr is one of several that have been used to study biological systems exact realizations of the underlying stochastic processes described by the sdlr model can be generated by the recently proposed first passage kinetic monte carlo fpkmc method this exactness relies on sampling analytical solutions to one and two body diffusion equations in simplified protective domains in this work we extend the fpkmc to allow for drift arising from fixed background potentials as the corresponding fokker planck equations that describe the motion of each molecule can no longer be solved analytically we develop a hybrid method that discretizes the protective domains the discretization is chosen so that the drift diffusion of each molecule within its protective domain is approximated by a continuous time random walk on a lattice new lattices are defined dynamically as the protective domains are updated hence we will refer to our method as dynamic lattice fpkmc or dl fpkmc we focus primarily on the one dimensional case in this manuscript and demonstrate the numerical convergence and accuracy of our method in this case for both smooth and discontinuous potentials we also present applications of our method which illustrate the impact of drift on reaction kinetics

effect of spatial concentration fluctuations on effective kinetics in diffusion reaction systems 1 the effect of spatial concentration fluctuations on the reaction of two solutes a b c is considered in the absence of fluctuations the concentration of solutes decays as adet bdet t 1 contrary to this experimental and numerical studies suggest that concentrations decay significantly slower existing theory suggests a t d 4 scaling in the asymptotic regime d is the dimensionality of the problem here we study the effect of fluctuations using the classical diffusion reaction equation with random initial conditions initial concentrations of the reactants are treated as correlated random fields we use the method of moment equations to solve the resulting stochastic diffusion reaction equation and obtain a solution for the average concentrations that deviates from t 1 to t d 4 behavior at characteristic transition time t we also derive analytical expressions for t as a function of damkohler number and the coefficient of variation of the initial concentration

discontinuous galerkin hp adaptive methods for multiscale chemical reactors quiescent reactors abstract we present a class of chemical reactor systems modeled numerically using a fractional multistep method between the reacting and diffusing modes of the system subsequently allowing one to utilize algebraic techniques for the resulting reactive subsystems a mixed form discontinuous galerkin method is presented with implicit and explicit imex timestepping strategies coupled to dioristic entropy schemes for h p adaptivity of the solution where the h and p are adapted based on an l 1 stability result finally we provide some numerical studies on the convergence behavior adaptation and asymptotics of the system applied to a pair of equilibrium problems as well as to general three dimensional nonlinear lotka volterra chemical systems

bayesian inversion of mualem van genuchten parameters in a multilayer soil profile a data driven assumption free likelihood function this paper introduces a hierarchical simulation and modeling framework that allows for inference and validation of the likelihood function in bayesian inversion of vadose zone hydraulic properties the likelihood function or its analogs objective functions and likelihood measures are commonly assumed to be multivariate gaussian in form however this assumption is not possible to verify without a hierarchical simulation and modeling framework in this paper we present the necessary statistical mechanisms for utilizing the hierarchical framework we apply the hierarchical framework to the inversion of the vadose zone hydraulic properties within a multilayer soil profile conditioned on moisture content observations collected in the uppermost four layers the key result of our work is that the goodness of fit validated likelihood function form provides empirical justification for the assumption of multivariate gaussian likelihood functions in past and future inversions at similar sites as an alternative the likelihood function need not be assumed to follow a parametric statistical distribution and can be computed directly using nonparametric methods the nonparametric methods are considerably more computationally demanding and to demonstrate this approach we present a smaller dimension synthetic case study of evaporation from a soil column the main drawback of our work is the increased computational expense of the inversion

error analysis of the supg finite element discretization of evolutionary convection diffusion reaction equations conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the streamline upwind petrov galerkin supg finite element stabilization of time dependent convection diffusion reaction equations exemplarily it is shown for the supg method combined with the backward euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step the stabilization vanishes in the time continuous limit however based on numerical experience this seems not to be the correct behavior for this reason the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step it is shown that such estimates can be obtained in the case of time independent convection and reaction an error estimate for the time continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady state problem analogous estimates are obtained for the fully discrete case using the backward euler method and the crank nicolson scheme numerical studies support the analytical results

a lagrangian transport eulerian reaction spatial laters markov model for prediction of effective bimolecular reactive transport prediction of effective transport for mixing driven reactive systems at larger scales requires accurate representation of mixing at small scales which poses a significant upscaling challenge depending on the problem at hand there can be benefits to using a lagrangian framework while in others an eulerian might have advantages here we propose and test a novel hybrid model which attempts to leverage benefits of each specifically our framework provides a lagrangian closure required for a volume averaging procedure of the advection diffusion reaction equation this hybrid model is a lagrangian transport eulerian reaction spatial markov model laters markov model which extends previous implementations of the lagrangian spatial markov model and maps concentrations to an eulerian grid to quantify closure terms required to calculate the volume averaged reaction terms the advantage of this approach is that the spatial markov model is known to provide accurate predictions of transport particularly at preasymptotic early times when assumptions required by traditional volume averaging closures are least likely to hold likewise the eulerian reaction method is efficient because it does not require calculation of distances between particles this manuscript introduces the laters markov model and demonstrates by example its ability to accurately predict bimolecular reactive transport in a simple benchmark 2d porous medium

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data we present a computational approach to quantify predictive uncertainty in complex phenomena which is modeled by partial differential equations with uncertain parameters exhibiting multi scale variability the approach is motivated by flow in random composites whose internal architecture spatial arrangement of constitutive materials and spatial variability of properties of each material are both uncertain the proposed two scale framework combines a random domain decomposition rdd and a probabilistic collocation method pcm on sparse grids to quantify these two sources of uncertainty respectively the use of sparse grid points significantly reduces the overall computational cost especially for random processes with small correlation lengths a series of one two and three dimensional computational examples demonstrate that the combined rdd pcm approach yields efficient robust and non intrusive approximations for the statistics of diffusion in random composites

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

time domain particle tracking methods for simulating transport with retention and first order transformation 1 particle tracking in the time domain has received increasing attention as a technique for robustly simulating transport along one dimensional subsurface pathways using a stochastic lagrangian perspective integral representations of transport including the effects of advection longitudinal dispersion and a broad class of retention models are derived monte carlo sampling of that integral leads directly to new time domain particle tracking algorithms that represent a wide range of physical phenomena retention time distributions are compiled for key retention models an extension to accommodate linear transformations such as decay chains is also introduced detailed testing using first order decay chains and four retention models equilibrium sorption limited diffusion unlimited diffusion and first order kinetic sorption demonstrate that the method is highly accurate simulations using flow fields produced by large scale discrete fracture network simulations a transport problem that is difficult for conventional algorithms demonstrate that the new algorithms are robust and highly efficient

spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction diffusion systems stochastic partial differential equations are introduced for the continuum concentration fields of reaction diffusion systems the stochastic partial differential equations account for fluctuations arising from the finite number of molecules which diffusively migrate and react spatially adaptive stochastic numerical methods are developed for approximation of the stochastic partial differential equations the methods allow for adaptive meshes with multiple levels of resolution neumann and dirichlet boundary conditions and domains having geometries with curved boundaries a key issue addressed by the methods is the formulation of consistent discretizations for the stochastic driving fields at coarse refined interfaces of the mesh and at boundaries methods are also introduced for the efficient generation of the required stochastic driving fields on such meshes as a demonstration of the methods investigations are made of the role of fluctuations in a biological model for microorganism direction sensing based on concentration gradients also investigated a mechanism for spatial pattern formation induced by fluctuations the discretization approaches introduced for spdes have the potential to be widely applicable in the development of numerical methods for the study of spatially extended stochastic systems

do we really need a large number of particles to simulate bimolecular reactive transport with random walk methods a kernel density estimation approach random walk particle tracking methods are a computationally efficient family of methods to solve reactive transport problems while the number of particles in most realistic applications is in the order of 10 6 10 9 the number of reactive molecules even in diluted systems might be in the order of fractions of the avogadro number thus each particle actually represents a group of potentially reactive molecules the use of a low number of particles may result not only in loss of accuracy but also may lead to an improper reproduction of the mixing process limited by diffusion recent works have used this effect as a proxy to model incomplete mixing in porous media in this work we propose using a kernel density estimation kde of the concentrations that allows getting the expected results for a well mixed solution with a limited number of particles the idea consists of treating each particle as a sample drawn from the pool of molecules that it represents this way the actual location of a tracked particle is seen as a sample drawn from the density function of the location of molecules represented by that given particle rigorously represented by a kernel density function the probability of reaction can be obtained by combining the kernels associated to two potentially reactive particles we demonstrate that the observed deviation in the reaction vs time curves in numerical experiments reported in the literature could be attributed to the statistical method used to reconstruct concentrations fixed particle support from discrete particle distributions and not to the occurrence of true incomplete mixing we further explore the evolution of the kernel size with time linking it to the diffusion process our results show that kdes are powerful tools to improve computational efficiency and robustness in reactive transport simulations and indicates that incomplete mixing in diluted systems should be modeled based on alternative mechanistic models and not on a limited number of particles

self correcting geometry in model based algorithms for derivative free unconstrained optimization several efficient methods for derivative free optimization are based on the construction and maintenance of an interpolation model for the objective function most of these algorithms use special geometry improving iterations where the geometry poisedness of the underlying interpolation set is made better at the cost of one or more function evaluations we show that such geometry improvements cannot be completely eliminated if one wishes to ensure global convergence but we also provide an algorithm where such steps occur only in the final stage of the algorithm where criticality of a putative stationary point is verified global convergence for this method is proved by making use of a self correction mechanism inherent to the combination of trust regions and interpolation models this mechanism also throws some light on the surprisingly good numerical results reported by fasano morales and nocedal optim methods softw 24 2009 pp 145 154 for a method where no care is ever taken to guarantee poisedness of the interpolation set

bayesian inference of uncertainties in precipitation streamflow modeling in a snow affected catchment 1 bayesian inference is used to study the effect of precipitation and model structural uncertainty on estimates of model parameters and confidence limits of predictive variables in a conceptual rainfall runoff model in the snow fed rudback catchment 142 ha in southern finland the ihacres model is coupled with a simple degree day model to account for snow accumulation and melt the posterior probability distribution of the model parameters is sampled by using the differential evolution adaptive metropolis dream zs algorithm and the generalized likelihood function precipitation uncertainty is taken into account by introducing additional latent variables that were used as multipliers for individual storm events results suggest that occasional snow water equivalent swe observations together with daily streamflow observations do not contain enough information to simultaneously identify model parameters precipitation uncertainty and model structural uncertainty in the rudback catchment the addition of an autoregressive component to account for model structure error and latent variables having uniform priors to account for input uncertainty lead to dubious posterior distributions of model parameters thus our hypothesis that informative priors for latent variables could be replaced by additional swe data could not be confirmed the model was found to work adequately in 1 day ahead simulation mode but the results were poor in the simulation batch mode this was caused by the interaction of parameters that were used to describe different sources of uncertainty the findings may have lessons for other cases where parameterizations are similarly high in relation to available prior information

numerical crosswind smear in the streamline diffusion method abstract the streamline diffusion method is a popular element method for approximating convection dominated transport equations the solution to such equations may develop boundary or internal layers in the crosswind direction orthogonal to the flow direction and this results in numerical crosswind layers it is known that the size of these layers is o h 1 2 or perhaps o h 3 4 h being the size of the mesh used in the numerical computation but in practice o h is observed if the gridlines of the method are aligned with the streamlines of the problem this connection is studied and an explanation is given

from analytical solutions of solute transport equations to multidimensional time domain random walk tdrw algorithms in this study new multi dimensional time domain random walk tdrw algorithms are derived from approximate one dimensional 1 d two dimensional 2 d and three dimensional 3 d analytical solutions of the advection dispersion equation and from exact 1 d 2 d and 3 d analytical solutions of the pure diffusion equation these algorithms enable the calculation of both the time required for a particle to travel a specified distance in a homogeneous medium and the mass recovery at the observation point which may be incomplete due to 2 d or 3 d transverse dispersion or diffusion the method is extended to heterogeneous media represented as a piecewise collection of homogeneous media the particle motion is then decomposed along a series of intermediate checkpoints located on the medium interface boundaries the accuracy of the multi dimensional tdrw method is verified against i exact analytical solutions of solute transport in homogeneous media and ii finite difference simulations in a synthetic 2 d heterogeneous medium of simple geometry the results demonstrate that the method is ideally suited to purely diffusive transport and to advection dispersion transport problems dominated by advection conversely the method is not recommended for highly dispersive transport problems because the accuracy of the advection dispersion tdrw algorithms degrades rapidly for a low peclet number consistent with the accuracy limit of the approximate analytical solutions the proposed approach provides a unified methodology for deriving multi dimensional time domain particle equations and may be applicable to other mathematical transport models provided that appropriate analytical solutions are available

an assessment of discretizations for convection dominated convection diffusion equations the performance of several numerical schemes for discretizing convection dominated convection diffusion equations will be investigated with respect to accuracy and efficiency accuracy is considered in measures which are of interest in applications the study includes an exponentially fitted finite volume scheme the streamline upwind petrov galerkin supg finite element method a spurious oscillations at layers diminishing sold finite element method a finite element method with continuous interior penalty cip stabilization a discontinuous galerkin dg finite element method and a total variation diminishing finite element method femtvd a detailed assessment of the schemes based on the hemker example will be presented

simulating diffusion processes in discontinuous media in this article we propose new monte carlo techniques for moving a diffusive particle in a discontinuous media in this framework we characterize the stochastic process that governs the positions of the particle the key tool is the reduction of the process to a skew brownian motion sbm in a zone where the coefficients are locally constant on each side of the discontinuity the new position of the particle after a constant time step is sampled from the exact distribution of the sbm process at the considered time to do so we propose two different but equivalent algorithms a two steps simulation with a stop at the discontinuity and a one step direct simulation of the sbm dynamic some benchmark tests illustrate their effectiveness

reaction diffusion waves of advance in the transition to agricultural economics in a previous paper j fort and v mendez phys rev lett 82 867 1999 the possible importance of higher order terms in a human population wave of advance has been studied however only a few such terms were considered here we develop a theory including all higher order terms results are in good agreement with the experimental evidence involving the expansion of agriculture in europe

the reaction diffusion master equation as an asymptotic approximation of diffusion to a small target the reaction diffusion master equation rdme has recently been used as a model for biological systems in which both noise in the chemical reaction process and diffusion in space of the reacting molecules is important in the rdme space is partitioned by a mesh into a collection of voxels there is an unanswered question as to how solutions depend on the mesh spacing to have confidence in using the rdme to draw conclusions about biological systems we would like to know that it approximates a reasonable physical model for appropriately chosen mesh spacings this issue is investigated by studying the dependence on mesh spacing of solutions to the rdme in mathbb r 3 for the bimolecular reaction mathrm a mathrm b to varnothing with one molecule of species mathrm a and one molecule of species mathrm b present initially we prove that in the continuum limit the molecules never react and simply diffuse relative to each other nevertheless we show that the rdme with nonzero lattice spacin

two dimensional exponential fitting and applications to drift diffusion models a class of methods is introduced that generalizes the so called exponential fitting method to two dimensional problems and more specifically to equations of the type operatorname div underset raise0 3em hbox smash scriptscriptstyle nabla u u underset raise0 3em hbox smash scriptscriptstyle nabla psi f with psi given and underset raise0 3em hbox smash scriptscriptstyle nabla psi big in a part of the domain the basic idea is the following i write the equation as operatorname div e psi underset raise0 3em hbox smash scriptscriptstyle nabla rho f through the change of variables u rho e psi ii discretize the symmetric equation by means of a method that uses some kind of harmonic average for the coefficient e psi mixed hybrid etc iii write the discrete scheme in terms of the nodal values of u e psi rho this produces an m matrix

use of an entropy based metric in multiobjective calibration to improve model performance parameter estimation for hydrological models is complicated for many reasons one of which is the arbitrary emphasis placed by most traditional measures of fit on various magnitudes of the model residuals recent research has called for the development of robust diagnostic measures that provide insights into which model structural components and or data may be inadequate in this regard the flow duration curve fdc represents the historical variability of flow and is considered to be an informative signature of catchment behavior here we investigate the potential of using the recently developed conditioned entropy difference metric ced in combination with the kling gupta efficiency kge the ced respects the static information contained in the flow frequency distribution and hence the fdc but does not explicitly characterize temporal dynamics the kge reweights the importance of various hydrograph components correlation bias variability in a way that has been demonstrated to provide better model calibrations than the commonly used nash sutcliffe efficiency while being explicitly time sensitive we employ both measures within a multiobjective calibration framework and achieve better performance over the full range of flows than obtained by single criteria approaches or by the common multiobjective approach that uses log transformed and untransformed data to balance fitting of low and high flow periods the investigation highlights the potential of ced to complement kge and vice versa during model identification it is possible that some of the complementarity is due to ced representing more information from moments 2 than kge or other common metrics we therefore suggest that an interesting way forward would be to extend kge to include higher moments i e use different moments as multiple criteria

field theory of branching and annihilating random walks we develop a systematic analytic approach to the problem of branching and annihilating random walks equivalent to the diffusion limited reaction processes 2a and a m 1 a where m 1 starting from the master equation a field theoretic representation of the problem is derived and fluctuation effects are taken into account via diagrammatic and renormalization group methods for d 2 the mean field rate equation which predicts an active phase as soon as the branching process is switched on applies qualitatively for both even and odd m but the behavior in lower dimensions is shown to be quite different for these two cases for even m and d near 2 the active phase still appears immediately but with nontrivial crossover exponents which we compute in an expansion in 2 d and with logarithmic corrections in d 2 however there exists a second critical dimension d c 4 3 below which a nontrivial inactive phase emerges with asymptotic behavior characteristic of the pure annihilation process this is confirmed by an exact calculation in d 1 the subsequent transition to the active phase which represents a new nontrivial dynamic universality class is then investigated within a truncated loop expansion which appears to give a correct qualitative picture the model with m 2 is also generalized to n species of particles which provides yet another universality class and which is exactly solvable in the limit n for odd m we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d 2 in this case the transition to the active phase is in the directed percolation universality class finally we study the modification when the annihilation reaction is 3a when m 0 mod 3 the system is always in its active phase but with logarithmic crossover corrections for d 1 while the other cases should exhibit a directed percolation transition out of a fluctuation driven inactive phase

mixing driven equilibrium reactions in multidimensional fractional advection dispersion systems we study instantaneous mixing driven bimolecular equilibrium reactions in a system where transport is governed by a multidimensional space fractional dispersion equation the superdiffusive nonlocal nature of the system causes the location and magnitude of reactions that take place to change significantly from a classical fickian diffusion model in particular regions where reaction rates would be zero for the fickian case become regions where the maximum reaction rate occurs when anomalous dispersion operates we also study a global metric of mixing in the system the scalar dissipation rate and compute its asymptotic scaling rates analytically the scalar dissipation rate scales asymptotically as t d where d is the number of spatial dimensions and is the fractional derivative exponent

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