fractional order legendre collocation method for solving fractional initial value problems in this paper we present a numerical algorithm for solving second order fractional initial value problems this numerical algorithm is based on a fractional legendre collocation spectral method the governing fractional differential equation is converted into a nonlinear system of algebraic equations the error analysis of the proposed numerical algorithm is presented comparisons with other numerical methods shows that the proposed algorithm is more accurate and simpler to implement several examples are discussed to illustrate the efficiency and accuracy of the present scheme

neumann boundary value problems for a time fractional diffusion wave equation in a half plane the time fractional diffusion wave equation with the caputo derivative of the order 0 a 2 is considered in a half plane two types of neumann boundary condition are examined the mathematical condition with the prescribed boundary value of the normal derivative and the physical one with the prescribed boundary value of the matter flux

geometric depletion of vortex stretch in 3d viscous incompressible flow new geometric constraints on vorticity are obtained which suppress possible development of finite time singularity from the nonlinear vortex stretching mechanism we find a new condition on the smoothness of the direction of vorticity in the vortical region which yields regularity we also detect a regularity condition of isotropy type on vorticity in the intensive vorticity region via a new cancellation principle this is in contrast with the one of isotropy type on the curl of vorticity obtained recently by a ruzmaikina and z grujic a ruzmaikina z grujic on depletion of the vortex stretching term in the 3d navier stokes equations comm math phys 247 2004 601 611 we improve as well all of their results by eliminating their assumption that the initial vorticity 0 is required to be in l1

application of the arnoldi algorithm to the solution of the advection dispersion equation the solution to the finite element matrix differential equations resulting from the discretization of the contaminant transport equation is normally carried out by a finite difference approximation to the time derivative the total computational effort in simulating a contaminant plume is then directly related to the number of unknowns and the number of time steps required to obtain accurate and stable solutions an alternative method is the arnoldi algorithm which uses orthogonal matrix transformations to reduce the finite element equations to a much smaller upper hessenberg system of first order differential equations this new system can be solved by a standard crank nicolson algorithm with very little computational effort a matrix vector multiplication is then used to obtain the original solution at desired time steps the algorithm is used to simulate accurately the contaminant plumes for a strip source areal aquifer a cross sectional problem and the borden landfill in ontario canada the arnoldi algorithm shows an impressive 613 increase in speed over the conventional crank nicolson scheme for this latter case the method affords an efficient means of solving large problems particularly when time durations are long

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

boundary integral equation method for linear porous elasticity with applications to soil consolidation for physical phenomena governed by the biot model of porous elasticity a reciprocal relation similar to the betti s recoprocal theorem in elasticity is constructed in laplace transformed space integrating the reciprocal relation enables one to formulate boundary integral equations the fundamental kernels for the integral equations are solved in closed forms for the case of isotropic material numerical implementation of two dimensional problems includes finite element ideas of discretization and polynomial interpolation and numerical inversion of a laplace transform practical applications of the method are found in consolidation problems in soils which contain compressible as well as incompressible pore fluids also as a numerical experiment consolidation of partially saturated soil is simulated and interesting phenomena are observed the currently developed boundary integral equation method biem for porous elasticity may be viewed as an efficient and accurate alternative of existing finite element and finite difference methods for linear consolidation problems application of biem is always preferred to the other numerical methods whenever possible

algebraic splitting for incompressible navier stokes equations fully discretized incompressible navier stokes equations are solved by splitting the algebraic system with an approximate factorization this splitting affects the temporal convergence order of velocity and pressure the splitting error is proportional to the pressure variable and a simple analysis shows that the original convergence order of the time integration scheme can be retained by solving for incremental pressure the combination of splitting and incremental pressure is shown to be equivalent to an error correcting method using the full pressure in numerical experiments employing a third order time integration scheme and various orders for the pressure increment the splitting error is shown to control the convergence order and the full order of the scheme is recaptured for both velocity and pressure the difference between perturbing the momentum or the continuity equation is also explored

the laplace transform finite difference method for simulation of flow through porous media a new numerical method the laplace transform finite difference ltfd method was developed to solve the partial differential equation pde of transient flow through porous media ltfd provides a solution which is semianalytical in time and numerical in space by solving the discretized pde in the laplace space and numerically inverting the transformed solution vector the effects of the traditional treatment of the time derivative on accuracy and stability are rendered irrelevant because time is no longer a consideration for a single time step ltfd requires no more than eight matrix solutions and an execution time eight times longer than the analogous finite difference fd requirement without an increase in storage this disadvantage is outweighed by an unlimited time step size without any loss of accuracy a superior accuracy and a stable nonincreasing round off error thus a problem in standard fd format may require several hundred time steps and matrix inversions between the initial condition and the desired solution time but ltfd requires only one time step and no more than eight matrix inversions to achieve a more accurate result

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

a modified nodal scheme for the time dependent incompressible navier stokes equations based on poisson equation for pressure a nodal numerical scheme is developed for the time dependent incompressible navier stokes equations derivation is based on local transverse integrations over finite size brick like cells that transform each partial differential equation to a set of ordinary differential equations odes solutions of these odes for the transverse averaged dependent variables are then utilized to develop the difference scheme the discrete variables are scalar velocities and pressure averaged over the faces of brick like cells in the x y t space cell interior variation of transverse averaged pressure in each spatial direction is quadratic cell interior variation of transverse averaged velocity in each spatial direction is a sum of a constant a linear and an exponential term due to the introduction of delayed coefficients the exponential functions are to be evaluated only once at each time step the semi implicit scheme has inherent upwinding results of applications to several test problems show that the scheme is very robust and leads to a second order error as expected in such coarse mesh schemes even relatively large size cells lead to small errors extension to three dimensions is straightforward

mathematical modeling of different types of instabilities in time fractional reaction diffusion systems abstract in this article we investigate possible scenarios of pattern formations in reaction diffusion systems with time fractional derivatives linear stability analysis is performed for different values of derivative orders results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations most attention is paid to two models a fractional order reaction diffusion system with bonhoeffer van der pol kinetics and to the brusselator model

finite analytic numerical method for unsteady two dimensional navier stokes equations abstract the main purpose of this paper is to develop a finite analytic fa numerical solution for unsteady two dimensional navier stokes equations the fa method utilizes the analytic solution in a small local element to formulate the algebraic representation of partial differential equations in this study the combination of linear and exponential functions that satisfy the governing equation is adopted as the boundary function thereby improving the accuracy of the finite analytic solution two flows one a starting cavity flow and the other a vortex shedding flow behind a rectangular block are solved by the fa method the starting square cavity flow is solved for reynolds numbers of 400 1000 and 2000 to show the accuracy and stability of the fa solution the fa solution for flow over a rectangular block h h 4 predicts the strouhal number for reynolds numbers of 100 and 500 to be 0 156 and 0 125 details of the flow patterns are given in addition to streamlines and vorticity distribution rest streamlines are given to illustrate the vortex motion downstream of the block

a generalized two dimensional analytical solution for hydrodynamic dispersion in bounded media with the first type boundary condition at the source a generalized two dimensional analytical solution is developed for hydrodynamic dispersion in a unidirectional bounded surface and subsurface media flow field from time and space dependent sources with the first type boundary condition at the source longitudinal and transverse dispersion radioactive decay and linear adsorption are considered in the mathematical analysis laplace transform and fourier analysis techniques are used simultaneously and a general equation in infinite series form for the unsteady state concentration distribution has been obtained solution for a single strip source is obtained as a special case from the general solution the equations for convective dispersive flux components are also derived the solution of bruch and street 1967 for a symmetrically located strip source is shown to be a special case of the solution expressions for concentration distribution and convective dispersive flux components for the steady state solute transport case are also presented the results are compared with two different finite element codes with a good agreement the results of this model may be used for solute transport in a unidirectional porous media flow field as well as in rivers and canals with some additional assumptions the solutions can also be used for numerical model validation

highly efficient oscillation free solution of the transport equation over long times and large spaces transport problems with sharp transitions or high convection often stress even the most advanced computational methods beyond feasibility limits one promising line of attack involves transforming problems to a moving deforming coordinate system mdcs using mdcs as recently developed one may concentrate computational attention when and where it is most appropriate also possibly greatly reducing the effective magnitude of terms which cause trouble in principle mdcs may be used with finite difference finite element or other numerical methods according to preference in the particular finite element system advanced here mesh movement may be variable in time and space so that physically fixed boundaries may remain fixed in the transformed coordinate system a simple finite difference system is used in time when the method is applied to the standard transport equation test case great efficiency and accuracy are obtained in early time even very steep fronts may be represented well with no oscillations using space and time step sizes well beyond conventional peclet courant number constraints more important problems are solved involving transport over much longer times and larger spaces when diffusion has some chance to operate and hence a parabolic model is more warranted the same accuracy and stability are achieved as during early time using no more nodes and time steps overall than during early time runs this constitutes orders of magnitude savings in effort over what the application of conventional constraints would require relative efficiency from application of mdcs to other problems would depend on the particulars of each case

pattern formation in a fractional reaction diffusion system abstract we investigate pattern formation in a fractional reaction diffusion system by the method of computer simulation of the model of excitable media with cubic nonlinearity we are able to show structure formation in the system with time and space fractional derivatives we further compare the patterns obtained by computer simulation with those obtained by simulation of the similar system without fractional derivatives as a result we are able to show that nonlinearity plays the main role in structure formation and fractional derivative terms change the transient dynamics so when the order of time derivative increases and approaches the value of 1 5 the special structure formation switches to homogeneous oscillations in the case of space fractional derivatives the decrease of the order of these derivatives leads to more contrast dissipative structures the variational principle is used to find the approximate solution of such fractional reaction diffusion model in addition we provide a detailed analysis of the characteristic dissipative structures in the system under consideration

an extension and simpler proof of berselli cordoba s geometric regularity condition for the navier stokes system abstract in this short note we provide an simpler proof of berselli cordoba s geometric regularity condition for the navier stokes system moreover some interesting extensions are obtained

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

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

discrete mittag leffler kernel type fractional difference initial value problems and gronwall s inequality abstract in this article we studied the caputo and riemann liouville type discrete fractional difference initial value problems with discrete mittag leffler kernels the existence and uniqueness of the solution is proved by using banach contraction principle the linear type equations are used to prove new discrete fractional versions of the gronwall s inequality the nabla discrete laplace transform is used to obtain solution representations the proven gronwall s inequality under a new defined lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions numerical examples are discussed to demonstrate the reliability of the theoretical results

finite difference spectral approximation for a time space fractional equation on two and three space dimensions abstract we consider the numerical method for solving a time space fractional equation whose solution has a weak initial singularity l 1 scheme on graded mesh is used to capture the initial weak singularity it is shown that the method can recover order 2 convergence by choosing some appropriate grading parameters where 0 1 is the order of temporal caputo fractional derivative a fully discrete scheme is proposed by combining spectral approximation for the discretization of space in 2d and 3d cases stability and convergence of the fully discrete scheme are rigorously established numerical results are presented to show the sharpness of the error estimate

comparison of the nodal integral method and nonstandard finite difference schemes for the fisher equation the relationship between the so called nonstandard finite difference schemes and the nodal integral method nim is investigated using the fisher equation as a model problem exact and best finite difference schemes are reviewed first next the nim for the fisher equation is developed it is shown that the nim leads to a nonstandard evaluation of the derivatives moreover the resulting scheme possesses the desirable characteristics of the nonstandard finite difference schemes such as the nonlocal evaluation of the nonlinear terms thus the nim provides a systematic framework to obtain schemes similar to the best finite difference schemes numerical results for a propagating front problem show that the nim can capture the shape and speed of the front very accurately results also show that the best finite difference scheme is stable for large grid sizes but only at the cost of inaccuracy in the front propagation speed additional results are obtained using the nim for symmetric and asymmetric initial conditions these describe the interaction of two fronts of advantageous genes that approach each other and merge it is noted that the traveling fronts evolving from certain asymmetric initial conditions become indistinguishable from those that evolve from symmetric initial conditions

efficient time stepping methods for miscible displacement problems in porous media efficient procedures for time stepping galerkin methods for approximating the solution of a coupled system for c c x t and p p x t with nonlinear neumann boundary conditions of the form begin gathered nabla cdot a x c nabla p y x c nabla g equiv nabla cdot u f 1 x t quad x in omega quad t in 0 t hfill nabla cdot b x c nabla p nabla c u cdot nabla c phi x frac partial c partial t f 2 x t c quad x in omega quad t in 0 t hfill u cdot v q 1 x t quad x in partial omega quad t in 0 t hfill b frac partial c partial v q x t c quad x in partial omega quad t in 0 t hfill c x 0 c 0 x quad x in omega hfill end gathered where omega subset mathbb r d 2 leqq d leqq 3 are presented and analyzed this system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions a

fast tensor product solvers for optimization problems with fractional differential equations as constraints fractional differential equations have recently received much attention within computational mathematics and applied science and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms an optimization problem with constraints given by fractional differential equations is considered which in its discretized form leads to a high dimensional tensor equation to reduce the computation time and storage the solution is sought in the tensor train format we compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned krylov solvers or tailored alternating schemes the competitiveness of these approaches is presented using several examples with constant and variable coefficients

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

transcendental bernstein series for solving nonlinear variable order fractional optimal control problems abstract this paper deals with finding an approximate solution of variable order fractional optimal control problems v focps based on lagrange multiplier optimization technique in combination with the new operational matrix of variable order vo fractional derivatives to carry out the proposed approach we firstly develop the well known bernstein polynomials to the new series of functions namely transcendental bernstein series tbs then we implement these basis functions to approximate the solutions of v focps in fact the series expansion in tbs with unknown free coefficients and control parameters is the novel idea for solving the fractional systems with less terms of approximation the convergence analysis of our proposed method will be guaranteed by proving a new theorem for the tbs to investigate the practical computational efficiency and accuracy of the presented method some numerical examples are provided the experimental results confirm the applicability of the method and a good agreement between the approximate and exact solutions

generalized taylor s formula abstract in this paper a new generalized taylor s formula of the kind f x j 0 n a j x a j r n x where a j r x a 0 1 is established such an expression is precisely the classical taylor s formula in case of 1 in addition detailed expressions for r n x and a j that involving caputo fractional derivatives are well constructed some applications including approximation of functions and solutions of fractional differential equations are also given

moving finite element model for one dimensional infiltration in unsaturated soil the numerical integration of richards equation for infiltration into homogeneous unsaturated soil is carried out by a moving finite element mfe method accurate solution of this one dimensional nonlinear partial differential equation by standard finite difference fd or finite element fe methods can be obtained by using a fine discretization on space and time this can lead to large computer times with the mfe method grid points are moved during computation along the wetting front so that accuracy can be maintained by using a small number of nodes for simulating cases several meters in depth for these cases the mfe method is faster than fd and fe methods the numerical mfe model is verified through comparison with the quasi analytical solution by philip 1957 moreover we compare for some other test problems the computational efficiency of this method with that of two efficient pressure head based fd and fe fixed grid formulations

finite analytic numerical solution for two dimensional groundwater solute transport a new numerical method called the finite analytic fa method is used to solve a groundwater solute transport problem the basic idea of the finite analytic method is the incorporation of local analytic solution in the numerical solution of the partial differential equation when the local analytic solution is evaluated at a given nodal point it gives an algebraic relationship between a nodal value in an element and its neighboring nodal points the assemble of the linear system equations results in a tridiagonal matrix like most finite difference method the advantages of using efficient iterative techniques for solving tridiagonal matrices are equally applicable to fa method the automatic localized upstream shift and the analytic property of the fa method eliminates the difficulty of numerical dispersion locally and suppresses the overall numerical dispersion for large peclet number for small peclet number fa method yields excellent results in comparison with the analytic solution for large peclet number fa solutions are oscillation free with some degree of numerical dispersion the results are comparable with those obtained using upstream weighted finite element method

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

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

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

simulation of contaminant transport in three dimensions 1 the alternating direction galerkin technique the alternating direction galerkin technique for the simulation of advective dispersive transport in three dimensions is developed the technique is designed for high efficiency in handling the large and detailed grids that are often required in simulations of natural groundwater systems it is comparable to an alternating direction finite difference scheme except for two advantages first the capability to handle certain curvilinear grids that may conform to either a flow net geometry or a natural stratigraphy and second the option to enhance solution accuracy through a choice between a finite difference or a finite element representation of the time derivative criteria for controlling numerical dispersion and accuracy can be applied easily three options for the time stepping algorithm are developed and analyzed for stability and their accuracy is investigated for simple flow systems the technique is shown to be far more efficient than a conventional three dimensional finite element model in a companion paper an application to a field scale contaminant transport problem is described

pseudopotential functions in construction of flow nets for contaminant transport modeling construction of flow nets in most hydrogeologic field problems is complicated by nonexistence of orthogonality between streamlines and hydraulic head lines in anisotropic porous media the nonexistence of orthogonality can be circumvented by transformation of the anisotropic porous medium into an isotropic one flow nets are valuable in generation of a numerical solution grid for a contaminant transport equation formulated along the direction of flow and the direction orthogonal to flow effect of flow domain transformation on contaminant transport simulation is not known the flow domain transformation can be avoided by constructing flow nets with streamlines and pseudopotential lines the pseudopotential lines are based on the theory of pseudopotential function a discussion of the theory in isotropic porous media exists in literature to my knowledge this is the first time that the theory is being extended to anisotropic porous media an integration procedure is used to evaluate values of pseudopotential function and hydraulic head along a streamline a comparison of flow nets based on pseudopotential lines and hydraulic head lines indicates that pseudopotential lines are more useful than hydraulic head lines in simulation of contaminant transport in saturated groundwater flow systems

front propagation in hyperbolic fractional reaction diffusion equations from the continuous time random walk scheme and assuming a l evy waiting time distribution typical of subdiffusive transport processes we study a hyperbolic reaction diffusion equation involving time fractional derivatives the linear speed selection of wave fronts in this equation is analyzed when the reaction diffusion dimensionless number and the fractional index satisfy a certain condition we find fronts exhibiting an unphysical behavior they travel faster in the subdiffusive than in the diffusive regime

an alternating direction galerkin technique for simulation of contaminant transport in complex groundwater systems a general contaminant transport simulation technique based on a galerkin finite element representation in space and an alternating direction time stepping scheme is developed the formulation which is in terms of natural coordinates following the principal directions of hydraulic conductivity of the medium decouples the spatial components of the equations leading to tridiagonal matrices the numerical accuracy can be easily controlled in each principal direction through the well established peclet and courant numbers results are found to be comparable to those from the classic finite element scheme but the cost of computation is generally much less because the computational effort varies only linearly with the grid size the technique is thus well suited for transport simulations with complex layered anisotropic groundwater systems requiring large and detailed grids

diffusive effects on dispersion in excitable media perturbation and numerical methods are used to study traveling and standing wave solutions for a system of reaction diffusion equations modeled after excitable media and the effect of variation of the diffusion coefficients it is shown that traveling waves and standing waves can coexist and that as a certain diffusion coefficient is varied a branch of traveling waves can bifurcate from a branch of standing waves providing a transition between only traveling solutions in one limit and only standing waves in the other limit key words traveling waves standing waves bifurcation dispersion reaction diffusion equations singular perturbation

factorization methods for the numerical approximation of navier stokes equations we investigate a general approach for the numerical approximation of incompressible navier stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve the splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations several schemes based on approximate factorization are investigated for some of these methods a formal analogy with well known time advancing schemes such as the projection chorin temam s can be pointed out features and limits of this analogy that was earlier introduced in b perot j comp phys 108 1993 51 8 are addressed other new methods can also be formulated starting from this approach in particular we introduce the so called yosida method which can be investigated in the framework of quasi compressibility schemes numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

the finite layer method for groundwater flow models the finite layer method flm is an extension of the finite strip method familiar in structural engineering the idea behind the method is to discretize two space dimensions using truncated fourier series approximating variations in the third via finite elements the eigenfunctions used in the fourier expansions are orthogonal and consequently the galerkin integrations decouple the weighted residual equations associated with different fourier modes the method therefore reduces three dimensional problems to sets of independent matrix equations that one can solve either sequentially on a microcomputer or concurrently on a parallel processor the latter capability makes the method suitable for such computationally intensive applications as optimization and inverse problems four groundwater flow applications are presented to demonstrate the effectiveness of flm as a forward solver

block preconditioning strategies for time space fractional diffusion equations abstract we present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time space fractional diffusion equations in contrast to the traditional time marching procedure the discretization via finite difference is considered in a fully coupled time space framework the resulting fully coupled discretized linear system is a summation of two kronecker products the four preconditioning methods are based on block diagonal banded block triangular and kronecker product splittings of the coefficient matrix all preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator numerical experiments show the efficiency of the four block preconditioners and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one

a variable explicit implicit numerical method for calculating advection on unstructured meshes a variable explicit implicit numerical method is presented for calculating fluid advection on hybrid unstructured meshes the method is intended for unsteady flows that have a large range of advective time scales with subregions where the flow may be quasi steady on time scales of interest the method generalizes the method of collins collela and glaz by providing an explicit to implicit switching criterion that is suitable for use on unstructured meshes no directional splitting is used upwinding is in the direction of the material velocity and thus we define and use a new multidimensional limited gradient that generalizes van leer s muscl limiting 7 after reviewing the one dimensional method of collins collela and glaz we present the generalization to hybrid unstructured grids and detail the results of four example calculations that demonstrate the accuracy and flexibility of the new method

geometric constrains for global regularity of 2d quasi geostrophic flows we study the two dimensional quasi geostrophic equations 2d qg in sobolev spaces we first prove a new analytic condition for global regularity which is both sufficient and necessary we then prove several new results on the geometric constraints on the 2d qg active scalar which suppress the development of singularity from the nonlinear stretching mechanism we focus mainly on the case with critical dissipation our results are also relevant to the inviscid case

algorithms for solving richards equation for variably saturated soils previous work has shown that a one dimensional water content based finite difference algorithm can offer improved cpu efficiency in modeling infiltration into very dry heterogeneous soil when compared to similar pressure based algorithms the usefulness of the water content based method is limited however since it cannot be applied to saturated soils in this paper we develop two new methods which retain the advantages of water content based methods while still permitting fully saturated conditions in the first method we define a new variable for the transformed richards equation which has the characteristics of water content when soil is unsaturated and of pressure when soil is at or near saturation in addition to the transformed richards equation method an improved pressure based method which uses flux updating is presented both methods are implemented in algorithms using a preconditioned conjugate gradient method equation solver the performance of the algorithms is compared to that of an algorithm based on a mixed form of richards equation using modified picard iteration two test cases are examined the first test case examines two dimensional infiltration into very dry heterogeneous soil which remains unsaturated throughout the simulation period the second test case examines a two dimensional developing perched water table results indicate that the new methods retain the advantages of the water content based method for fully unsaturated heterogeneous problems while allowing for fully saturated conditions for the unsaturated problem the new algorithms were from 16 to 59 times faster than the algorithm based on the mixed form of richards equation using modified picard iteration for the saturated problem the new algorithms were from 6 to 25 times faster than the mixed base algorithm results also indicate that the performance of the mixed base algorithm could be significantly improved if a more effective adaptive time step were developed

eulerian lagrangian solution of the convection dispersion equation in natural coordinates the vast majority of numerical investigations of transport phenomena use an eulerian formulation for the convenience that the computational grids are fixed in space an eulerian lagrangian method elm of solution for the convection dispersion equation is discussed and analyzed the elm uses the lagrangian concept in an eulerian computational grid system the values of the dependent variable off the grid are calculated by interpolation when a linear interpolation is used the method is a slight improvement over the upwind difference method at this level of approximation both the elm and the upwind difference method suffer from large numerical dispersion however if second order lagrangian polynomials are used in the interpolation the elm is proven to be free of artificial numerical dispersion for the convection dispersion equation the concept of the elm is extended for treatment of anisotropic dispersion in natural coordinates in this approach the anisotropic properties of dispersion can be conveniently related to the properties of the flow field several numerical examples are given to further substantiate the results of the present analysis

mathematical modeling of time fractional reaction diffusion systems we study a fractional reaction diffusion system with two types of variables activator and inhibitor the interactions between components are modeled by cubical nonlinearity linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives it is shown that by combining the fractional derivatives index with the ratio of characteristic times it is possible to find the marginal value of the index where the oscillatory instability arises the increase of the value of fractional derivative index leads to the time periodic solutions the domains of existing periodic solutions for different parameters of the problem are obtained a computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented for the fractional reaction diffusion systems it is established that there exists a set of stable spatio temporal structures of the one dimensional system under the neumann and periodic boundary conditions the characteristic features of these solutions consist of the transformation of the steady state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system

sensitivity analysis of flow in unsaturated heterogeneous porous media theory numerical model and its verification sensitivity analysis is one of the tools available for analyzing the effects of parameter uncertainty and soil heterogeneity on the transport of moisture in the unsaturated similar porous media direct differentiation of the discretized richards equation with respect to parameters defining spatial variability leads to linear systems of equations for elementary sensitivities that are readily solved in conjunction with the original equation these elementary sensitivities can be easily transformed into approximations of functional sensitivities and into sensitivities of boundary fluxes a numerical implementation of this technique in one space dimension yields results that are consistent with exact analytical solutions and with numerical perturbation calculations the effects of a given heterogeneity can be modeled adequately provided that the maximum relative change of the scale factor from one grid point to the next not exceed a number on the order of unity

an upwind finite element method via numerical intergration upwind finite element schemes remove spurious oscillations that occur in the solution of diffusion convection equations up to now these schemes lose part of their accuracy when the peclet is large as an improvement it is proposed to move the integration nodes along the streamlines before evaluating the elementary convection matrices the displacements of the nodes along the streamlines which are one dimensional manifolds are calculated analogously to well known one dimensional formulae the last section of this paper illustrates this new method with the help of four examples which show its validity

hybrid finite analytic solution of lateral subsurface flow a hybrid finite analytic method a numerical technique was used to solve lateral subsurface storm flow described by the extended dupuit forchheimer equation in subregions the local linearized one dimensional dupuit forchheimer equation was solved analytically in space and discretized in time by a simple difference formula this is called a hybrid finite analytic approach the system of algebraic equations obtained by the hybrid finite analytic approach can approximately preserve the overall nonlinear effect because the term h x 2 and the coefficient of diffusion are treated as constants only in the local regions a four point numerical formula can provide results that are stable and acceptably accurate without complex calculation and without small time steps the comparison between steady state profiles of the water table provided by this study with profiles provided by previous investigations shows that the method performs very well in the analysis of lateral subsurface storm flow

a kronecker product splitting preconditioner for two dimensional space fractional diffusion equations abstract we study preconditioned iterative methods for the linear system arising in the numerical discretization of a two dimensional space fractional diffusion equation our approach is based on a formulation of the discrete problem that is shown to be the sum of two kronecker products by making use of an alternating kronecker product splitting iteration technique we establish a class of fixed point iteration methods theoretical analysis shows that the new method converges to the unique solution of the linear system moreover the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real the basic iteration is accelerated by a krylov subspace method like gmres the corresponding preconditioner is in a form of a kronecker product structure and requires at each iteration the solution of a set of discrete one dimensional fractional diffusion equations we use structure preserving approximations to the discrete one dimensional fractional diffusion operators in the action of the preconditioning matrix numerical examples are presented to illustrate the effectiveness of this approach

a modified method of characteristics technique and mixed finite elements method for simulation of groundwater solute transport a comprehensive groundwater solute transport simulator is developed based on the modified method of characteristics mmoc combined with the galerkin finite element method for the transport equation and the mixed finite element mfe method for the groundwater flow equation the preconditioned conjugate gradient algorithm is used to solve the two large sparse algebraic system of equations arising from the mmoc and mfe discretizations the mmoc takes time steps in the direction of flow along the characteristics of the velocity field of the total fluid the physical diffusion and dispersion terms are treated by a standard finite element scheme the crucial aspect of the mmoc technique is that it looks backward in time along an approximate flow path instead of forward in time as in many method of characteristics or moving mesh techniques the mfe procedure involves solving for both the hydraulic head and the specific discharge simultaneously one order of convergence is gained by the mfe method as compared with other standard finite element methods and therefore more accurate velocity fields are simulated the overall advantages of the mmoc mfe method include minimum numerical oscillation or grid orientation problems under steep concentration gradient simulations and material balance errors are greatly reduced due to a very accurate velocity simulation by the mfe method in addition much larger time steps with courant number well in excess of 1 as compared with the standard galerkin finite element method can be taken on a fixed spatial grid system without significant loss of accuracy

conservative versions of the locally exact consistent upwind scheme of second order lecusso scheme two conservative versions of the recently introduced locally exact consistent upwind scheme of second order lecusso scheme are presented which both use four points two upstream one downstream and do not suffer from wiggles test calculations using a flow problem with an exact analytical solution show that the two schemes called lecusso c and quick plus are superior to the original lecusso scheme and to several other standard difference schemes

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

simulation of contaminant plumes with large dispersive contrast evaluation of alternating direction galerkin models the evolution of narrow sharply defined contaminant plumes corresponding in shape to those often observed in the field is examined it is shown that the transverse dispersion mechanism consistent with such shapes can be as low as of the order of molecular diffusion and that dispersivity contrasts of as much as 4 orders of magnitude are possible even at the lowest physically realistic values of transverse dispersion parameters however transverse mass spreading is capable of significantly retarding the advance of the plume front the principal direction and alternating direction galerkin models together are found to be capable of handling all cases of dispersive contrast including those with dispersivity ratio of infinity the models are formulated in curvilinear coordinates with certain restrictions on the element deformation a three way comparison between principal direction alternating direction galerkin and conventional finite element models is performed with respect to accuracy and efficiency the effects of longitudinal and transverse numerical dispersion occuring in the various models are examined and related to the discretization visual indications of the presence of numerical error are discussed

the cell analytical numerical method for solution of the advection dispersion equation two dimensional problems in this paper we present the development and evaluation of a new numerical scheme for efficient solution of groundwater solute transport problems the scheme which we have named the cell analytical numerical can method is an extension of the so called nodal methods which have been developed in the nuclear engineering area for solving neutron diffusion problems the can method is based upon decomposition of the solution domain into a number of rectangular cells volume subdomains each cell is homogeneous so that a local analytic solution to the solute transport equation can be obtained a first order accurate finite difference approximation of the time derivative followed by a transverse averaging procedure is used to transform the governing partial differential equation into a set of coupled one dimensional ordinary differential equations for the transverse moments of the concentration the local analytical solution is therefore found in terms of these moments solute mass flux continuity across cell surfaces along with the local analytical solution is used to construct an algebraic relationship between concentration moment values at adjacent cell surfaces assembling all the cells together results in a set of coupled tridiagonal matrix equations one set for each spatial direction which can be solved very efficiently the can method is applied to several simple test problems involving uniform flow in homogeneous aquifers while the method lacks geometrical flexibility it offers a spatial approximation that is demonstrated to have high accuracy and minimal grid orientation error even when applied to coarse meshes

existence of solutions of abstract fractional integrodifferential equations of sobolev type this paper deals with the study of existence of solutions of nonlinear fractional integrodifferential equations of sobolev type with nonlocal condition in banach spaces the results are obtained by using resolvent operators fractional calculus and fixed point technique an example is provided to illustrate the theory

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

the laplace transform galerkin technique a time continuous finite element theory and application to mass transport in groundwater a time continuous numerical technique referred to as the laplace transform galerkin ltg method is developed and applied to the problem of solute transport in porous media after application of galerkin s procedure and subdivision of the domain into finite elements the method involves a simple application of the laplace transformation to eliminate the temporal derivatives appearing in the space discretized set of ordinary differential equations then by solving the resulting transformed system of algebraic equations in laplace p space numerical inversion of the laplace transformed nodal concentration is performed using the robust and accurate crump 1976 algorithm the crump algorithm permits the concentration to be evaluated from a range of time values from a single set of laplace p space solutions because each of the needed p space solutions are independent the algorithm is well suited for execution on multiprocessor parallel computers it is demonstrated by means of a series of examples that the ltg scheme is capable of providing highly accurate solutions essentially devoid of numerical dispersion for grid peclet numbers in excess of 30 examination of the complex valued laplace domain concentration profiles reveal that they are generally smooth well behaved oscillatory functions compared to the profiles in the time domain thus permitting the use of a coarse finite element grid because of the nature of the laplace transformation the ltg method is particularly well suited to the problem of transient groundwater flow and solute transport in fractured porous media or multiple aquifer aquitard systems based on the dual porosity integrodifferential equation approach

a space time accurate method for solving solute transport problems solute transport studies frequently rely on numerical solutions of the classical advection diffusion equation unfortunately solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates recently developed variants of these techniques such as the finite analytic method chen and li 1979 chen and chen 1984 and the optimal test function method celia et al 1989a b c perform well for steady state problems extensions of these methods to the transient case have however not been successful primarily because of inadequate approximations of the temporal derivative the new numerical method proposed in this paper avoids this difficulty by taking the laplace transform of the transient equation the transformed expression behaves like a steady state advection diffusion equation with a first order decay term this expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse laplace transform algorithm the result is an accurate and robust transient solution which performs well over a very wide range of peclet numbers we demonstrate this approach by applying the finite analytic method to a laplace transformed one dimensional model problem a comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection dominated problems a good space approximation combined with a poor temporal approximation or vice versa does not give satisfactory results the method we propose provides a balanced space time approximation which works very well for one dimensional problems extensions to multiple dimensions are conceptually straightforward and briefly discussed

stability of the static spike autosolitons in the gray scott model we performed an asymptotic linear stability analysis of the static spike autosolitons ass self sustained solitary inhomogeneous states in the gray scott model of an autocatalytic chemical reaction we found that in one dimension these ass destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough in higher dimensions the one dimensional static spike ass are always unstable with respect to corrugation and wriggling the higher dimensional radially symmetric static spike ass may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow

on depletion of the vortex stretching term in the 3d navier stokes equations certain new cancellation properties in the vortex stretching term are detected leading to new geometric criteria for preventing finite time blow up in the 3d navier stokes equations

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

approximate solutions of nonlinear fractional equations fractional exponential that are invariant under fractional derivatives elementary and special fractional functions are introduced approximate solutions to fractional burgers equation by using the homotopy perturbation method are obtained furthermore real integral representations for some h functions are found that may be very helpful in numerical computations

variational representation of the projection dynamics and random motion of highly dissipative systems we propose the method of investigation of highly dissipative systems which are based on the approximation of the attractor by some manifold the projection dynamic equations for the general form of such manifold of the dissipative dynamic system are obtained variational principles for the projection dynamics are considered on the basis of the projection dynamic equations we investigate the influence of the random forces on the behaviour of the system

a lagrangian porous media mass transport model the mathematical simulation of advective dispersive contaminant transport in groundwater involving large peclet numbers is subject to numerical difficulties for most numerical models the computational cost and computer core requirements escalate as the peclet number increases a two dimensional galerkin finite element model for flow and lagrangian mass transport in porous media has been developed to alleviate numerical and computational difficulties the solution methodology involves a linear triangular mesh which tracks along streamlines calculated from a flow equation having the stream function as the dependent variable a comparison of this model to eulerian or fixed coordinate type models showed this model to be accurate and stable at a reduced computational effort for an actual field contamination problem which contains large vertical concentration gradients the simulated results compared with observed data

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

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

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

iterative solutions for a coupled system of fractional differential integral equations with two point boundary conditions abstract in this paper we deal with a coupled system of fractional differential integral equations with two point boundary conditions the maximal and minimal solutions are obtained by using the monotone iterative technique combined with the method of upper and lower solutions meantime the error estimate of the solutions are given in the end some examples are given to illustrate the results

on boundary conditions and point sources in the finite element integration of the transport equation in the finite element integration of the convection dispersion equation over a groundwater domain boundary conditions of the third or cauchy type may or may not be incorporated with the variational statement while the former development formulation a requires the application of green s first identity to both the convective and the dispersive terms the latter formulation b does not although both approaches are equivalent at the theoretical level they behave differently from a computational standpoint the numerical solution proves more accurate and stable if green s lemma is not applied to the convective term since significant errors arising from the representation of the darcy velocity over the elements are avoided point sources or sinks representing injection or pumping wells can be treated as either sources sinks distributed over a small volume or external boundaries with prescribed contaminant flux it is shown that when the well bore is excluded from the flow field formulation a yields the same matrix addition for an injection abstraction well as formulation b does for an abstraction injection well reciprocity of the results the computational behaviors of these alternative modeling procedures are illustrated with a number of significant numerical examples

fractional diffusion equations coupled by reaction terms we investigate the behavior for a set of fractional reaction diffusion equations that extend the usual ones by the presence of spatial fractional derivatives of distributed order in the diffusive term these equations are coupled via the reaction terms which may represent reversible or irreversible processes for these equations we find exact solutions and show that the spreading of the distributions is asymptotically governed by the same the long tailed distribution furthermore we observe that the coupling introduced by reaction terms creates an interplay between different diffusive regimes leading us to a rich class of behaviors related to anomalous diffusion

a characteristic based finite analytic method for solving the two dimensional steady state advection diffusion equation 1 this paper develops an improved finite analytic fa solution method to the advection diffusion equation ade for solving advection dominated steady state transport problems the fa method solves the ade analytically in localized discrete elements with each element linked through local boundary conditions previous fa methods have suffered from complex solution formulations as well as from numerical dispersion stemming from inaccuracies in the local boundary estimations here we use finite difference approximations of the dispersion terms to reduce solution complexity implement an improved particle tracking scheme to account for velocity variations within each element and use a hermite interpolation scheme to estimate the local boundary conditions the new fa method previous fa methods and a finite difference upwinding method are compared in both homogeneous and heterogeneous velocity fields at different peclet numbers results show the new fa method exhibits little numerical dispersion without undue complexity or computational effort across all tested flow conditions

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence