a eulerian lagrangian numerical scheme for the dispersion convection equation using conjugate space time grids abstract a new numerical scheme is proposed for the dispersion convection equation which combines the utility of a fixed grid in eulerian coordinates with the computational power of the lagrangian method convection is formally decoupled from dispersion in a manner which does not leave room for ambiguity the resulting convection problem is solved by the method of characteristics on a grid fixed in space dispersion is handled by finite elements on a separate grid which may but need not coincide wit the former at selected points in spacetime information is projected from one grid to another by local interpolation the conjugate grid method is implemented by linear finite elements in conjunction with piecewise linear interpolation functions and applied to five problems ranging from predominant dispersion to pure convection the results are entirely free of oscillations numerical dispersion exists but can be brought under control either by reducing the spatial increment or by increasing the time step size of the grid used to solve the convection problem contrary to many other methods best results are often obtained when the courant number exceeds 1

a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media abstract we develop a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media by utilizing the intrinsic mathematical numerical and physical properties of the mathematical model we model the process in which co2 is injected into a deep aquifer spreads within the aquifer and upon reaching a leaky well rises up to a shallower aquifer to quantify the leakage rate which depends on the pressure build up in the aquifer due to injection and the buoyancy of co2 the underlying eulerian lagrangian framework has high potential to improve the efficiency and accuracy for the numerical simulation of complex flow and transport processes in co2 sequestration the sparse grid probabilistic collocation framework adds computationally efficient uncertainty quantification functionality onto pre existing eulerian lagrangian methods in a nonintrusive manner it also provides a scalable framework to consider uncertainty in a straightforward parallel manner preliminary numerical experiments show the feasibility and potential of the method

on conditional mckean lagrangian stochastic models this paper is motivated by a new class of sdes pdes systems the so called lagrangian stochastic models which are commonly used in the simulation of turbulent flows we study a position velocity system which is nonlinear in the sense of mckean as the dynamics of the velocity depends on the conditional expectation with respect to its position the interaction kernel is singular we prove existence and uniqueness of the solution to the system by solving a nonlinear martingale problem and showing that the corresponding interacting particle system propagates chaos

a non hybrid method for the pdf equations of turbulent flows on unstructured grids in probability density function pdf methods of turbulent flows the joint pdf of several flow variables is computed by numerically integrating a system of stochastic differential equations for lagrangian particles a set of parallel algorithms is proposed to provide an efficient solution of the pdf transport equation modeling the joint pdf of turbulent velocity frequency and concentration of a passive scalar in geometrically complex configurations in the vicinity of walls the flow is resolved by an elliptic relaxation technique down to the viscous sublayer explicitly modeling the high anisotropy and inhomogeneity of the low reynolds number wall region without damping or wall functions an unstructured eulerian grid is employed to extract eulerian statistics to solve for quantities represented at fixed locations of the domain i e the mean pressure and the elliptic relaxation tensor and to track particles all three aspects regarding the grid make use of the finite element method employing the simplest linear shapefunctions to model the small scale mixing of the transported scalar the interaction by exchange with the conditional mean iecm model is adopted an adaptive algorithm to compute the velocity conditioned scalar mean is proposed that homogenizes the statistical error over the sample space with no assumption on the shape of the underlying velocity pdf compared to other hybrid particle in cell approaches for the pdf equations the current methodology is consistent without the need for consistency conditions the algorithm is tested by computing the dispersion of passive scalars released from concentrated sources in two different turbulent flows the fully developed turbulent channel flow and a street canyon or cavity flow algorithmic details on estimating conditional and unconditional statistics particle tracking and particle number control are presented in detail relevant aspects of performance and parallelism on cache based shared memory machines are discussed

the hybrid method for the pdf equations of turbulent reactive flows consistency conditions and correction algorithms the hybrid method solves the modeled transport equation for the joint pdf of velocity turbulence frequency and compositions for turbulent reactive flows a finitevolume fv method is used to solve the mean conservation equations for mass momentum and energy and the mean equation of state and a particle method is used to solve the modeled pdf equation the method is completely consistent at the level of the governing equations solved by the fv and particle algorithms in this work the conditions to be fulfilled for full consistency at the numerical solution level are examined and the independent consistency conditions are identified then correction algorithms are developed to enforce these independent consistency conditions to achieve full consistency at the numerical solution level in addition a new formulation of the energy equation and the equation of state is developed which is both general and simple the hybrid method is applied to a non premixed piloted jet flame the numerical results show that the correction algorithms are completely successful in achieving consistency the convergence of the method is demonstrated and in particular it is shown that the bias error is dramatically reduced compared to that in previous pdf calculations in addition the results are shown to be in a good agreement with some earlier pdf calculations and also with the available experimental data because of the substantially reduced numerical error for given grid size and number of particles the present hybrid method represents a significant advance in the computational efficiency of particle mesh method for the solution of pdf equations c 2001 academic press

characteristic local grid refinement techniques for reservoir flow problems using a sequential time marching approach we describe and implement effective solution strategies for the equations governing immiscible displacement in a porous medium by combining adaptive grid refinement with an operator splitting technique based on the modified method of characteristics the saturation equation is treated in a consistent and accurate way for the pressure equation we construct accurate piecewise linear velocity components from a piecewise linear pressure approximation

a numerical source of small scale number density fluctuations in eulerian lagrangian simulations of multiphase flows eulerian lagrangian simulations of multiphase flow are known to suffer from two errors that can introduce small scale fluctuations in the number density of the dispersed phase these errors can be reduced by increasing the number of particles in the simulation here we present results to demonstrate that a third error exists that can also generate small scale number density fluctuations in contrast to the two known errors this error cannot be lowered by increasing the number of particles analysis shows that this error is caused by spatial variation at the subgrid scale in the interpolation error of the fluid velocity to the particle location if the particle velocity divergence is zero the particle concentration error remains at the subgrid scale however if particles preferentially accumulate either due to their inertia or due to divergence of the underlying fluid velocity field this error manifests as number density fluctuations on the grid scale the only mechanism of reducing these errors is through higher order accurate interpolation by studying two model problems estimates for the errors are derived these estimates are shown to be quite accurate for simulations of shock and expansion waves interacting with particles

constructing an entropy stable upwind scheme for compressible fluid flow computations a relationship between entropy processes and numerical upwinding is examined in the context of computational gasdynamics a discretized form of the entropy inequality is constructed at the integration point where convection diffusion modeling occurs in finite volume methods conventional upwinding schemes may violate the local form of the second law of thermodynamics but a modified upwinding scheme uses additional momentum constraints and pressure terms to provide a positive entropy production rate the second law is seen as an important quantitative measure of nonphysical numerical results as well as a sound basis for error analysis applications to converging diverging nozzle and blunt body flow problems demonstrate the promising performance of the overall numerical algorithm

pdf model calculations of compressible turbulent flows using smoothed particle hydrodynamics a particle method which applies the probability density function pdf method to compressible turbulent flows is presented solution of the pdf equation is achieved using a lagrangian monte carlo approach a unique feature of the method is its ability to calculate the mean pressure gradient directly from the particles using a grid free approach this is accomplished by applying techniques borrowed from the field of smoothed particle hydrodynamics furthermore these techniques have been implemented using a recently discovered algorithm which greatly reduces the computational work in 1d the particle method also incorporates a variance reduction technique which can significantly reduce statistical error in first and second moments of selected mean flow quantities when combined with a second order accurate predictor corrector scheme the resulting particle method provides a feasible way to obtain accurate pdf solutions to compressible turbulent flow problems results have been obtained for a variety of quasi 1d flows to demonstrate the method s robustness these include solutions to both statistically stationary and nonstationary problems and use both periodic and characteristic based inflow outflow boundary conditions convergence of the method with respect to four different kinds of numerical errors has also been studied detailed results are presented which confirm the expected convergence behavior of each error

a comparison of hybrid and quadratic upstream differencing in high reynolds number elliptic flows abstract numerical computations have been performed of various two dimensional elliptic flows at high reynolds number with a view to assessing the relative merits of the widely used hybrid i e upwind central interpolation and the recently proposed quadratic upstream interpolation of leonard known as quick the latter scheme possesses less intrinsic stability than hybrid differencing by selectively evaluating some of the convective elements at a previous iteration level however satisfactorily rapid convergence has been obtained it is concluded from the flows tested that the extra computational effort per node required for quick over the hybrid difference scheme is more than repaid by the greater numerical accuracy that results

iterative multiscale finite volume method the multiscale finite volume msfv method for the solution of elliptic problems is extended to an efficient iterative algorithm that converges to the fine scale numerical solution the localization errors in the msfv method are systematically reduced by updating the local boundary conditions with global information this iterative multiscale finite volume i msfv method allows the conservative reconstruction of the velocity field after any iteration and the msfv method is recovered if the velocity field is reconstructed after the first iteration both the i msfv and the msfv methods lead to substantial computational savings where an approximate but locally conservative solution of an elliptic problem is required in contrast to the msfv method the i msfv method allows a systematic reduction of the error in the multiscale approximation line relaxation in each direction is used as an efficient smoother at each iteration this smoother is essential to obtain convergence in complex highly anisotropic heterogeneous domains numerical convergence of the method is verified for different test cases ranging from a standard poisson equation to highly heterogeneous anisotropic elliptic problems finally to demonstrate the efficiency of the method for multiphase transport in porous media it is shown that it is sufficient to apply the iterative smoothing procedure for the improvement of the localization assumptions only infrequently i e not every time step this result is crucial since it shows that the overall efficiency of the i msfv algorithm is comparable with the original msfv method at the same time the solutions are significantly improved especially for very challenging cases

a solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion in this paper a stochastic model is presented to simulate the flow of gases which are not in thermodynamic equilibrium like in rarefied or micro situations for the interaction of a particle with others statistical moments of the local ensemble have to be evaluated but unlike in molecular dynamics simulations or dsmc no collisions between computational particles are considered in addition a novel integration technique allows for time steps independent of the stochastic time scale the stochastic model represents a fokker planck equation in the kinetic description which can be viewed as an approximation to the boltzmann equation this allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations the fluid dynamic equations of navier stokes and fourier are fully recovered for small relaxation times while for larger values the new model extents into the kinetic regime numerical studies demonstrate that the stochastic model is consistent with navier stokes in that limit but also that the results become significantly different if the conditions for equilibrium are invalid the application to the knudsen paradox demonstrates the correctness and relevance of this development and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages moreover results of a test case with geometrically complex boundaries are presented

fundamental solution of the heat and schrodinger equations with point interaction abstract we find explicit formulae for the fundamental solution for the heat and time dependent schrodinger equations with point interaction in dimension n 3 we discuss in detail the small time asymptotics of these fundamental solutions the case n 2 has the peculiarity of involving logarithmic terms in the expansion

derivation of the time dependent propagator for the three dimensional schrodinger equation with one point interaction an explicit formula for the time dependent propagator of the schrodinger equation with one point interaction in three dimensions is given the derivation is based on the inverse laplace transformation applied to the corresponding resolvent

heterogeneous multiscale method for the modeling of complex fluids and micro fluidics the framework of the heterogeneous multiscale method hmm is used to develop numerical methods for the study of macroscale dynamics of fluids in situations where either the constitutive relation or the boundary conditions are not explicitly available and have to be inferred from microscopic models such as molecular dynamics continuum hydrodynamics is used as the macroscopic model while molecular dynamics serves as the microscopic model and is used to supply the necessary data e g the stress or the boundary condition for the macroscopic model scale separation is exploited so that the macroscopic variables can be evolved in macroscopic spatial temporal scales using data that are estimated from molecular dynamics simulation on microscale spatial temporal domains this naturally decouples the micro and macrospatial and temporal scales whenever possible applications are presented for models of complex fluids contact line dynamics and a simple model of non trivial fluid solid interactions

a computational strategy for the regularized 13 moment equations with enhanced wall boundary conditions the challenge of modeling low speed rarefied gas flow in the transition regime is well known in this paper we propose a numerical solution procedure for the regularized 13 moment equations within a finite volume framework the stress and heat flux equations arising in the method of moments are transformed into the governing equations for the stress and heat flux deviators based on their first order approximation to model confined flows a complete set of wall boundary conditions for the 13 moment equations are derived based on the maxwell wall boundary model this has been achieved by expanding the molecular distribution function to fourth order accuracy in hermite polynomials empirical correction factors are introduced into the boundary conditions and calibrated against direct simulation monte carlo data the numerical predictions obtained from the regularized 13 moment equations and the navier stokes fourier equations are compared with data generated using the direct simulation monte carlo method for planar couette flow for a range of wall velocities and knudsen numbers 0 012 1 0 the results indicate that the regularized 13 moment equations are in good qualitative agreement with the direct simulation monte carlo data the results also highlight limitations that are caused by the use of a first order expansion of the third moment

evaluation of methods for calculating volume fraction in eulerian lagrangian multiphase flow simulations the present work addresses numerical methods required to compute particle volume fraction or number density local volume fraction of the lth particle l is the quantity of foremost importance in calculating the gas mediated particle particle interaction effect in multiphase flows a general multiphase flow with a distribution of lagrangian particles inside a fluid flow discretized on an eulerian grid is considered particle volume fraction is needed both as a lagrangian quantity associated with each particle and also as an eulerian quantity associated with the grid cell for eulerian lagrangian simulations in grid based gb methods the particle volume fraction is first obtained within each grid cell as an eulerian quantity and then the local particle volume fraction associated with any lagrangian particle can be obtained from interpolation the second class of methods presented are particle based pb methods where particle volume fraction will first be obtained at each particle as a lagrangian quantity which then can be projected onto the eulerian grid traditionally the gb methods are used in multiphase flow but sub grid resolution can be obtained through use of the pb methods by evaluating the total error and its discretization bias and statistical error components the performance of the different pb methods is compared against several common gb methods of calculating volume fraction the standard von neumann error analysis technique has been adapted for evaluation of rate of convergence of the different methods the discussion and error analysis presented focus on the volume fraction calculation but the methods can be extended to obtain field representations of other lagrangian quantities such as particle velocity and temperature

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

consistent inflow and outflow boundary conditions for transported probability density function methods in transported probability density function pdf methods the pdf transport equations are most often solved with realization based techniques using particles in this paper a consistent treatment of particle in and outflow boundary conditions for transported pdf methods is devised it is shown that the presented approach is simple and most important consistent with the underlying eulerian pdf transport equation this is not the case for other boundary condition implementations discussed in the literature especially if the fluctuating particle velocities are high compared with the averaged ones

time dependent schrodinger like equation with nonlocal term we investigate a time dependent schrodinger like equation in presence of a nonlocal term by using the method of variable separation and the green function approach we analyze the green function for different forms of the memory kernel and the nonlocal term results for delta potential energy function are presented distributed order memory kernels are also considered and the asymptotic behaviors of the green function are derived by using tauberian theorem the obtained results for the green function for the considered schrodinger like equation may be transformed to those for the probability distribution function of a diffusion like equation with memory kernel and can be used to explain various anomalous diffusive behaviors

stochastic modeling of evaporating sprays within a consistent hybrid joint pdf framework in the present study a framework for modeling two phase evaporating flow is presented which employs an eulerian lagrangian lagrangian approach for the continuous phase a joint velocity composition probability density function pdf method is used opposed to other approaches such pdf methods require no modeling for turbulent convection and chemical source terms for the dispersed phase the pdf of velocity diameter temperature seen gas velocity and seen gas composition is calculated this provides a unified formulation which allows to consistently address the different modeling issues associated with such a system because of the high dimensionality particle methods are employed to solve the pdf transport equations to further enhance computational efficiency a local particle time stepping algorithm is implemented and a particle time averaging technique is employed to reduce statistical and bias errors in comparison to previous studies a significantly smaller number of droplet particles per grid cell can be employed for the computations which rely on two way coupling between the droplet and gas phases the framework was validated using established experimental data and a good overall agreement can be observed

time dependent solutions for a fractional schrodinger equation with delta potentials we investigate for an arbitrary initial condition the time dependent solutions for a fractional schrodinger equation in the presence of delta potentials by using the green function approach the solutions obtained show an anomalous spreading asymptotically characterized by a power law behavior which is governed by the order of the fractional spatial operator present in the schrodinger equation

convergence of a fully conservative volume corrected characteristic method for transport problems we consider the convergence of a volume corrected characteristics mixed method vccmm for advection diffusion systems it is known that without volume correction the method is first order convergent provided there is a nondegenerate diffusion term we consider the advective part of the system and give some properties of the weak solution with these properties we prove that the volume corrected method with no diffusion term gives a lower order l 1 convergence rate of mathcal o h sqrt delta t h delta t r where r is related to the accuracy of the characteristic tracing this result compares favorably to godunov s method but avoids the cfl constraint so large time steps can be taken in practice in fact godunov s method converges at mathcal o h 1 2 which is our result for delta t ch where now c is not limited however the optimal choice delta t ch 2 2r 1 gives a better rate mathcal o h 2r 2r 1 than godunov s method e g mathcal o h 2 3 if r 1 with a nondegenerate diffusion term we obtain an l 2 error estimate for the problem we also prove the existence of and give an error estimate for a perturbed velocity field for which the volume is locally conserved finally some convergence tests are given to verify the optimal convergence rate for r 1

numerical implementation of mixing and molecular transport in les pdf studies of turbulent reacting flows probability density function pdf methods in combination with large eddy simulations les are a powerful tool for studying turbulent reacting flow problems and we are interested in the implementation of mixing and molecular transport in les pdf methods the numerical methodology used for solution procedure is the hybrid particle mesh method and a fractional step scheme is used to solve for transport reaction and mixing sequentially mixing is modeled using the interaction by exchange with the mean iem model and the effects of molecular transport are incorporated as a mean drift term in the mixing step this methodology avoids spurious production of scalar variance and also allows direct incorporation of differential diffusion effects in this study various numerical implementations of mixing and molecular transport are presented and evaluated using the method of manufactured solutions mms for 1 accuracy 2 detailed conservation 3 realizability and 4 stability moreover the methodology is shown to be successful in capturing the effects of differential diffusion accurately with the additional property of ensuring realizability of species mass fractions finally and most importantly we describe a new variance reduction technique by way of an implicit smoothing methodology this smoothing scheme is shown to satisfy conservation boundedness and regularity criteria moreover for an appropriate choice of the smoothing length scale significant improvements in accuracy can be achieved for an incremental increase in computational cost also it is shown that with smoothing on a length scale greater than the grid size the bias and statistical errors due to there being a finite number of particles in the lagrangian monte carlo simulations scale as n t o t 1 and n t o t 1 2 respectively where n t o t is the total number of particles in the computational domain whereas without smoothing these errors scale as n p c 1 and n p c 1 2 where n p c is the much smaller number of particles in a computational cell

sharp interface immersed boundary level set method for wave body interactions a sharp interface cartesian grid method for the large eddy simulation of two phase turbulent flows interacting with moving bodies is presented the overall approach uses a sharp interface immersed boundary formulation and a level set ghost fluid method for solid fluid and fluid fluid interface treatments respectively a four step fractional step method is used for velocity pressure coupling and a lagrangian dynamic smagorinsky subgrid scale model is adopted for large eddy simulations a simple contact angle boundary condition treatment that conforms to the immersed boundary formulation is developed a variety of test cases of different scales ranging from bubble dynamics water entry and exit landslide generated waves to ship hydrodynamics are performed for validation extensions for high reynolds number ship flows using wall layer models are also considered

local mesh refinement within a multi block structured grid scheme for general flows abstract a multi level flow adaptive mesh refinement strategy has been formulated and implemented within a structured grid multiblock scheme for 2d laminar transitional and turbulent recirculating flow although the locally refined irregular grid portions are organised as block structured patches within coarse grid blocks a special data structure allows high storage economy to be achieved issues of particular interest in the study include the performance of different flow adaptation sensors based on solution properties and error and the response of the solution accuracy to inter grid interpolation practices based on bi linear bi quadratic and compact interpolation the effectiveness of the local refinement strategy especially in terms of cpu and memory resources is examined by reference to four test cases ranging from an analytically prescribed scalar field over a square domain to a transitional flow over a highly loaded compressor blade the study demonstrates that cpu and memory savings with a three level refinement for the compressor blade case are of order 70 and 90 respectively relative to the requirements of globally fine grids returning similar accuracy levels

multidisciplinary interactions in energy and environmental modeling abstract mathematical models have been widely used to understand predict or optimize many complex physical processes from semiconductor or pharmaceutical design to large scale applications such as global weather models to astrophysics in particular simulation of environmental effects of air polution is extensive here we address the need for using similar models to understand the fate and transport of groundwater contaminants and to design in situ remediation strategies three basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination one must first obtain an effective model to describe the complex fluid fluid and fluid rock interactions that control the transport of contaminants in groundwater this includes the problem of obtaining accurate reservoir descriptions at various length scales and modeling the effects of this heterogeneity in the reservoir simulators next one must develop accurate discretization techniques that retain the important physical properties of the continuous models finally one should develop efficient numerical solution algorithms that utilize the potential of the emerging computing architectures we will discuss recent advances in each of these three areas

semi implicit second order schemes for numerical solution of level set advection equation on cartesian grids abstract a new parametric class of semi implicit numerical schemes for a level set advection equation on cartesian grids is derived and analyzed an accuracy and a stability study is provided for a linear advection equation with a variable velocity using partial lax wendroff procedure and numerical von neumann stability analysis the obtained semi implicit scheme is 2nd order accurate in space and time in any dimensional case when using a dimension by dimension extension of the one dimensional scheme that is not the case for analogous fully explicit or fully implicit schemes a further improvement is obtained by using so called corner transport upwind extension in two dimensional case the extended semi implicit scheme with a specific velocity dependent value of is 3rd order accurate in space and time for a constant advection velocity and it is unconditional stable according to the numerical von neumann stability analysis for the linear advection equation in general

fractional schrodinger equation with riesz feller derivative for delta potentials the fractional schrodinger equation with the riesz feller derivative is discussed and solved when the potential involves delta functions some results in the literature are generalized

local time stepping algorithm for solving probability density function turbulence model equations a local time stepping algorithm has been developed to improve the numerical efe ciency of lagrangian particlebased monte carlo methods for obtaining the steady state solutions of the modeled probability density function equations of turbulent reacting e ows on each step in the pseudo time marching algorithm the properties of each particle are advanced by a time step the magnitude of which depends on the particle s spatial location this algorithm has been incorporated into the consistent hybrid e nite volume particle method the performance of the localtime steppingmethodisevaluated intermsofnumericalefe ciencyandaccuracy throughapplication toa nonreacting bluff body e ow forthistest case itisfound that localtime stepping can acceleratetheglobal convergence of the hybrid method by as much as an order of magnitude depending on the grid stretching additionally local time stepping is found to improve signie cantly the robustness of the hybrid method mainly due to the accelerated convection of error waves out of thecomputational domain the method is very simpleto implement and the small increase in cpu time per step typically 3 is a negligible penalty compared to the substantial reduction in the number of time steps required to reach convergence

a finite volume method with linearisation in time for the solution of advection reaction diffusion systems the numerical solution in one space dimension of advection reaction diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models this work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest the basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method the resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations

an efficient algorithm for scalar pdf modelling in incompressible turbulent flow numerical analysis with evaluation of iem and iecm micro mixing models in this paper we investigate the application of probability density function pdf monte carlo methods to scalar release from small sources in a turbulent flow spanning a large physical domain this is a typical situation encountered when modeling the dispersion of a gaseous substance in the atmosphere monte carlo pdf methods have recently been applied to atmospheric modeling responding to the need for predicting the higher statistics and the concentration pdf generated by the continuous release of reactive and non reactive substances in this work we introduce some optimized numerical techniques based on the paradigm that the main field of interest is the scalar field and not the fluid dynamic field the scalar are considered dynamically passive and the statistical characteristics of the turbulence velocity field are assumed known these techniques are a block structured grid coupled with a particle splitting erasing algorithm and a localized time stepping the proposed technique is different from others presented before since the particle splitting and erasing is done in a more straightforward and consistent manner this method has been applied to the study of scalar dispersion from localized line sources in a canopy generated boundary layer the line source has been treated as a point source in a two dimensional space but the extension to three dimensions is straightforward our framework allows for an evaluation of the effects induced by different levels of discretization in the velocity space of the involved micro mixing model starting from the interaction by the exchange with the mean iem toward the more physically consistent interaction by exchange with the conditional mean iecm therefore aside from the algorithm description and a complete numerical analysis of the code a comparison between the iem and iecm micro mixing models has been investigated

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

a consistently formulated quick scheme for fast and stable convergence using finite volume iterative calculation procedures abstract previous applications of quick for the discretization of convective transport terms in finite volume calculation procedures have failed to employ a rigorous and systematic approach for consistently deriving this finite difference scheme instead earlier formulations have been established numerically by trial and error the new formulation for quick presented here is obtained by requiring that it satisfy four rules that guarantee physically realistic numerical solutions having overall balance careful testing performed for the wall driven square enclosure flow configuration shows that the consistently derived version of quick is more stable and converges faster than any of the formulations previously employed this testing includes the relative evaluation of boundary conditions approximated by second and third order finite difference schemes as well as calculations performed at higher reynolds numbers than previously reported

stable numerical method applying a total variation diminishing scheme for incompressible flow to solve incompressible navier stokes equations a stable finite difference method applying a total variation diminishing tvd upwind scheme originally designed for compressible flow is proposed the validity and the receptivity of this method employing such a tvd scheme for application to incompressible flow analysis are investigated by the evaluation of the accuracy stability and convergence rate using the generalized implicit simplified marker and cell smac solution method combined with the tvd scheme three dimensional developing entry flows through a square curved duct are calculated and compared with available experimental data and some computational results obtained by quadratic upstream interpolations for the convective kinematics quick schemes as well as by third order upwind schemes

particle velocity interpolation in block centered finite difference groundwater flow models a block centered finite difference model of two dimensional groundwater flow yields velocity values at the midpoints of interfaces between adjacent blocks method of characteristics random walk and particle tracking models of solute transport require velocities at arbitrary particle locations within the finite difference grid particle path lines and travel times are sensitive to the spatial interpolation scheme employed particularly in heterogeneous aquifers this paper briefly reviews linear and bilinear interpolation of velocity and introduces a new interpolation scheme linear interpolation of velocity is consistent with the numerical solution of the flow equation and preserves discontinuities in velocity caused by abrupt blocky changes in transmissivity or hydraulic conductivity however linear interpolation yields discontinuous and somewhat unrealistic velocities in homogeneous aquifers bilinear interpolation of velocity yields continuous and realistic velocities in homogeneous and smoothly heterogeneous aquifers but does not preserve discontinuities in velocity at abrupt transmissivity boundaries the new scheme uses potentiometric head gradients and offers improved accuracy for nonuniform flow in heterogeneous aquifers with abrupt changes in transmissivity the new scheme is equivalent to bilinear interpolation in homogeneous media and is equivalent to linear interpolation where gradients are uniform selecting the best interpolation scheme depends in part on the conceptualization of aquifer heterogeneity that is whether changes in transmissivity occur abruptly or smoothly

a lagrangian stochastic modeling framework for multi phase flow in porous media many of the complex physical processes relevant for compositional multi phase flow in porous media are well understood at the pore scale level in order to study co 2 storage in sub surface formations however it is not feasible to perform simulations at these small scales directly and effective models for multi phase flow description at darcy scale are needed unfortunately in many cases it is not clear how the micro scale knowledge can rigorously be translated into consistent macroscopic equations here we present a new methodology which provides a link between lagrangian statistics of phase particle evolution and darcy scale dynamics unlike in finite volume methods the evolution of lagrangian particles representing small fluid phase volumes is modeled each particle has a state vector consisting of its position velocity fluid phase information and possibly other properties like phase composition while the particles are transported through the computational domain according to their individual velocities the properties are modeled via stochastic processes honoring specified lagrangian statistics note that the conditional expectations of the particle velocities are different for different fluid phases the goal of this paper is to present the general framework for this alternative modeling approach various one and two dimensional numerical experiments demonstrate that with appropriate stochastic rules the particle solutions are consistent with a standard two phase darcy flow formulation in the end we demonstrate how to model non equilibrium phenomena within the stochastic particle framework which will be the main focus of the future work

green s functions and energy eigenvalues for delta perturbed space fractional quantum systems starting from the propagator we introduced a time ordered perturbation expansion and employed wick rotation to obtain a general energy dependent green s function expressions for space fractional quantum systems with dirac delta function perturbation we then obtained the green s functions and equations for the bound state energies for the space fractional schrodinger equation with single and double dirac delta well potentials and the delta perturbed infinite well

partially stirred reactor model analytical solutions and numerical convergence study of a pdf monte carlo method we investigate the partially stirred reactor pasr which is based on a simplified joint composition probability density function pdf transport equation analytical solutions for the first four moments of the mass density function mdf obtained from the pasr model are presented the monte carlo particle method with first order time splitting algorithm is implemented to obtain the first four moments of the mdf numerically the dynamics of the stochastic particle system is determined by inflow outflow chemical reaction and mixing events three different inflow outflow algorithms are investigated an algorithm based on the inflow outflow event modeled as a poisson process an inflow outflow algorithm mentioned in the literature and a novel algorithm derived on the basis of analytical solutions it is demonstrated that the inflow outflow algorithm used in the literature can be explained by considering a deterministic waiting time parameter of a corresponding stochastic process and also forms a specific case of the new algorithm the number of particles in the ensemble n the nondimensional time step delta t ratio of the global time step to the characteristic time of an event and the number of independent simulation trials l are the three sources of the numerical error the split analytical solutions and the numerical experiments suggest that the systematic error converges as delta t and n 1 the statistical error scales as l 1 2 and n 1 2 the significance of the numerical parameters and the inflow outflow algorithms is also studied by applying the pasr model to a practical case of premixed kerosene and air combustion

particle method for turbulent flows integration of stochastic model equations abstract a numerical method is developed to integrate the stochastic differential equations that arise in a particle method for modelling turbulent flows these equations present several challenges the foremost being the presence of multiple time scales the smallest of which can be significantly less than an acceptable time step size t the essence of the approach adopted is to transform and decompose the equations so that the stochastic components which contain the small time scales appear as strictly linear stochastic differential equations analytic solutions to these equations with frozen coefficients are then exploited to produce a stable and accurate scheme when the method is used to advance the properties of n particles the resulting numerical error can be decomposed into three contributions statistical error bias and time stepping error comprehensive tests to study these errors are reported for two test cases a novel variance reduction technique is described that significantly reduces the statistical error which scales as n 1 2 in general the bias is smaller and scales as n 1 in accord with a simple analysis the time stepping error is less than 1 for a non dimensional time step of 1 32 which may be several times larger than the smallest time scale over the range of time step size investigated the dominant time stepping error varies as t 3 2 the method has the requisite stability accuracy and efficiency for incorporation in multi dimensional particle methods

a finite volume eulerian lagrangian localized adjoint method for solution of the advection dispersion equation a new mass conservative method for solution of the one dimensional advection dispersion equation is derived and discussed test results demonstrate that the finite volume eulerian lagrangian localized adjoint method fvellam outperforms standard finite difference methods in terms of accuracy and efficiency for solute transport problems that are dominated by advection for dispersion dominated problems the performance of the method is similar to that of standard methods like previous ellam formulations fvellam systematically conserves mass globally with all types of boundary conditions fvellam differs from other ellam approaches in that integrated finite differences instead of finite elements are used to approximate the governing equation this approach in conjunction with a forward tracking scheme greatly facilitates mass conservation the mass storage integral is numerically evaluated at the current time level and quadrature points are then tracked forward in time to the next level forward tracking permits straightforward treatment of inflow boundaries thus avoiding the inherent problem in backtracking as used by most characteristic methods of characteristic lines intersecting inflow boundaries fvellam extends previous ellam results by obtaining mass conservation locally on lagrangian space time elements details of the integration tracking and boundary algorithms are presented test results are given for problems in cartesian and radial coordinates

simple accurate and efficient revisions to maccormack and saulyev schemes high peclet numbers abstract stream water quality modeling often involves numerical methods to solve the dynamic one dimensional advection dispersion reaction equations adre there are numerous explicit and implicit finite difference schemes for solving these problems and two commonly used schemes are the maccormack and saulyev schemes this paper presents simple revisions to these schemes that make them more accurate without significant loss of computation efficiency using advection dominated high peclet number problems as test cases performances of the revised schemes are compared to performances of five classic schemes forward time centered space ftcs backward time centered space btcs crank nicolson and the traditional maccormack and saulyev schemes all seven of the above numerical schemes were tested against analytical solutions for pulse and step inputs of mass to a steady flow in a channel and performances were considered with respect to stability accuracy and computational efficiency results indicated that both the modified saulyev and the maccormack schemes which are named the saulyevc and maccormackc schemes respectively greatly improved the prediction accuracy over the original ones the computation efficiency in terms of cpu time was not impacted for the saulyevc scheme the maccormackc scheme demonstrated increased time consumption but was still much faster than implicit schemes

kinetic and dynamic probability density function descriptions of disperse turbulent two phase flows this article analyzes the status of two classical one particle probability density function pdf descriptions of the dynamics of discrete particles dispersed in turbulent flows the first pdf formulation considers only the process made up by particle position and velocity z p x p u p and is represented by its pdf p t y p v p which is the solution of a kinetic pdf equation obtained through a flux closure based on the furutsu novikov theorem the second pdf formulation includes fluid variables into the particle state vector for example the fluid velocity seen by particles z p x p u p u s and consequently handles an extended pdf p t y p v p v s which is the solution of a dynamic pdf equation for high reynolds number fluid flows a typical formulation of the latter category relies on a langevin model for the trajectories of the fluid seen or conversely on a fokker planck equation for the extended pdf in the present work a new derivation of the kinetic pdf equation is worked out and new physical expressions of the dispersion tensors entering the kinetic pdf equation are obtained by starting from the extended pdf and integrating over the fluid seen this demonstrates that under the same assumption of a gaussian colored noise and irrespective of the specific stochastic model chosen for the fluid seen the kinetic pdf description is the marginal of a dynamic pdf one however a detailed analysis reveals that kinetic pdf models of particle dynamics in turbulent flows described by statistical correlations constitute incomplete stand alone pdf descriptions and moreover that present kinetic pdf equations are mathematically ill posed this is shown to be the consequence of the non markovian characteristic of the stochastic process retained to describe the system and the use of an external colored noise furthermore developments bring out that well posed pdf descriptions are essentially due to a proper choice of the variables selected to describe physical systems and guidelines are formulated to emphasize the key role played by the notion of slow and fast variables

on meromorphic solutions of some algebraic partial differential equations oncn abstract applying the techniques from nevanlinna theory of value distribution theory of meromorphic functions oncn we investigate the existence problem of some meromorphic solutions and obtain a satisfactory malmquist type theorem for a class of algebraic partial differential equations oncnwhich improves some earlier main results we give several examples to complement our results

a hybrid algorithm for the joint pdf equation of turbulent reactive flows in this paper a new particle finite volume hybrid algorithm for the joint velocityfrequency composition pdf method for turbulent reactive flows is presented this method is a combination of a finite volume scheme and a particle method the finitevolume scheme is used to solve the reynolds averaged navier stokes equations and the particle method to solve the joint pdf transport equation the motivation is to reduce the bias and the statistical error and to have an algorithm which is more efficient than stand alone particle mesh methods therefore in the particle method we use the smoother mean densityh i and favre averaged velocity u fields computed

the corrected operator splitting approach applied to a nonlinear advection diffusion problem so called corrected operator splitting methods are applied to a 1 d scalar advection diffusion equation of buckley leverett type with general initial data front tracking and a 2nd order godunov method are used to advance the solution in time diffusion is modelled by piece wise linear finite elements at each new time level to obtain correct structure of shock fronts independently of the size of the time step a dynamically defined residual flux term is grouped with diffusion different test problems are considered and the methods are compared with respect to accuracy and runtime finally we extend the corrected operator splitting to 2 d equations by means of dimensional splitting and we apply it to a buckley leverett type problem including gravitational effects

level set reinitialization at a contact line when a level set signed distance function is reinitialized in the vicinity of a contact line there is a blind spot that prevents an accurate reconstruction of a signed distance function the numerical method can create parasitic velocity currents near this region if additional contact line physics are included the parasitic velocity currents would pollute the solution and alter the physical behavior in this study a modified reinitialization routine is proposed that combines the standard hamilton jacobi equation with a relaxation equation for those grid cells along a wall in the blind spot two test cases an angled fluid wedge zero curvature and a circular fluid arc constant curvature are used to evaluate the numerical error induced by different methods the proposed method has less numerically induced interface distortion than other techniques examined furthermore this routine can be easily extended to three dimensions drops sliding on a wall are simulated in both two and three dimensions to demonstrate the advantages of this method a spreading fluid interface further shows that this method allows contact lines to merge naturally

on the computation of ensemble averages for spatially non uniform particle systems when there is no clear separation between micro and macro scales ergodicity cannot be invoked to transform ensemble into volume averages in such cases it is necessary to use ensemble averaging directly a straightforward calculation of such averages converges slowly and therefore requires a large number of realizations of the system this paper describes a much more efficient method based on the use of a fourier expansion of the quantity to be averaged the advantages of the fourier approach are estimated in general terms and demonstrated explicitly with several examples for the specific problem of equal spheres in a viscous fluid the analytical estimates suggest that similar results can be expected for other situations as well it is shown both analytically and numerically that the variance of the fourier coefficients is in many cases significantly smaller than that of the direct method which leads to a much faster convergence of the former the paper also describes a method by which the probability distribution of a uniform ensemble can be biased so as to mimic that of a non uniform one with prescribed properties

the fractional schrodinger equation for delta potentials the fractional schrodinger equation is solved for the delta potential and the double delta potential for all energies the solutions are given in terms of fox s h function

evaluation of a bounded high resolution scheme for combustor flow computations the present paper is concerned with the application of a high resolution scheme for a finite volume discretization method the capability of the chosen scheme is evaluated for the numerical simulation of three dimensional flowfields special emphasis lies in the calculation of the processes in the mixing zone of gas turbine combustors thus numerical results are compared with detailed measurements of velocity and temperature in a model mixing zone the velocity field as well as the temperature field strongly depend on the scheme used for the discretization of the momentum and energy equations the proposed high resolution scheme and the well known quick scheme yield similar results which are superior to those obtained with the simple upwind scheme for the simulation of turbulent transport the standard fc model is employed the effect of numerical diffusion caused by the usual upwind discretization of the convective transport of a and e is assessed by the application of the high resolution scheme of the discretization of the a and e equations as a result it is found that the velocity and temperature fields are insensitive to the scheme that is applied for the discretization of the a and e equations

an optimal order estimate for eulerian lagrangian localized adjoint methods for variable coefficient advection reaction problems previously we developed eulerian lagrangian localized adjoint methods ellams to solve advection reaction problems ellams provide a systematic approach to treat boundary conditions and to conserve mass and have proven to be powerful methods for advection dominated problems in this paper the authors conduct theoretical analysis for these ellams and prove that they have an optimal order convergence rate moreover the estimates involve only the derivatives of the exact solution in space and along the characteristics in contrast many existing methods have only suboptimal order error estimates for advection reaction problems and the estimates involve the temporal derivatives of the exact solution which are usually much larger than the derivatives along the characteristics key words advection reaction equations eulerian lagrangian method convergence analysis optimal order error estimates

bound states for multiple dirac δ wells in space fractional quantum mechanics using the momentum space approach we obtain bound states for multiple dirac wells in the framework of space fractional quantum mechanics introducing first an attractive dirac comb potential i e dirac comb with strength g g 0 in the space fractional schrodinger equation we show that the problem of obtaining eigenenergies of a system with n dirac wells can be reduced to a problem of obtaining the eigenvalues of an n n matrix as an illustration we use the present matrix formulation to derive expressions satisfied by the bound state energies of n 1 2 3 delta wells we also obtain the corresponding wave functions and express them in terms of fox s h function

a general mass consistency algorithm for hybrid particle finite volume pdf methods an algorithm is devised to maintain a correct spatial distribution of computational particles in hybrid particle finite volume fv probability density function pdf methods for chemically reacting turbulent flows the approach is by design compatible with existing fv computational fluid dynamics cfd codes that are used to model practical engineering flows in complex geometric configurations the algorithm is suitable for general three dimensional incompressible or compressible steady or time dependent flows using structured or unstructured stationary or deforming computational meshes it is compatible with a variety of element shapes commonly used in research and engineering cfd codes including hexahedra prisms and tetrahedra robustness accuracy and efficiency of the approach are demonstrated via computations for several two and three dimensional steady and unsteady flow configurations using computational meshes that vary in element type and in mesh quality both composition pdf and velocity pdf methods are employed this work broadens the accessibility of pdf methods for practical turbulent combustion systems

assessment of finite difference approximations for the advection terms in the simulation of practical flow problems abstract an assessment of seven discretization schemes for the advection terms of the transport equation to reduce numerical diffusion in practical flow problems has been established the schemes have been evaluated using three test cases for laminar flow problems the test cases consist of the transport of a scalar step in a uniform velocity field two interacting parallel streams and a slot jet the performance of the schemes is evaluated for advection dominated flows in transient and steady state solutions the considered schemes include four that have not been evaluated before for practical flow problems in general schemes which produced less numerical diffusion suffered from more numerical dispersion or oscillations two bounding techniques considered in the study were effective in significantly elliminating numerical dispersion

operator spltting methods for systems of convection diffusion equations nonlinear error mechanisms and correction strategies abstract many numerical methods for systems of convection diffusion equations are based on an operator splitting formulation where convective and diffusive forces are accounted for in separate substeps we describe the nonlinear mechanism of the splitting error in such numerical methods in the one dimensional case a mechanism that is intimately linked to the local linearizations introduced implicitly in the hyperbolic convection steps by the use of an entropy condition for convection dominated flows we demonstrate that operator splitting methods typically generate a numerical widening of viscous fronts unless the splitting step is of the same magnitude as the diffusion scale to compensate for the potentially damaging splitting error we propose a corrected operator splitting cos method for general systems of convection diffusion equations with the ability of correctly resolving the nonlinear balance between the convective and diffusive forces in particular cos produces viscous shocks with a correct structure also when the splitting step is large a front tracking method for systems of conservation laws which in turn relies heavily on a riemann solver constitutes an important part of our cos strategy the proposed cos method is successfully applied to a system modeling two phase multicomponent flow in porous media and a triangular system modeling three phase flow

on fractional powers of singular perturbations of the laplacian abstract we qualify a relevant range of fractional powers of the so called hamiltonian of point interaction in three dimensions namely the singular perturbation of the negative laplacian with a contact interaction supported at the origin in particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts we also qualify the norms of the resulting singular fractional sobolev spaces and their mutual control with the corresponding classical sobolev norms

a multiblock joint pdf finite volume hybrid algorithm for the computation of turbulent flows in complex geometries a new hybrid joint probability density function jpdf solution algorithm for turbulent flows in complex 3d geometries is presented the main focus is to demonstrate the applicability of jpdf methods for complex flows as observed in industrial applications all elements of the algorithm are explained in detail and extensive validation studies are presented a multiblock finite volume solver capable of handling globally unstructured locally structured grids was implemented together with a lagrangian particle method efficient and robust particle management and accurate schemes for estimation and interpolation of particle statistics have been developed for numerical efficiency particle sub time stepping and an implicit finite volume solver are applied a fast coupling strategy was developed together with a multigrid method that allows very fast convergence on refined grids comparison with an established jpdf code for a bluff body stabilized flow shows very good agreement furthermore robustness and consistency of the algorithm for turbulent flow simulations with complex geometry is demonstrated

a fractional step ellam approach to high dimensional convection diffusion problems with forward particle tracking in this paper a fractional step method combined with an eulerian lagrangian localized adjoint method ellam is proposed to solve high dimensional convection diffusion problems the method reduces high dimensional problems to a series of uncoupled one dimensional problems in each time step interval in which one dimensional ellam is used to solve the one dimensional splitting equations the approach takes the attractive advantages of the ellam method and the fractional step technique it reduces computational complexities large memory requirements and long computation durations due to the application of the splitting technique it reduces temporal errors and generates accurate numerical solutions even if large time and coarse spatial step sizes are used in computation it effectively eliminates non physical oscillation or excessive numerical dispersion and treats boundary conditions well and in a natural way numerical experiments show the efficient performance of the approach

some issues in discrete approximate solution for stochastic differential equations abstract an evaluation method for numerical schemes of stochastic differential equations is treated discussing the source of errors in the discrete numerical solution we highlight the effect of pseudo random numbers upon the numerical solution and point out the significance of the independencies of the series of them required in the numerical schemes to discriminate the stochastic and deterministic parts in the errors more clearly we propose a new two dimensional linear test equation of multiplicative type whose analytical solution can be obtained readily our results are illustrated through some numerical examples

fractional powers and singular perturbations of quantum differential hamiltonians we consider the fractional powers of singular point like perturbations of the laplacian and the singular perturbations of fractional powers of the laplacian and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains in application to the linear and non linear schrodinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated we consider the fractional powers of singular point like perturbations of the laplacian and the singular perturbations of fractional powers of the laplacian and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains in application to the linear and non linear schrodinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated

assessment of numerical accuracy of pdf monte carlo methods for turbulent reacting flows this study is to explore the numerical features of a particle mesh algorithm developed for a stand alone joint velocity frequency composition pdf method for turbulent reactive flows numerical experiments are performed on a piloted jet nonpremixed turbulent flame of methane to characterize and quantify various numerical errors in terms of numerical parameters number of particles per cellnpc number of cells m2 and time step t first a stationary solution is obtained and is verified to be independent of the time step t then the total numerical error is identified as statistical error bias and discretization error it is revealed that the statistical error converges as npc 1 2 and the bias as npc 1 the statistical error can be reduced by time averaging or by performing multiple independent simulation e g with a parallelized program finally the scheme is shown to be second order accurate the spatial discretization error converging as m 2 a modified turbulence frequency model based on the turbulence production to dissipation ratio is shown to improve the numerical behavior of the turbulence model these results demonstrate that the particle mesh method is convergent also the optimal numerical parameters minimizing computational cost subject to a specified error tolerance are estimated an error reduction scheme similar to richardson extrapolation is proposed and shown to be quite effective in reducing the deterministic error

numerical simulation of filamentary discharges with parallel adaptive mesh refinement direct simulation of filamentary gas discharges like streamers or dielectric barrier micro discharges requires the use of an adaptive mesh the objective of this paper is to develop a strategy which can use a set of grids with suitable local refinements for the continuity equations and poisson s equation in 2d and 3d geometries with a high order discretization the advantages of this approach are presented with a filamentary discharge simulation in plane plane geometry in nitrogen within the diffusion drift approximation

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

eulerian lagrangian localized adjoint methods for a nonlinear advection diffusion equation eulerian lagrangian localized adjoint methods ellam are developed for the non linear buckley leverett equation which is characterized by degenerate diffusion and sharpening near shock solutions the ellam methodology employs space time finite elements with edges oriented along flow paths and space time test functions that satisfy a local adjoint condition this combination extends eulerian lagrangian concepts in a systematic mass conservative fashion to problems with general boundary conditions various kinds of boundary conditions are considered and a local time stepping procedure is developed for a no flow outlet condition that leads to a boundary layer numerical experiments illustrate the potential of these methods

numerical integration of stochastic differential equations weak second order mid point scheme for application in the composition pdf method in this paper a weak second order accurate mid point scheme for the stochastic differential equations sdes arising in the composition pdf method for turbulent reactive flows is proposed and tested the results are compared with two other schemes which are commonly used in the composition pdf method in contrast to most higher order schemes for sdes the present scheme uses a mid point which makes it especially suitable for the implementation of the position advance fractional step in the composition pdf method the scheme can also be applied to the pdf method used in conjunction with large eddy simulation les since the sdes considered in this paper include explicit time dependence of the drift and diffusion coefficients test calculations are reported including a 2d unsteady case to demonstrate the weak second order accuracy of the scheme and to compare its performance to that of two first order schemes a new methodology for developing higher order scheme for sdes by comparing the moments of the increments of the numerical approximation with those of the exact increments is used to develop this scheme

stability monotonicity maximum and minimum principles and implementation of the volume corrected characteristic method we consider the volume corrected characteristics mixed method vccmm for tracer transport problems the volume correction adjustment maintains the local volume conservation of bulk fluids and the numerical convergence of the method we discuss some details of implementation by considering the scheme from an algebraic point of view we show that the volume correction adjustment is important for stability and necessary for the monotonicity and the maximum and minimum principles of the method we also derive a relatively weaker stability property for the uncorrected characteristic mixed method cmm some numerical experiments of a quarter five spot pattern of wells are given to verify our theoretical results and compare the concentration errors of vccmm and cmm due to random perturbations set up in the computation of the algorithm more numerical tests including one related to long time nuclear waste storage are given to compare vccmm with cmm and godunov s method showing that vccmm exhibits no overshoots or undershoots and less numerical diffusion

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

regular article a consistent hybrid finite volume particle method for the pdf equations of turbulent reactive flows the paper describes a new hybrid finite volume fv particle method developed for the solution of the pdf equations for statistically stationary turbulent reactive flows in this approach the conservation equations for mean mass momentum and energy conservation are solved by a fv method while a particle algorithm is employed to solve the fluctuating velocity turbulence frequency compositions joint pdf transport equation the mean velocity and pressure are supplied to the particle code by the fv code which in turn obtains all the reynolds stresses the scalar fluxes and the reaction terms needed in the fv code an important feature of the method is the complete consistency between the set of equations solved by the fv and particle methods the algorithmic and numerical issues arising in the development of the hybrid method are studied in the simple setting of the stochastic ideal flow equations numerical results are obtained for 1d reactive stochastic ideal flow to demonstrate numerical properties of the method the total numerical error is identified as statistical error bias spatial truncation error and temporal truncation error in contrast to the self contained particle method the bias is found to be negligibly small it is shown that all the numerical errors converge at the expected rates finally the global convergence of the hybrid method is demonstrated and the optimal strategy for time averaging that gives the best global convergence rate is investigated