stabilization effect of magnetic fields on two dimensional compressible current vortex sheets we analyze the linear stability of rectilinear compressible current vortex sheets in two dimensional isentropic magnetohydrodynamics which is a free boundary problem with the boundary being characteristic in the case when the magnitude of the magnetic field has no jump on the current vortex sheets we find a necessary and sufficient condition of linear stability for the rectilinear current vortex sheets showing that magnetic fields exert a stabilization effect on compressible vortex sheets in addition a loss of regularity with respect to the source terms both in the interior domain and on the boundary occurs in a priori estimates of solutions to the linearized problem for a rectilinear current vortex sheet as the kreiss lopatinskii determinant associated with this linearized boundary value problem has roots on the boundary of frequency spaces in this study the construction of symmetrizers for a reduced differential system which has poles at which the kreiss lopatinskii condition may fail simultaneously plays a crucial role in the a priori estimates

subordinated advection dispersion equation for contaminant transport a mathematical method called subordination broadens the applicability of the classical advection dispersion equation for contaminant transport in this method the time variable is randomized to represent the operational time experienced by different particles in a highly heterogeneous aquifer the operational time captures the fractal properties of the medium this leads to a simple parsimonious model of contaminant transport that exhibits many of the features heavy tails skewness and non fickian growth rate typically seen in real aquifers we employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives applied to a one dimensional approximation of the made 2 data set the model shows excellent agreement

a fast finite difference method for distributed order space fractional partial differential equations on convex domains abstract fractional partial differential equations pdes provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long range spatial interactions which cannot be modeled accurately by classical second order diffusion equations however numerical methods for space fractional pdes usually generate dense or full stiffness matrices for which a direct solver requires o n 3 computations per time step and o n 2 memory where n is the number of unknowns the significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three dimensional space fractional diffusion equations computationally intractable fast numerical methods were previously developed for space fractional pdes on multidimensional rectangular domains without resorting to lossy compression but rather via the exploration of the tensor product form of the toeplitz like decompositions of the stiffness matrices in this paper we develop a fast finite difference method for distributed order space fractional pdes on a general convex domain in multiple space dimensions the fast method has an optimal order storage requirement and almost linear computational complexity without any lossy compression numerical experiments show the utility of the method

error estimates of a spectral petrov galerkin method for two sided fractional reaction diffusion equations abstract we study regularity and the spectral method for two sided fractional diffusion equations with a reaction term we show that the regularity of the solution in weighted sobolev spaces can be greatly improved compared to that in standard sobolev spaces with this regularity we prove an optimal error estimate for the spectral petrov galerkin method numerical results are presented to verify our theoretical convergence orders

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

heavy tailed log hydraulic conductivity distributions imply heavy tailed log velocity distributions 1 equations of contaminant transport describing non gaussian dispersion of solute in heterogeneous porous media have been developed by several authors e g berkowitz and scher 1995 1998 benson 1998 2001 berkowitz et al 2000 and baeumer et al 2005 the equations assume that solute particle movement in time and or space can be described by heavy tailed probability distributions e g levy stable distributions this assumption requires a mechanism for solute particles to move through heterogeneous porous media in a manner that can be approximated with heavy tailed probability distributions this work investigates whether a heavy tailed levy stable distribution of increments in log hydraulic conductivity results in a heavy tailed distribution of increments in log velocity a heavy tailed distribution of increments in log velocity would be a mechanism for solute particle movement in time or space that can be described by a heavy tailed probability distribution a computer algorithm was written to generate realistic hydraulic conductivity fields the algorithm modifies the spectral synthesis method for generating fractional brownian motion i e saupe 1988 and brewer and wheatcraft 1994 and produces self similar hydraulic conductivity fields with a heavy tailed levy stable distribution of increments in log hydraulic conductivity a monte carlo approach was adopted in which 100 hydraulic conductivity fields were generated for each of the following values of 0 8 1 1 1 4 and 1 7 the usgs finite difference groundwater code modflow was used to calculate the velocity fields and the increments in longitudinal log hydraulic conductivity and log velocity from the fields were analyzed to determine whether they were consistent with a heavy tailed levy stable probability density function pdf after determining that the distributions of increments in log hydraulic conductivity and log velocity were consistent with a levy stable pdf the tail parameters describing each pair of hydraulic conductivity and velocity fields were estimated with the nolan 1997 maximum likelihood estimator the relationship between the two tail parameters was then determined there are three conclusions of this research first mandelbrot and pp plots confirm that levy stable distributions of increments in log hydraulic conductivity give rise to levy stable distributions of increments in log velocity second the tail parameter for the increments in log velocity is typically smaller indicating a heavier probability tail than the corresponding tail parameter for the increments in log hydraulic conductivity third there is a positive statistical relation between the two tail parameters

the existence of current vortex sheets in ideal compressible magnetohydrodynamics we prove the local in time existence of solutions with a surface of current vortex sheet tangential discontinuity of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity this paper is a natural completion of our previous analysis trakhinin in arch ration mech anal 177 331 366 2005 where a sufficient condition for the weak stability of planar current vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities the original nonlinear problem is a free boundary hyperbolic problem since the free boundary is characteristic the functional setting is provided by the anisotropic weighted sobolev spaces h m the fact that the kreiss lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates therefore we prove our existence theorem by a suitable nash moser type iteration scheme

a priori estimates for 3d incompressible current vortex sheets we consider the free boundary problem for current vortex sheets in ideal incompressible magneto hydrodynamics it is known that current vortex sheets may be at most weakly neutrally stable due to the existence of surface waves solutions to the linearized equations the existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms however under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front we prove an a priori estimate in sobolev spaces for smooth solutions with no loss of derivatives the result of this paper gives some hope for proving the local existence of smooth current vortex sheets without resorting to a nash moser iteration such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instabilities which is well known in astrophysics

a mass balance based numerical method for the fractional advection dispersion equation theory and application 1 the inherent heterogeneity of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement and one approach to model solute transport through such media is the continuous time random walk ctrw one special asymptotic case of the ctrw is the fractional advection dispersion equation fade which has proven to be a promising alternative to model anomalous dispersion and has been increasingly used in hydrology to model chemical transport in both surface and subsurface water most practical problems in hydrology have complicated initial and boundary conditions and need to be solved numerically but the numerical solution of the fade is not trivial in this paper we present a finite volume approach to solve the fade where the spatial derivative of the dispersion term is fractional we also give methods to solve different boundary conditions often encountered in practical applications the linear system resulting from the temporal spatial discretization is solved using a semi implicit scheme the numerical method is derived on the basis of mass balance and its accuracy is tested against analytical solutions the method is then applied to simulate tracer movement in a stream and a near saturated hillslope in a naturally structured upland podzol field in northeast scotland

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

eulerian lagrangian theory of transport in space time nonstationary velocity fields exact nonlocal formalism by conditional moments and weak approximation a unified eulerian lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally v x t f x t where f x t is a random function including sources and or the time derivative of head solute concentration satisfies locally the eulerian equation c x t t j x t g x t where j x t is advective solute flux and g x t is a random source independent of f x t we consider the prediction of c x t and j x t by means of their unbiased ensemble moments c x t and j x t conditioned as implied by the subscript on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate x t of v x t these predictors satisfy c x t v t j x t g x t where j x t x t c x t q x t and q x t is a dispersive flux we show that q is given exactly by three space time convolution integrals of conditional lagrangian kernels with q with c and with c for a broad class of v x t fields including fractals this implies that q x t is generally nonlocal and non fickian rendering c x t non gaussian the direct contribution of random variations in f to q depends on c rather than on c we elucidate the nature of the above kernels discuss conditions under which the convolution of and c becomes pseudo fickian with a lagrangian dispersion tensor similar to that derived in 1921 by taylor recall a 1952 result by batchelor which yields an exact expression for c at early time use the latter to conclude that linearizations which predict that c bifurcates at early time when the probability density function of v is unimodal cannot be correct propose instead a weak approximation which leads to a nonlinear integro differential equation for c due to an instantaneous point source and which improves with the quantity and quality of hydraulic data demonstrate that the weak approximation is analogous to the direct interaction closure of turbulence theory offer non fickian and pseudo fickian weak approximations for the second conditional moment of the concentration prediction error demonstrate that it yields the so called two particle covariance as a special case conclude that the conditional variance of c does not become infinite merely as a consequence of disregarding local dispersion and discuss how to estimate explicitly the cumulative release of a contaminant across a compliance surface together with the associated estimation error

application of vibrational control to linear quadratic control problems abstract the application of the theory of vibrational control to linear quadratic control problems is developed the solution of the matrix riccati differential equation rde and the optimal trajectory are found approximately as power series of a small parameter the minimizing effect of vibrations on the maximal solution of rde and the cost functional is studied the region of attraction of the maximal solution of rde for the case of hamiltonian matrix with imaginary axis eigenvalues is found special attention is paid to the application of vibrations to the linear quadratic problem of stabilization with respect to a part of variables and transfer of the other variables to a given position a problem of vibrational stabilization and optimal control of a carriage with an inverted pendulum is solved as an example

on diffusion in fractal porous media generalized hydrodynamics and irreversible thermodynamics are used to derive the relation between a general wave vector and frequency dependent diffusion tensor and the power spectral density the results are valid for porous media with evolving heterogeneities an integrodifferential transport equation valid in porous media with evolving heterogeneities is presented

crank nicolson adi galerkin finite element method for two dimensional fractional fitzhugh nagumo monodomain model in this paper a two dimensional fractional fitzhugh nagumo monodomain model 2d ffhnmm with zero dirichlet boundary condition is considered the model consists of a coupled two dimensional space fractional nonlinear reaction diffusion model 2d sfnrdm and an ordinary differential equation the 2d sfnrdm and ordinary differential equation are decoupled at each time step a new crank nicolson alternating direction implicit adi galerkin finite element method for the 2d sfnrdm is developed the stability and convergence of the numerical method are discussed finally some numerical examples on 2d sfnrdm and 2d ffhnmm are given for verification of our theoretical analysis

existence of compressible current vortex sheets variable coefficients linear analysis we study the initial boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the rankine hugoniot relations about an unsteady piecewise smooth solution this solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity current vortex sheet under some assumptions on the unperturbed flow we prove an energy a priori estimate for the linearized problem since the tangential discontinuity is characteristic the functional setting is provided by the anisotropic weighted sobolev space w21 despite the fact that the constant coefficients linearized problem does not meet the uniform kreiss lopatinskii condition the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current vortex sheets the result of this paper is a necessary step in proving the local in time existence of current vortex sheet solutions of the nonlinear equations of magnetohydrodynamics

fractional vector calculus for fractional advection dispersion abstract we develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient divergence and curl and a fractional divergence theorem and stokes theorem these basic tools are then applied to provide a physical explanation for the fractional advection dispersion equation for flow in heterogeneous porous media

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

fast upwind and eulerian lagrangian control volume schemes for time dependent directional space fractional advection dispersion equations abstract we develop control volume methods for two dimensional time dependent advection dominated directional space fractional advection dispersion equations with the directional space fractional derivative weighted in all the directions by a probability measure in the unit circle which are used to model the anisotropic superdiffusive transport of solutes in groundwater moving through subsurface heterogeneous porous media we develop a fast upwind control volume method for the governing equation to eliminate the spurious numerical oscillations that often occur in space centered numerical discretizations of advection term which are relatively straightforward to implement we also develop a eulerian lagrangian control volume method for the governing equation which symmetrizes the governing equation by combining the time derivative term and the advection term into a material derivative term along characteristic curves both methods are locally mass conservative which are essential in these applications due to the nonlocal nature of the directional space fractional differential operators corresponding numerical discretizations usually generate full stiffness matrices conventional direct solvers tend to require o n 2 memory requirement and have o n 3 computational complexity per time step where n is the number of spatial unknowns which is computationally significantly more expensive than the numerical approximations of integer order advection diffusion equations based on the analysis of the structure of stiffness matrix we propose a fast krylov subspace iterative solver to accelerate the numerical approximations of both the upwind and eulerian lagrangian control volume methods which reduce computational complexity from o n 3 by a direct solver to o n log n per krylov subspace iteration per time step and a memory requirement from o n 2 to o n numerical results are presented to show the utility of the methods

multiscaling fractional advection dispersion equations and their solutions 1 the multiscaling fractional advection dispersion equation ade is a multidimensional model of solute transport that encompasses linear advection fickian dispersion and super fickian dispersion the super fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions the directions need not be orthogonal so the model can be applied to irregular noncontinuum fracture networks the statistical model underlying multiscaling fractional dispersion is a continuous time random walk ctrw in which particles have arbitrary jump length distributions and finite mean waiting time distributions the meaning of the parameters in a compound poisson process a subset of ctrws is used to develop a physical interpretation of the equation variables the green s function solutions are the densities of operator stable probability distributions the limit distributions of normalized sums of independent and identically distributed random vectors these densities can be skewed heavy tailed and scale nonlinearly resembling solute plumes in granular aquifers they can also have fingers in any direction resembling transport along discrete pathways such as fractures

a stochastic differential equation model with jumps for fractional advection and dispersion the path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation sde driven by brownian noise this model may not be adequate for highly heterogeneous media this paper extends the model to a general sde driven by a levy noise particle paths follow a markov process with long jumps their transition probability density solves a forward equation derived here via pseudo differential operator theory and fourier analysis in particular the sde with stable driving noise has a space fractional advection dispersion equation fade with variable coefficients as the forward equation this result provides a stochastic solution to anomalous diffusion models and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

spectral approximation of a variable coefficient fractional diffusion equation in one space dimension abstract in this article we consider the approximation of a variable coefficient two sided fractional diffusion equation fde having unknown u by introducing an intermediate unknown q the variable coefficient fde is rewritten as a lower order constant coefficient fde a spectral approximation scheme using jacobi polynomials is presented for the approximation of q q n the approximate solution to u u n is obtained by post processing q n an a priori error analysis is given for q q n and u u n two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates

reply to comment by d h lim on diffusion processes in composite porous media and their numerical integration by random walks generalized stochastic differential equations with discontinuous coefficients 1 we thank lim 2005 for providing this opportunity to clarify the restrictions and confirm the applicability of the results of labolle et al 2000 labolle et al 2000 develop generalized stochastic differential equations sde that converge to advection dispersion equations with discontinuous dispersion tensor d the problem of interest here relates to equation 15a of labolle et al 2000 and its application in the numerical simulation of random walks converging to advection dispersion equations with discontinuous coefficients in one dimension this equation can be written as

a fast semi implicit difference method for a nonlinear two sided space fractional diffusion equation with variable diffusivity coefficients in this paper we derive a new nonlinear two sided space fractional diffusion equation with variable coefficients from the fractional fick s law a semi implicit difference method sidm for this equation is proposed the stability and convergence of the sidm are discussed for the implementation we develop a fast accurate iterative method for the sidm by decomposing the dense coefficient matrix into a combination of toeplitz like matrices this fast iterative method significantly reduces the storage requirement of o n 2 and computational cost of o n 3 down to n and o n log n where n is the number of grid points the method retains the same accuracy as the underlying sidm solved with gaussian elimination finally some numerical results are shown to verify the accuracy and efficiency of the new method

linear stability of compressible vortex sheets in two dimensional elastodynamics abstract the linear stability of rectilinear compressible vortex sheets is studied for two dimensional isentropic elastic flows this problem has a free boundary and the boundary is characteristic a necessary and sufficient condition is obtained for the linear stability of the rectilinear vortex sheets more precisely it is shown that besides the stable supersonic zone the elasticity exerts an additional stable subsonic zone we also find that there is a class of states in the interior of subsonic zone where the stability of such states is weaker than the stability of other states in the sense that there is an extra loss of tangential derivatives with respect to the source terms this is a new feature that the euler flow does not possess one of the difficulties for the elastic flow is that the non differentiable points of the eigenvalues may coincide with the roots of the lopatinskii determinant as a result the kreiss symmetrization cannot be applied directly instead we perform an upper triangularization of the system to separate only the outgoing modes at all points in the frequency space so that an exact estimate of the outgoing modes can be obtained moreover all the outgoing modes are shown to be zero due to the l 2 regularity of solutions the estimates for the incoming modes can be derived directly from the lopatinskii determinant this new approach avoids the lengthy computation and estimates for the outgoing modes when kreiss symmetrization is applied this method can also be applied to the euler flow and mhd flow

the fractional fick s law for non local transport processes fick s law is extensively adopted as a model for standard diffusion processes however requiring separation of scales it is not suitable for describing non local transport processes we discuss a generalized non local fick s law derived from the space fractional diffusion equation generating the levy feller statistics this means that the fundamental solutions can be interpreted as levy stable probability densities in the feller parameterization with index 1 2 and skewness 2 we explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities flux and gradient it turns out that the presence of asymmetry 0 plays a fundamental role it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter gradient transport

stability of incompressible current vortex sheets abstract we revisit the study in y trakhinin on the existence of incompressible current vortex sheets study of a linearized free boundary value problem math methods appl sci 28 2005 917 945 where an energy a priori estimate for the linearized free boundary value problem for planar current vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in s i syrovatskij the stability of tangential discontinuities in a magnetohydrodynamic medium zh eksper teor fiz 24 1953 622 629 in russian w i axford note on a problem of magnetohydrodynamic stability canad j phys 40 1962 654 655 in this paper we derive an a priori estimate in the whole stability domain the crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure this symmetrizer is an analogue of kreiss type symmetrizers as in hyperbolic theory the failure of the uniform lopatinski condition i e the fact that current vortex sheets are only weakly neutrally stable yields loss of derivatives in the energy estimate the result of this paper is a necessary step to prove the local in time existence of stable nonplanar incompressible current vortex sheets by a suitable nash moser type iteration scheme

stability and convergence of a finite volume method for the space fractional advection dispersion equation we consider the space fractional advection dispersion equation which is obtained from the classical advection diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order we derive a finite volume method that utilises fractionally shifted grunwald formulae for the discretisation of the fractional derivative to numerically solve the equation on a finite domain with homogeneous dirichlet boundary conditions we prove that the method is stable and convergent when coupled with an implicit timestepping strategy results of numerical experiments are presented that support the theoretical analysis

modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit lagrangian scheme this study develops an explicit two step lagrangian scheme based on the renewal reward process to capture transient anomalous diffusion with mixed retention and early arrivals in multidimensional media the resulting 3 d anomalous transport simulator provides a flexible platform for modeling transport the first step explicitly models retention due to mass exchange between one mobile zone and any number of parallel immobile zones the mobile component of the renewal process can be calculated as either an exponential random variable or a preassigned time step and the subsequent random immobile time follows a hyper exponential distribution for finite immobile zones or a tempered stable distribution for infinite immobile zones with an exponentially tempered power law memory function the second step describes well documented early arrivals which can follow streamlines due to mechanical dispersion using the method of subordination to regional flow applicability and implementation of the lagrangian solver are further checked against transport observed in various media results show that although the time nonlocal model parameters are predictable for transport with retention in alluvial settings the standard time nonlocal model cannot capture early arrivals retention and early arrivals observed in porous and fractured media can be efficiently modeled by our lagrangian solver allowing anomalous transport to be incorporated into 2 d 3 d models with irregular flow fields extensions of the particle tracking approach are also discussed for transport with parameters conditioned on local aquifer properties as required by transient flow and nonstationary media

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

theory of anomalous chemical transport in random fracture networks we show that dominant aspects of chemical particle transport in fracture networks char22 non gaussian propagation char22 result from subtle features of the steady flow field distribution through the network this is an outcome of a theory based on a continuous time random walk formalism structured to retain the key space time correlations of particles as they are advected across each fracture segment the approach is designed to treat the complex geometries of a large variety of fracture networks and multiscale interactions monte carlo simulations of steady flow in these networks are used to determine the distribution of velocities in individual fractures as a function of their orientation the geometry and velocity distributions are used in conjunction with particle mixing rules to map the particle movement between fracture intersections onto a joint probability density ensuremath psi mathbf r t the chemical concentration plume and breakthrough curves can then be calculated analytically particle tracking simulations on these networks exhibit the same non gaussian profiles demonstrating quantitative agreement with the theory the analytic plume shapes display the same basic behavior as extensive field observations at the columbus air force base mississippi the quantitative correlation between the time dependence of the mean and standard deviation of the field plumes and their shape is predicted by the theory

operator levy motion and multiscaling anomalous diffusion the long term limit motions of individual heavy tailed power law particle jumps that characterize anomalous diffusion may have different scaling rates in different directions operator stable motions y t tg 0 are limits of d dimensional random jumps that are scale invariant according to c h y t y mathrm ct where h is a d ifmmode times else texttimes fi d matrix the eigenvalues of the matrix have real parts 1 ensuremath alpha j with each positive ensuremath alpha j l 2 in each of the j principle directions the random motion has a different fickian or super fickian diffusion dispersion rate proportional to t 1 ensuremath alpha j these motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions subsets of the generalized fractional operator include i a fractional laplacian with a single order ensuremath alpha and a general directional mixing measure m ensuremath theta and ii a fractional laplacian with uniform mixing measure the riesz potential the motivation for the generalized dispersion is the observation that tracers in natural aquifers scale at different super fickian rates in the directions parallel and perpendicular to mean flow

numerical simulations of preasymptotic transport in heterogeneous porous media departures from the gaussian limit 1 the objective of this work is to determine whether conventional solute macrodispersion theories adequately predict the behavior of individual subsurface plumes over time and length scales relevant to practical risk assessment and remediation activities this issue is studied with a set of high resolution numerical simulations of conservative tracers moving through saturated two dimensional heterogeneous conductivity fields the simulation experiments are designed to mimic long term field studies spatially correlated statistically stationary lognormal conductivity realizations are generated with a fourier transform procedure steady state velocity solutions are calculated for these fields using an accurate darcian solver velocity contour plots reveal the presence of disconnected networks of preferential pathways over a range of correlation lengths reverse flow cells are rare the velocity probability density functions have exponential tails and strong longitudinal asymmetries solute concentrations are derived from the simulated velocity fields with an accurate taut spline transport code that minimizes numerical dispersion the resulting plumes are tracked for travel distances of over one hundred spatial correlation lengths corresponding to scales of practical interest first moments and macrodispersivities which measure the location and extent of the plume are reasonably well approximated by conventional fickian theories but continue to vary after long travel distances this temporal variability highlights the slow convergence of the plume moments to gaussian limits calculations of the relative dilution index which is a measure of the plume mixing state indicate strongly non gaussian behavior other measures of plume structure suggested by anomalous dispersion theories also reveal the persistence of non gaussian behavior after long travel distances these measures appear to be more sensitive to non gaussian behavior than the spatial moments taken together the simulation results suggest that conservative solute plumes moving through statistically stationary random media may not converge to gaussian limits even after traveling hundreds of log conductivity correlation scales

generic types and transitions in hyperbolic initial boundary value problems the stability of linear initial boundary value problems for hyperbolic systems with constant coefficients is linked to the zeros of the so called lopatinskii determinant depending on the location of these zeros problems may be either unstable strongly stable or weakly stable the first two classes are known to be open in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and or in the boundary condition here we show that a third open class exists which we call weakly stable of real type many examples of physical or mathematical interest depend on one or more parameters and the determination of the stability class as a function of these parameters usually needs an involved computation we simplify it by characterizing the transitions from one open class to another one these boundaries are easier to determine since they must solve some overdetermined algebraic system applications to the wave equation linear elasticity shock waves and phase boundaries in fluid mechanics are given

diffusion theory for transport in porous media transition probability densities of diffusion processes corresponding to advection dispersion equations local scale spatial averaging of pore scale advection diffusion equations in porous media leads to advection dispersion equations ades while often used to describe subsurface transport ades may pose special problems in the context of diffusion theory standard diffusion theory applies only when characteristic coefficients velocity porosity and dispersion tensor are smooth functions of space subsurface porous material properties however naturally exhibit spatial variability transitions between material types are often abrupt rather than smooth such as sand in contact with clay in such composite porous media characteristic coefficients in spatially averaged transport equations may be discontinuous although commonly called on to model transport in these cases standard diffusion theory does not apply herein we develop diffusion theory for ades of transport in porous media derivation of ades from probabilistic assumptions yields 1 necessary conditions for convergence of diffusion processes to ades even when coefficients are discontinuous and 2 general probabilistic definitions of physical quantities velocity and dispersion tensor as examples of how the new theory can be applied to theoretical and numerical problems of transport in porous media we evaluate several random walk methods that have appeared in the water resources literature

boundary value problems of fractional fokkerplanck equations this paper is devoted to solving boundary value problems for important fractional differential equations of the fokkerplanck family in particular to studying fractional differential equation for advectiondispersion the consideration is carried out by the separation of variables the fourier method most part of this paper is devoted to justification of this method to proof of a basis of the system of eigenfunctions for the basic equation aleroev etal 2015 for modeling the random walk of a point particle which starts to move at the origin of coordinates in t 0 on a self similar fractal set

an unstructured mesh control volume method for two dimensional space fractional diffusion equations with variable coefficients on convex domains in this paper we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains which to the best of our knowledge is a new contribution to the literature firstly we present the finite volume scheme for the two dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements secondly we explore the property of the stiffness matrix generated by the integral of the space fractional derivative we find that the stiffness matrix is sparse and not regular therefore we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system which is more efficient than using the gaussian elimination method finally we present several examples to verify our method in which we make a comparison of our method with the finite element method for solving a riesz space fractional diffusion equation on a circular domain the numerical results demonstrate that our method can reduce cpu time significantly while retaining the same accuracy and approximation property as the finite element method the numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

nonlinear stability of the current vortex sheet to the incompressible mhd equations in this paper we solve a long standing open problem nonlinear stability of the current vortex sheet in the ideal incompressible magnetohydrodynamics under the syrovatskij stability condition this result gives the first rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instability 2017 wiley periodicals inc

linear stability of compressible vortex sheets in 2d elastodynamics variable coefficients the linear stability with variable coefficients of the vortex sheets for the two dimensional compressible elastic flows is studied as in our earlier work chen et al in adv math 311 18 60 2017b on the linear stability with constant coefficients the problem has a free boundary which is characteristic and also the kreiss lopatinskii condition is not uniformly satisfied in addition the roots of the lopatinskii determinant of the para linearized system may coincide with the poles of the system such a new collapsing phenomenon causes serious difficulties when applying the bicharacteristic extension method of coulombel siam j math anal 34 1 142 172 2002 ann inst h poincare anal non lineaire 21 4 401 443 2004 and coulombel and secchi indiana univ math j 53 4 941 1012 2004 motivated by our method introduced in the constant coefficient case adv math 311 18 60 2017b we perform an upper triangularization to the para linearized system to separate the outgoing mode into a closed form where the outgoing mode only appears at the leading order this procedure results in a gain of regularity for the outgoing mode which allows us to overcome the loss of regularity of the characteristic components at the poles and hence to close all the energy estimates we find that analogous to the constant coefficient case elasticity generates notable stabilization effects and there are additional stable subsonic regions compared with the isentropic euler flows moreover since our method does not rely on the construction of the characterisic curves it can also be applied to other fluid models such as the non isentropic euler equations and the mhd equations

hydraulic conductivity velocity and the order of the fractional dispersion derivative in a highly heterogeneous system 1 a one dimensional fractional order advection dispersion equation accurately models the movement of the core of the tritium plume at the highly heterogeneous made site an a priori estimate of the parameters in that equation including the order of the fractional dispersion derivative was based on the assumption that the observed power law heavy tail of the hydraulic conductivity k field would create a similarly distributed velocity field monte carlo simulations were performed to test this hypothesis results from the monte carlo analysis show that heavy tailed k fields do give rise to heavy tailed velocity fields however the exponent of the power law the tail parameter describing these two distributions is not necessarily the same the tail parameter that characterizes a velocity distribution is not solely dependent on the tail parameter that characterizes the k distribution the k field must also have long range dependence so that water may flow through relatively continuous high k channels

fractional advection dispersion equation a classical mass balance with convolution fickian flux the classical form of the partial differential equation governing advective dispersive transport of a solute in idealized porous media is the advection dispersion equation ade this equation is built on a fickian constitutive theory that gives the local dispersive flux as the inner product of a constant dispersion tensor and the spatial gradient of the solute concentration over the past decade investigators in subsurface transport have been increasingly focused on anomalous i e non fickian dispersion in natural geologic formations it has been recognized that this phenomenon involves spatial and possibly temporal nonlocality in the constitutive theory describing the dispersive flux most recently several researchers have modeled anomalous dispersion using a constitutive theory that relies on fractional as opposed to integer derivatives of the concentration field the resulting ade itself is expressed in terms of fractional derivatives and they are described as fractional ades here we show that the fractional ade is obtained as a special case of the authors convolution fickian nonlocal ade cushman and ginn 1993 to obtain the fractional ade from the convolution fickian model requires only a judicial choice of the wave vector and frequency dependent dispersion tensor

local existence for the free boundary problem for nonrelativistic and relativistic compressible euler equations with a vacuum boundary condition we study the free boundary problem for the equations of compressible euler equations with a vacuum boundary condition our main goal is to recover in eulerian coordinates the earlier well posedness result obtained by lindblad 11 for the isentropic euler equations and extend it to the case of full gas dynamics for technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain we prove the local in time existence in sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities 5 21 it contains in particular the reduction to a fixed domain using the good unknown of alinhac 1 and a suitable nash moser type iteration scheme a certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic this approach is still directly applicable to the relativistic version of our problem in the setting of special relativity and we briefly discuss its extension to general relativity 2009 wiley periodicals inc

a complete 2d stability analysis of fast mhd shocks in an ideal gas an algorithm of numerical testing of the uniform lopatinski condition for linearized stability problems for 1 shocks is suggested the algorithm is used for finding the domains of uniform stability neutral stability and instability of planar fast mhd shocks a complete stability analysis of fast mhd shock waves is first carried out in two space dimensions for the case of an ideal gas main results are given for the adiabatic constant 5 3 mono atomic gas that is most natural for the mhd model the cases 7 5 two atomic gas and 5 3 are briefly discussed not only the domains of instability and linear in the usual sense stability but also the domains of uniform stability for which a corresponding linearized stability problem satisfies the uniform lopatinski condition are numerically found for different given angles of inclination of the magnetic field behind the shock to the planar shock front as is known uniform linearized stability implies the nonlinear stability that is local existence of discontinuous shock front solutions of a quasilinear system of hyperbolic conservation laws

an explicit high order method for fractional advection diffusion equations we propose a high order explicit finite difference method for fractional advection diffusion equations these equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order with 1 2 this operator is defined by a combination of the left and right riemann liouville fractional derivatives we study the convergence of the numerical method through consistency and stability the order of convergence varies between two and three and for advection dominated flows is close to three although the method is conditionally stable the restrictions allow wide stability regions the analysis is confirmed by numerical examples

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

on the well posedness of a linearized plasma vacuum interface problem in ideal compressible mhd abstract we study the initial boundary value problem resulting from the linearization of the plasma vacuum interface problem in ideal compressible magnetohydrodynamics mhd we suppose that the plasma and the vacuum regions are unbounded domains and the plasma density does not go to zero continuously but jumps for the basic state upon which we perform linearization we find two cases of well posedness of the frozen coefficient problem the gas dynamical case and the purely mhd case in the gas dynamical case we assume that the jump of the normal derivative of the total pressure is always negative in the purely mhd case this condition can be violated but the plasma and the vacuum magnetic fields are assumed to be non zero and non parallel to each other everywhere on the interface for this case we prove a basic a priori estimate in the anisotropic weighted sobolev space h 1 for the variable coefficient problem

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

a superfast preconditioned iterative method for steady state space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second order diffusion equations because of the nonlocal property of fractional differential operators the corresponding numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 for a problem of size n we develop a superfast preconditioned conjugate gradient squared method for the efficient solution of steady state space fractional diffusion equations the method reduces the computational work from o n 2 to o nlogn per iteration and reduces the memory requirement from o n 2 to o n furthermore the method significantly reduces the number of iterations to be mesh size independent preliminary numerical experiments for a one dimensional steady state diffusion equation with 2 1 3 nodes show that the fast method reduces the overall cpu time from 3h and 27min for the gaussian elimination to 0 39s for the fast method while retaining the accuracy of gaussian elimination in contrast the regular conjugate gradient squared method diverges after 2days of simulations and more than 20 000 iterations

random walk approximation of fractional order multiscaling anomalous diffusion random walks are developed to approximate the solutions of multiscaling fractional order anomalous diffusion equations the essential elements of the diffusion are described by the matrix order scaling indexes and the mixing measure which describes the diffusion coefficient in every direction two forms of the governing equation also called the multiscaling fractional diffusion equation based on fractional flux and fractional divergence are considered where the diffusion coefficient and the drift vary in space the particle tracking algorithm is also extended to approximate anomalous diffusion with a streamline dependent mixing measure using a streamline projection technique in this and other general cases the random walk method is the only known way to solve the nonhomogeneous equations five numerical examples demonstrate the flexibility simplicity and efficiency of the random walk method

well posedness of characteristic symmetric hyperbolic systems we consider the initial boundary value problem for quasi linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity we show the well posedness in hadamard s sense i e existence uniqueness and continuous dependence of solutions on the data of regular solutions in suitable functions spaces which take into account the loss of regularity in the normal direction to the characteristic boundary

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

well posedness of the free boundary problem in incompressible elastodynamics abstract in this paper we prove the local well posedness of the free boundary problem in incompressible elastodynamics under a natural stability condition which ensures that the evolution equation describing the free boundary is strictly hyperbolic our result gives a rigorous confirmation that the elasticity has a stabilizing effect on the rayleigh taylor instability

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

symmetric positive systems with boundary characteristic of constant multiplicity the theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity no such hypothesis is needed on a neighborhood of the boundary both regularity theorems and mixed initial boundary value problems are discussed many classical ideas are sharpened in the process

well posedness of the linearized problem for mhd contact discontinuities abstract we study the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics mhd they are characteristic discontinuities with no flow across the discontinuity for which the pressure the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump under the rayleigh taylor sign condition p n 0 on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity we prove the well posedness in sobolev spaces of the linearized problem for 2d planar mhd flows

on the initial boundary value problem for the linearized equations of magnetohydrodynamics we study the initial boundary value problem for the linearized equations of ideal magnetohydrodynamics defined on a bounded domain whose boundary is a regular magnetic surface we show that it is well posed in h 1 omega and that the loss of regularity of solutions always occurs for some initial data in h l omega l geq 2

moment analysis for spatiotemporal fractional dispersion 1 the evolution of the first five nonnegative integer order spatial moments corresponding to the mass mean variance skewness and kurtosis are investigated systematically for spatiotemporal nonlocal fractional dispersion three commonly used fractional order transport equations including the time fractional advection dispersion equation time fade the fractal mobile immobile mim equation and the fully fractional advection dispersion equation ffade are considered analytical solutions verify our numerical results and reveal the anomalous evolution of the moments following adams and gelhar s 1992 work on the classical ade we find that a simultaneous analysis of all moments is critical in discriminating between different nonlocal models the evolution of dispersion among the subdiffusive to superdiffusive rates is then further explored numerically by a non markovian random walk particle tracking method that can be used for any heterogeneous boundary or initial value problem in three dimensions both the analytical and the numerical results also show the similarity at the early time and the difference at the late time of moment growth for solutes in different phases mobile versus total described by the mim models further simulations of the 1 d bromide snapshots measured at the made experiments using all three models with parameters fitted by the observed zeroth to fourth moments indicate that 1 both the time and space nonlocality strongly affect the solute transport at the made site 2 all five spatial moments should be considered in transport model selection and calibration because those up to the variance cannot effectively discriminate between nonlocal models and 3 the log concentration should be used when evaluating the plume leading edge and the effects of space nonlocality

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

finite element method for two dimensional space fractional advection dispersion equations the backward euler and crank nicolson galerkin fully discrete approximate schemes for two dimensional space fractional advection dispersion equations are established firstly we prove that the corresponding variational problem has a unique solution and the proposed fully discrete schemes are unconditionally stable whose solutions are all unique secondly the optimal error estimates are derived by use of properties of projection operator and fractional derivatives finally numerical examples demonstrate effectiveness of numerical schemes and confirm the theoretical analysis

random walk simulation of transport in heterogeneous porous media local mass conservation problem and implementation methods the random walk method for simulating solute transport in porous media is typically based on the assumption that the velocity and velocity dependent dispersion tensor vary smoothly in space however in cases where sharp interfaces separate materials with contrasting hydraulic properties these quantities may be discontinuous normally velocities are interpolated to arbitrary particle locations when finite difference or finite element methods are used to solve the flow equation the use of interpolation schemes that preserve discontinuities in velocity at material contacts can result in a random walk model that does not locally conserve mass unless a correction is applied at these contacts test simulations of random walk particle tracking with and without special treatment of material contacts demonstrate the problem techniques for resolving the problem including interpolation schemes and a reflection principle are reviewed and tested results from simulations of transport in porous media with discontinuities in the dispersion tensor show which methods satisfy continuity simulations of transport in two dimensional heterogeneous porous media demonstrate the potentially significant effect of using a nonconservative model to compute spatial moments and breakthrough of a solute plume

the fixed boundary value problems for the equations of ideal magneto hydrodynamics with a perfectly conducting wall condition the equations of ideal magneto hydrodynamics are investigated concerning initial boundary value problems with a perfectly conducting wall condition the local in time solution is proved to exist uniquely provided that the normal component of the initial magnetic field vanishes everywhere or nowhere on the boundary

monte carlo simulation of superdiffusion and subdiffusion in macroscopically heterogeneous media 1 monte carlo simulations are developed to approximate one dimensional superdiffusion and subdiffusion in macroscopically heterogeneous media with discontinuous or continuous transport parameters for superdiffusion characterized by a space fractional order derivative model one empirical reflection scheme is built to track particle trajectory across an interface with discontinuous dispersion coefficient d where the reflection probability depends on both and the ratio of d different from the superdiffusive case anomalous diffusion described by a time fractional derivative model can be decomposed into a motion component and a hitting time process where the discontinuity affects only the motion process implying an efficient monte carlo simulation of decoupled continuous time random walks the discontinuity of effective porosity n is also discussed and results show the influence of the ratio of n on solute particle dynamics in addition for anomalous superdiffusion and subdiffusion in heterogeneous media with spatially continuous d and n langevin analysis reveals that the corresponding particle dynamics contain three independent stable levy noises scaled by d the gradient of d and the gradient of ln n a new implicit eulerian finite difference method is also developed to solve the spatiotemporal fractional derivative models and then extensively cross verify the lagrangian solutions further testing against one field example of mixed superdiffusion and subdiffusion reveals the applicability and flexibility of the novel monte carlo approach in simulating realistic plumes in macroscopically heterogeneous media with locally variable transport parameters

backward fractional advection dispersion model for contaminant source prediction the forward fractional advection dispersion equation fade provides a useful model for non fickian transport in heterogeneous porous media the space fade captures the long leading tail skewness and fast spreading typically seen in concentration profiles from field data this paper develops the corresponding backward fade model to identify source location and release time the backward method is developed from the theory of inverse problems and then explained from a stochastic point of view the resultant backward fade differs significantly from the traditional backward advection dispersion equation ade because the fractional derivative is not self adjoint and the probability density function for backward locations is highly skewed finally the method is validated using tracer data from a well known field experiment where the peak of the backward fade curve predicts source release time while the median or a range of percentiles can be used to determine the most likely source location for the observed plume the backward ade cannot reliably identify the source in this application since the forward ade does not provide an adequate fit to the concentration data

possible problems of scale dependency in applications of the three dimensional fractional advection dispersion equation to natural porous media 1 a three dimensional 3 d analysis of transport and macrodispersion at the macrodispersion experiment made site boggs et al 1993 using the fractional advection dispersion equation fade developed by meerschaert et al 1999 2001 shows that the levy dispersion process is scale dependent levy dispersion may be superior to gaussian dispersion on a sufficiently small scale on larger scales both theories are likely to suffer from the fact that because of depositional structures most flow fields display an evolving nonstationary structure motion in such fields is advection dominated displays a lot of memory and therefore is not modeled well by markov random processes which underlie the derivation of both the gaussian and levy advection dispersion equations berkowitz et al 2002 to improve plume simulation of an advection dominated transport process one would have to bring in more advective irregularity while simultaneously decreasing the levy dispersion coefficient therefore on a 3 d basis first order levy dispersion has limitations similar to gaussian dispersion however this and related theories such as the continuous time random walk ctrw formalism are in the early stages of development and thus may be fruitful areas for further research