an adaptive grid finite difference method for conservation laws abstract adaptive grid finite difference methods for computing time accurate solutions of nonlinear hyperbolic conservation laws in one space dimension are studied the basic approach is to decouple the determination of the moving grid from the solution of the differential equation on the moving grid the grid is determined by an elliptic grid generation technique suitably modified for time accurate computations two methods godunov s scheme and an artificial viscosity method are used to discretize the differential equation the technique is applied to problems involving the buckley leverett equation modelling the flow of two immiscible fluids in a porous medium substantial improvement in computational efficiency is obtained by using the adaptive grid technique extensions of this approach to several space dimensions and systems of equations are discussed

mesh independent spectra in the moving finite element equations we derive the moving finite element mfe equations for the solution of a scalar evolutionary equation in d space dimensions d geqq 1 and introduce the elementwise approach to mfe this approa

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

a segmentation approach to grid generation using biharmonics abstract we present a method for the numerical generation of finite difference grids in this method a complicated shape is divided into simpler parts then each part is transformed to a rectangle and is individually gridded it is demonstrated that requiring the transformation function defining each subgrid to satisfy a linear fourth order pde system provides the flexibility necessary to smoothly patch together the subgrids the ability to prescribe mesh point locations on subgrid boundaries gives a simple method for controlling the locations of mesh points which are interior with respect to the composite grid

algebraic dynamic multilevel adm method for fully implicit simulations of multiphase flow in porous media this paper presents the development of an algebraic dynamic multilevel method adm for fully implicit simulations of multiphase flow in homogeneous and heterogeneous porous media built on the fine scale fully implicit fim discrete system adm constructs a multilevel fim system describing the coupled process on a dynamically defined grid of hierarchical nested topology the multilevel adaptive resolution is determined at each time step on the basis of an error criterion once the grid resolution is established adm employs sequences of restriction and prolongation operators in order to map the fim system across the considered resolutions several choices can be considered for prolongation interpolation operators e g constant bilinear and multiscale basis functions all of which form partition of unity the adaptive multilevel restriction operators on the other hand are constructed using a finite volume scheme this ensures mass conservation of the adm solutions and as such the stability and accuracy of the simulations with multiphase transport for several homogeneous and heterogeneous test cases it is shown that adm applies only a small fraction of the full fim fine scale grid cells in order to provide accurate solutions the sensitivity of the solutions with respect to the employed fraction of grid cells determined automatically based on the threshold value of the error criterion is investigated for all test cases adm is a significant step forward in the application of dynamic local grid refinement methods in the sense that it is algebraic allows for systematic mapping across different scales and applicable to heterogeneous test cases without any upscaling of fine scale high resolution quantities it also develops a novel multilevel multiscale method for fim multiphase flow simulations in natural subsurface formations

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

an adaptive grid solution procedure for convection diffusion problems abstract a computationally efficient and stable adaptive grid solution procedure is developed for convection diffusion problems in this method grid refinement and adaptation is based on an equidistribution law but is only performed in regions with high error estimates that are flagged from a preliminary coarse grid solution the equidistribution law is implicit in the grid generation procedure which requires the solution of two poisson equations with control functions that are obtained directly from the error estimates or weighting functions at the grid points solution on the refined equidistributed mesh in the flagged region is obtained with boundary conditions interpolated from the coarse grid results accurate solutions in both the flagged region and the coarse grid regions of the domain are obtained with a multigrid approach that requires successive solutions on the refined equidistributed mesh in the flagged region and on the coarse mesh in the entire domain the adaptive grid method including the multigrid calculations can be extended to several levels of refinement the acronym lame is coined for this procedure in view of its local adaptation multigridding and equidistribution features the method is shown to be stable computationally efficient and accurate by applying it to three test problems and comparing with conventional calculations on a fixed curvilinear grid

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

a two dimensional mesh moving technique for time dependent partial differential equations abstract we discuss an adaptive mesh moving technique that can be used with a finite difference or finite element scheme to solve initial boundary value problems for systems of partial differential equations in two space dimensions and time the mesh moving technique is based on an algebraic node movement function determined from the geometry and propagation of regions having significant discretization error indicators our procedure is designed to be flexible so that it can be used with many existing finite difference and finite element methods to test the mesh moving algorithm we implemented it in a system code with an initial mesh generator and a maccormack finite difference scheme on quadrilateral cells for hyperbolic vector systems of conservation laws results are presented for several computational examples the moving mesch scheme reduces dispersive errors near shocks and wave fronts and thereby reduces grid requirements necessary to compute accurate solutions while increasing computational efficiency

modeling persistence in hydrological time series using fractional differencing the class of autoregressive integrated moving average arima time series models may be generalized by permitting the degree of differencing d to take fractional values models including fractional differencing are capable of representing persistent series d 0 or short memory series d 0 the class of fractionally differenced arima processes provides a more flexible way than has hitherto been available of simultaneously modeling the long term and short term behavior of a time series in this paper some fundamental properties of fractionally differenced arima processes are presented methods of simulating these processes are described estimation of the parameters of fractionally differenced arima models is discussed and an approximate maximum likelihood method is proposed the methodology is illustrated by fitting fractionally differenced models to time series of streamflows and annual temperatures

fractal estimation from noisy data via discrete fractional gaussian noise dfgn and the haar basis the authors show that the application of the discrete wavelet transform dwt using the haar basis to the increments of fractional brownian motion fbm also known as discrete fractional gaussian noise dfgn yields coefficients which are weakly correlated and have a variance that is exponentially related to scale similar results were derived by flandrin 1989 tewfik and kim for a continuous time fbm going through a continuous wavelet transform cwt the new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by wornell and oppenheim the performance of the new algorithm is compared with that of wornell and oppenheim s see ieee trans signal processing vol 40 p 611 623 mar 1992 algorithm in numerical simulation

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

fractional brownian motion a maximum likelihood estimator and its application to image texture fractals have been shown to be useful in characterizing texture in a variety of contexts use of this methodology normally involves measurement of a parameter h which is directly related to fractal dimension in this work the basic theory of fractional brownian motion is extended to the discrete case it is shown that the power spectral density of such a discrete process is only approximately proportional to f a instead of in direct proportion as in the continuous case an asymptotic cramer rao bound is derived for the variance of an estimate of h subsequently a maximum likelihood estimator mle is developed to estimate h it is shown that the variance of this estimator nearly achieves the minimum bound a generation algorithm for discrete fractional motion is presented and used to demonstrate the capabilities of the mle when the discrete fractional brownian process is contaminated with additive gaussian noise the results show that even at signal to noise ratios of 30 db significant errors in estimation of h can result when noise is present the mle is then applied to x ray images of the human calcaneus to demonstrate how the line to line formulation can be applied to the two dimensional case these results indicate that it has strong potential for quantifying texture

determination of the hurst exponent by use of wavelet transforms we propose a method for global hurst exponent determination based on wavelets using this method we analyze synthetic data with predefined hurst exponents fracture surfaces and data from economy the results are compared to those obtained with fourier spectral analysis when many samples are available the wavelet and fourier methods are comparable in accuracy however when one or only a few samples are available the wavelet method outperforms the fourier method by a large margin

grid evolution in time asymptotic problems the selection of the proper coordinate system in solving any fluid flow or heat transfer problem is a very important consideration a new technique of moving mesh points in physical space is introduced so as to reduce the error in a computed asymptotic solution relative to that obtained using a fixed mesh the technique has been used to solve the simple viscous burgers equation in one and two dimensions substantial error reductions are demonstrated the treatment of boundary points and the effect of using different error criteria in generating grids are discussed

on the stability of mesh equidistribution strategies for time dependent partial differential equations abstract we study the stability of several mesh equidistribution schemes for time dependent partial differential equations in one space dimension the schemes move a finite difference or finite element mesh so that a given quantity is uniform over the domain we consider mesh moving methods that are based on solving a system of ordinary differential equations for the mesh velocities and show that some of these methods are unstable with respect to an equidistributing mesh when the partial differential system is dissipative using linear perturbation techniques we are able to develop simple criteria for determining the stability of a particular method and show how to construct stable differential systems for the mesh velocities several examples illustrating stable and unstable mesh motions are presented

statistical analysis of dwt coefficients of fgn processes using arfima p d q models abstract fractional gaussian noise fgn provides an important parametric representation for the data recorded from long memory processes also it has been well established in literature that the orthogonal wavelet transforms prove to be the optimal bases to represent the data as fgn or fbm fractional brownian motion models this paper highlights the statistical properties of discrete wavelet transform dwt coefficients in the wavelet expansion of fgn statistical analysis was carried out by analyzing the inter scale and intra scale correlations of the dwt coefficients for wavelets with varying vanishing moments two types of auto regressive moving average arma models were fit to the wavelet coefficients of fgn namely i arma p q and ii arfima p d q models the latter represents the arma models with fractional differencing using the akaike information criteria aic and the bayesian information criteria bic it has been shown that arfima models best represent the wavelet coefficients of fgn the above observation holds good when wavelets with increasing number of vanishing moments are used for obtaining dwt coefficients after estimating the optimal model and its parameters different properties pertaining to the inter scale and intra scale correlations were verified using these models

adaptive mesh refinement processes for finite element solutions the nature of adaptive processes is reviewed using as an example a specific finite element problem several possible formulations for the objective of mesh refinement processes are given for the most natural of these objectives the a algorithms of artificial intelligence turn out to provide a solution process which is known to be optimal in a certain sense but practically there is insufficient information and these processes tend to be inefficient on the other hand if the objective is replaced by a strictly local objective based on pareto optimality the error estimates of the resulting mesh sequence are shown to exhibit the order of convergence which is known to be best possible for the type of elements used

maximum likelihood estimation of stationary univariate fractionally integrated time series models abstract to estimate the parameters of a stationary univariate fractionally integrated time series the unconditional exact likelihood function is derived this allows the simultaneous estimation of all the parameters of the model by exact maximum likelihood issues involved in obtaining maximum likelihood estimates are discussed particular attention is given to efficient procedures to evaluate the likelihood function obtaining starting values and the small sample properties of the estimators limitations of previous estimation procedures are also discussed

characterization of lee yang polynomials the lee yang circle theorem describes complex polynomials of degree n in z with all their zeros on the unit circle jzj d 1 these polynomials are obtained by taking z1 d d zn d z in certain multiaffine polynomials z1 zn which we call lee yang polynomials they do not vanish when jz1j jznj 1 we characterize the lee yang polynomials in nc1 variables in terms of polynomials in n variables those such that z1 zn 0 when jz1j jznj 1 including jzi j d 1 in a sense to be made precise later our current understanding of lee yang polynomials is based on the concept of asano contraction 1 we shall define an inner radius associated with a multiaffine polynomial and see that it behaves supermultiplicatively with respect to asano contraction proposition 2 using the properties of the inner radius we shall characterize the 2 lync1 nc 1 variables in terms of polynomials in n variables such that z1 zn 0 when jz1j jznj 1 theorem 3 this characterization will give us a good understanding of lync1 proposition 5 and allow us to exhibit elements of lync1 outside of the pair interaction class originally considered by lee and yang see in particular example 7 d the original

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

algebraic dynamic multilevel method for compositional flow in heterogeneous porous media abstract this paper presents the algebraic dynamic multilevel method adm for compositional flow in three dimensional heterogeneous porous media in presence of capillary and gravitational effects as a significant advancement compared to the adm for immiscible flows cusini et al 2016 33 here mass conservation equations are solved along with k value based thermodynamic equilibrium equations using a fully implicit fim coupling strategy two different fine scale compositional formulations are considered 1 the natural variables and 2 the overall compositions formulation at each newton s iteration the fine scale fim jacobian system is mapped to a dynamically defined in space and time multilevel nested grid the appropriate grid resolution is chosen based on the contrast of user defined fluid properties and on the presence of specific features e g well source terms consistent mapping between different resolutions is performed by the means of sequences of restriction and prolongation operators while finite volume restriction operators are employed to ensure mass conservation at all resolutions various prolongation operators are considered in particular different interpolation strategies can be used for the different primary variables and multiscale basis functions are chosen as pressure interpolators so that fine scale heterogeneities are accurately accounted for across different resolutions several numerical experiments are conducted to analyse the accuracy efficiency and robustness of the method for both 2d and 3d domains results show that adm provides accurate solutions by employing only a fraction of the number of grid cells employed in fine scale simulations as such it presents a promising approach for large scale simulations of multiphase flow in heterogeneous reservoirs with complex non linear fluid physics