trust region based solver for nonlinear transport in heterogeneous porous media we describe a new nonlinear solver for immiscible two phase transport in porous media where viscous buoyancy and capillary forces are significant the flux fractional flow function f is a nonlinear function of saturation and typically has inflection points and can be non monotonic the non convexity and non monotonicity of f are major sources of difficulty for nonlinear solvers of coupled multiphase flow and transport in natural porous media we describe a modified newton algorithm that employs trust regions of the flux function to guide the newton iterations the flux function is divided into saturation trust regions delineated by the inflection unit flux and end points the updates are performed such that two successive iterations cannot cross any trust region boundary if a crossing is detected the saturation value is chopped back to the appropriate trust region boundary the proposed trust region newton solver which is demonstrated across the parameter space of viscous buoyancy and capillary effects is a significant extension of the inflection point strategy of jenny et al jcp 2009 5 for viscous dominated flows we analyze the discrete nonlinear transport equation obtained using finite volume discretization with phase based upstream weighting then we prove convergence of the trust region newton method irrespective of the timestep size for single cell problems numerical results across the full range of the parameter space of viscous gravity and capillary forces indicate that our trust region scheme is unconditionally convergent for 1d transport that is for a given choice of timestep size the unique discrete solution is found independently of the initial guess for problems dominated by buoyancy and capillarity the trust region newton solver overcomes the often severe limits on timestep size associated with existing methods to validate the effectiveness of the new nonlinear solver for large reservoir models with strong heterogeneity we compare it with state of the art nonlinear solvers for two phase flow and transport using the spe 10 model compared with existing nonlinear solvers our trust region solver results in superior convergence performance and achieves reduction in the total newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost

a finite volume method for a two phase multicomponent polymer flooding multicomponent polymer flooding used in enhanced oil recovery is governed by a system of coupled non strictly hyperbolic conservation laws in the presence of gravity the flux functions need not be monotone and hence designing godunov type upwind schemes is difficult and computationally expensive to overcome this difficulty we use the basic idea of discontinuous flux to reduce the coupled system into an uncoupled system of scalar conservation laws with discontinuous coefficients for these scalar equations we use the dflu flux developed in 5 to construct a second order scheme the scheme is shown to satisfy a maximum principle and the performance of the scheme is shown on both one and two dimensional test problems

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

implicit hybrid upwinding for two phase flow in heterogeneous porous media with buoyancy and capillarity abstract we consider the numerical solution of the partial differential equations governing multiphase flow in porous media for highly nonlinear problems the temporal discretization of choice is often the unconditionally stable fully implicit method however the nonlinear systems often solved with newton s method are difficult to solve thus the computational cost is strongly dependent on the nonlinear convergence rate and enhancing this convergence property is key to speed up subsurface flow simulation we focus on the case of spatially discontinuous capillary pressure between rock regions to efficiently and accurately simulate the flow dynamics in heterogeneous porous media the flux computation combines implicit hybrid upwinding with transmission conditions between different rock regions this leads to a scheme that correctly represents the trapping mechanisms while improving the nonlinear convergence we extend our previous results hamon et al 2016 18 by generalizing the scheme to fully implicit coupled flow and transport to address realistic problems in multiple dimensions the generalized scheme is supported by an analysis of its mathematical properties our multidimensional numerical examples which range from buoyancy driven flow with capillary barriers to viscous dominated flow demonstrate that the implicit hybrid upwinding scheme improves the accuracy compared to the standard phase based upwinding scheme while leading to significant reductions in the number of nonlinear iterations in multiple dimensions

monotonicity considerations for saturated unsaturated subsurface flow it is demonstrated that monotonicity is a sufficient condition to ensure that no new nonphysical local maxima and minima can be produced in the discrete nonlinear unsaturated flow equation monotonicity conditions are derived for various types of weighting for the mobility term the basic discretization is of finite element type but the results can be extended to finite volume discretizations central weightings are only conditionally monotone while upstream weightings are unconditionally monotone sample computations are given for highly heterogeneous problems

a donoho stark criterion for stable signal recovery in discrete wavelet subspaces abstract we derive a sufficient condition by means of which one can recover a scale limited signal from the knowledge of a truncated version of it in a stable manner following the canvas introduced by donoho and stark 1989 4 the proof follows from simple computations involving the zak transform well known in solid state physics geometric harmonics in the terminology of coifman and lafon 2006 22 for scale limited subspaces of l 2 r are also displayed for several test cases finally some algorithms are studied for the treatment of zero angle problems

numerical analysis of a finite volume scheme for two incompressible phase flow with dynamic capillary pressure abstract in this paper we propose and analyze the convergence of a tpfa two points flux approximation finite volume scheme to approximate the two incompressible phase flow with dynamic capillary pressure in porous media the fully implicit scheme is based on nonstandard approximation on mobilities and capillary pressure on the dual mesh we derive a discrete variational formulation and we present a new result of convergence in a two or three dimensional porous medium in comparison with static capillary pressure the non equilibrium capillary model requires more powerful techniques especially the discrete energy estimates are not standard

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

convergence of implicit monotone schemes with applications in multiphase flow in porous media phase based upstreaming which is a commonly used spatial discretization for multiphase flow in reservoir simulation naturally gives rise to implicit monotone schemes when implicit time stepping is used the nonlinear gauss seidel and jacobi algorithms are shown to converge to a unique bounded solution when applied to the resulting system of equations thus for one dimensional problems we obtain an alternate constructive proof that such schemes are well defined and converge to the entropy solution of the conservation law for arbitrary cfl numbers the accuracy of phase based upstream solutions is studied for various spatial and temporal grids revealing the importance of unconditional stability when nonuniform grids and or variable porosity is involved

on nonlocal monotone difference schemes for scalar conservation laws analyse d erreur d une classe de methodes aux differences finies monotones pour la solution approchee des lois de conservation hyperboliques

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

an efficient method for an ill posed problem band limited extrapolation by regularization in this paper a regularized spectral estimation formula and a regularized iterative algorithm for band limited extrapolation are presented the ill posedness is taken into account first the fredholm equation is regularized then it is transformed to a differential equation in the case where the time interval is r a fast algorithm to solve the differential equation is given by the finite difference and a regularized spectral estimation formula is obtained then a regularized iterative extrapolation algorithm is introduced and compared with the papoulis and gerchberg algorithm

potential based reduced newton algorithm for nonlinear multiphase flow in porous media we present a phase based potential ordering that is an extension of the cascade ordering introduced by appleyard and cheshire john r appleyard ian m cheshire the cascade method for accelerated convergence in implicit simulators in european petroleum conference 1982 pp 113 122 the proposed ordering is valid for both two phase and three phase flow and it can handle countercurrent flow due to gravity and or capillarity we show how this ordering can be used to reduce the nonlinear algebraic system that arises from the fully implicit method fim into one with only pressure dependence the potential based reduced newton algorithm is then obtained by applying newton s method to this reduced order system numerical evidence shows that our potential based reduced newton solver is able to converge for time steps that are much larger than what the standard newton s method can handle in addition whenever standard newton converges so does the reduced newton algorithm and the number of global nonlinear iterations required for convergence is significantly reduced compared with the standard newton s method

discrete approximations of bv solutions to doubly nonlinear degenerate parabolic equations in this paper we present and analyse certain discrete approximations of solutions to scalar doubly nonlinear degenerate parabolic problems of the form

a combined finite volume nonconforming finite element scheme for compressible two phase flow in porous media we propose and analyze a combined finite volume nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media the diffusion term which can be anisotropic and heterogeneous is discretized by piecewise linear nonconforming triangular finite elements the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh the relative permeability of each phase is decentred according the sign of the velocity at the dual interface this technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities next a priori estimates on the pressures and a function of the saturation that denote capillary terms are established these stabilities results lead to some compactness arguments based on the use of the kolmogorov compactness theorem and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations provided the mesh size tends to zero the proof is given for the complete system when the density of the each phase depends on its own pressure

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

simulation of incompressible two phase flow in porous media with large timesteps abstract multiphase flow in porous media occurs in several disciplines including petroleum reservoir engineering petroleum systems analysis and co 2 sequestration while simulations often use a fully implicit discretization to increase the time step size restrictions on the time step often exist due to non convergence of the nonlinear solver e g newton s method here this problem is addressed for the buckley leverett equations which model incompressible immiscible two phase flow with no capillary potential the equations are recast as a gradient flow using the phase field method and a convex energy splitting scheme is applied to enable large timesteps even for high degrees of heterogeneity in permeability and viscosity by using the phase field formulation as a homotopy map the underlying hyperbolic flow equations can be solved with large timesteps for a heterogeneous test problem the new homotopy method allows the timestep to be increased by more than six orders of magnitude relative to the unmodified equations while maintaining convergence

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

on the extrapolation of band limited functions with energy constraints a modification of the algorithm proposed by papoulis and gerchberg for extrapolating band limited functions is suggested to extend its applicability over data corrupted by noise we assume that energy constraints are known either for the band limited signal or for the noise in addition the discrete formulation of the iterative algorithm is derived and the transition from the continuous algorithm to its digital implementation is presented

unconditionally convergent nonlinear solver for hyperbolic conservation laws with s shaped flux functions this paper addresses the convergence properties of implicit numerical solution algorithms for nonlinear hyperbolic transport problems it is shown that the newton raphson nr method converges for any time step size if the flux function is convex concave or linear which is in general the case for cfd problems in some problems e g multiphase flow in porous media the nonlinear flux function is s shaped not uniformly convex or concave as a result a standard nr iteration can diverge for large time steps even if an implicit discretization scheme is used to solve the nonlinear system of equations in practice when such convergence difficulties are encountered the current time step is cut previous iterations are discarded a smaller time step size is tried and the nr process is repeated the criteria for time step cutting and selection are usually based on heuristics that limit the allowable change in the solution over a time step and or nr iteration here we propose a simple modification to the nr iteration scheme for conservation laws with s shaped flux functions that converges for any time step size the new scheme allows one to choose the time step size based on accuracy consideration only without worrying about the convergence behavior of the nonlinear solver the proposed method can be implemented in an existing simulator e g for co 2 sequestration or reservoir flow modeling quite easily the numerical analysis is confirmed with simulation studies using various test cases of nonlinear multiphase transport in porous media the analysis and numerical experiments demonstrate that the modified scheme allows for the use of arbitrarily large time steps for this class of problems

a priori error estimates for numerical methods for scalar conservation laws part iii multidimensional flux splitting monotone schemes on non cartesian grids this paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we consider multidimensional flux splitting monotone schemes defined on non cartesian grids we identify those schemes which are consistent and prove that the l infty 0 t l 1 bbb r d norm of the error goes to zero as delta x 1 2 when the discretization parameter delta x goes to zero moreover we show that nonconsistent schemes can converge at optimal rates of delta x 1 2 because i the conservation form of the schemes and ii the so called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency

limit boundary conditions for finite volume approximations of some physical problems in the industrial context finite volume schemes are used to compute an approximation of the solution of a system of equations set on a certain domain when this domain is bounded some numerical boundary conditions have to be implemented in order to complete the computation of the finite volume scheme this is a tricky step in the elaboration of the scheme which is still not mastered in fact at a closer sight it appears that there is a deep interaction between the understanding of the physical phenomena at the boundary of the domain and the implementation of the numerical boundary conditions unfortunately this link is not always completely intelligible and a reason for this lack of clarity is the fact that whereas the the continuous equation satisfied by the limit of the numerical solution is known the boundary conditions satisfied by this very limit are not well understood the purpose of this paper is to clarify this point in three industrial situations of one dimensional two phase flows

nonlinear analysis of multiphase transport in porous media in the presence of viscous buoyancy and capillary forces nonlinear convergence problems in numerical reservoir simulation can lead to unacceptably large computational time and are often the main impediment to performing simulation studies of large scale problems we analyze the nonlinearity of the discrete transport mass conservation equation for immiscible incompressible two phase flow in porous media in the presence of viscous buoyancy and capillary forces although simulation problems are multi dimensional with large numbers of cells and variables we find that the essence of the nonlinear behavior can be understood by studying the discretized numerical flux function for the interface between two cells the numerical flux is expressed in terms of the saturations of the two cells discontinuities in the first order derivative of the flux function referred to as kinks and inflection lines are identified as the cause of convergence difficulty these critical features kinks and inflections change the curvature of the numerical flux function abruptly and can lead to overshoots oscillations or divergence in newton iterations based on our understanding of the nonlinearity a nonlinear solver is developed referred to as the numerical trust region ntr solver the solver is able to guide the newton iterations safely and efficiently through the different saturation trust regions delineated by the kinks and inflections specifically overshoots and oscillations that often lead to convergence failure are avoided numerical examples demonstrate that our ntr solver has superior convergence performance compared with existing methods in particular convergence is achieved for a wide range of timestep sizes and courant friedrichs lewy cfl numbers spanning several orders of magnitude in addition a discretization scheme is proposed for handling heterogeneities in capillary pressure saturation relationship the scheme has less degree of nonlinearity compared with the standard single point phase based upstream weighting scheme leading to an improved nonlinear convergence performance especially when used together with our ntr solver our proposed numerical solution strategy that is based on the numerical flux and handles capillarity extends the previous work by jenny et al 2009 6 and wang and tchelepi 2013 7 significantly the nonlinearity of multiphase transport in porous media is analyzed the analysis considers viscous buoyancy and capillary forces the critical features that cause nonlinear convergence failures are identified a nonlinear solver that converges without oscillations and divergence is developed the solver converges faster than existing methods and allows large timesteps

implicit time dependent variable grid finite difference methods for the approximation of a linear waterflood an implicit time dependent variable grid finite difference method based on the addition of an artificial diffusivity is introduced and analyzed for approximating the solution of a scalar conservation law in a single space variable no relation between the grids at successive time steps is required for convergence two adaptive grid selection procedures are shown to be covered by the analysis analogous results are also established for an implicit upwinding procedure at ax provided that capillary effects are ignored this equation is derived from darcy s law and the conservation of mass as follows the subscripts o and w refer to oil and water respectively the symbols 4 k pi p q p and s represent porosity permeability viscosity density velocity pressure and saturation respectively with 4 being a rock property constant by the assumption of homogeneity and ko k x kro so being the product of the gross permeability k x again a constant and the relative permeability kro so of the oil phase the reservoir can be inclined it will be assumed to have height z x linear in x g is a gravitational constant then darcy s law states that

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

signal extrapolation in wavelet subspaces the papoulis gerchberg pg algorithm is well known for band limited signal extrapolation the authors consider the generalization of the pg algorithm to signals in the wavelet subspaces in this research the uniqueness of the extrapolation for continuous time signals is examined and sufficient conditions on signals and wavelet bases for the generalized pg gpg algorithm to converge are given a discrete gpg algorithm is proposed for discrete time signal extrapolation and its convergence is investigated numerical examples are given to illustrate the performance of the discrete gpg algorithm

a fast convergence algorithm for band limited extrapolation by sampling in this paper an algorithm for band limited extrapolation is presented this uses shannon s sampling theorem and fourier series error analysis is given by proof the convergence rate is much faster than that of the papoulis gerchberg algorithm the procedure of the fast convergence algorithm is described in detail the result is demonstrated by numerical examples

operator based linearization approach for modeling of multiphase multi component flow in porous media a new approach for the linearization of governing equations that describe flow and transport in porous media is proposed in this work it is based on an approximate representation of the exact physics of the problem which is similar to an approximate representation of space and time discretization performed in conventional simulation the governing equations are introduced as a combination of operators dependent on spatially altered properties and operators fully controlled by nonlinear properties of fluid and rock next a parametrization in the physics space of the problem is introduced the property based operators are approximated using direct interpolation in the space of nonlinear unknowns the discrete version of the governing equations is constructed as a combination of operators that approximate both nonlinear physics and discretization in time and space this approach is applied to the reservoir simulation of miscible and immiscible displacement processes the performance of the method demonstrates a convergence of simulation results by resolution in physical space with the improved performance

the dflu flux for systems of conservation laws the dflu numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space we show how this flux can be used to solve a certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering furthermore these results are extended to the case where the flux function is discontinuous in the space variable such a situation arises for example while dealing with oil reservoirs which are heterogeneous numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility lax friedrichs and force schemes

upstream differencing for multiphase flow in reservoir simulation upstream weighting for multiphase flow in reservoir simulation is analyzed the associated numerical flux is shown to be well defined monotone lipschitz continuous and consistent in the case of a two phase flow the corresponding numerical flux is compared to that of godunov and engquist and osher finally simple way to obtain a higher order scheme is outlined

time invariant orthonormal wavelet representations a simple construction of an orthonormal basis starting with a so called mother wavelet together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications an orthonormal basis may not however always be a suitable representation of a signal particularly when time or space invariance is a required property the conventional way around this problem is to use a redundant decomposition we address the time invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions we show that it is possible to achieve time invariance and preserve the orthonormality we subsequently propose an efficient approach to obtain such a decomposition we demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

extrapolation of bandlimited signals in linear canonical transform domain the linear canonical transform lct has been shown to be a powerful analyzing tool in signal processing many results of this transform are already known including bandlimited extrapolation the existing algorithm for solving the problem of lct bandlimited extrapolation is based on signal expansion into a series of generalized prolate spheroidal wave functions gpswfs however the requirement to compute and store the gpswfs and the errors due to the series truncation render this algorithm ill suitable for a practical implementation in this correspondence we first propose a new formulation of the gerchberg papoulis gp algorithm for lct bandlimited extrapolation then we present a fast convergence algorithm for the new formulation the classical gp algorithm related to fourier bandlimited signals is noted as a special case moreover the comparison between the proposed algorithm and the one based on gpswfs expansion is provided and the validity of the theoretical derivations is demonstrated via simulations several potential applications of the achieved theory are also presented

analysis of hybrid upwinding for fully implicit simulation of three phase flow with gravity the systems of algebraic equations arising from implicit backward euler finite volume discretization of the conservation laws governing multiphase flow in porous media are quite challenging for nonlinear solvers in the presence of countercurrent flow due to buoyancy the numerical flux obtained with single point phase potential upwinding ppu is not differentiable which causes convergence difficulties for nonlinear solvers recently lee efendiev and tchelepi adv water resour 82 2015 pp 27 38 proposed a hybrid upwinding strategy for two phase flow that gives rise to a differentiable numerical flux across the entire viscous gravity parameter space here we first present an implicit hybrid upwinding ihu scheme for hyperbolic conservation laws extending the work of lee efendiev and tchelepi to an arbitrary number of fluid phases we show that the numerical flux obtained with the ihu is consistent and a monotone function of its own saturation it is also a differentiable function of t

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a translation invariant wavelet representation algorithm with applications we address the time varying problem of wavelet transforms and a new translation invariant wavelet representation algorithm is proposed using the algorithm introduced by beylkin see siam j numer anal vol 29 p 1716 1740 1992 we compute the wavelet transform for all the circular time shifts of a length n signal in o n log n operations the wavelet coefficients of the time shift with minimal cost are selected as the best representation of the signal using a binary tree search algorithm with an appropriate cost function we apply the translation invariant representation algorithm to a geoacoustic data compression application the results show that the new algorithm can reduce the distortion the squared error in our case substantially if the input signals are transients that are sensitive to time shifts

generalized sampling theorems for bessel type transforms of bandlimited functions and distributions this paper adapts the ideas of helms and thomas proc ire 50 1962 pp 179 184 and jerri applicable anal 14 1982 pp 11 25 to derive bessel type sampling expansions and determine upper bounds on the truncation errors in deriving these expansions it is shown that the fourier transform property of additivity of supports under convolution also holds for h 0 hankel transforms under certain conditions

some aspects of band limited extrapolations in this paper some problems in band limited extrapolations are discussed these aspects include the computation of the inverse fourier transform the accuracy of the extrapolation the ill posedness and regularization method

on convergence of a domain decomposition method for a scalar conservation law in this paper we prove convergence of a domain decomposition method for one dimensional scalar conservation laws by dealing carefully with nonconservative terms at the interface of subdomains the method consists of an explicit scheme in some subdomains and an implicit scheme in other subdomains with a numerical flux being the same as the one used in the explicit scheme although such a multidomain algorithm is not strictly conservative the conservation error ce 0 n delta t is equal to mathcal o delta t regardless of the smoothness of the solution finally two test examples are given to validate convergence and the computational efficiency of the present method

frequency shift invariant orthonormal wavelet packet representations it is commonly known that the dyadic structure of wavelet expansions results in both time and frequency translation sensitivity of an input signal we develop the first efficient method to reduce frequency alignment sensitivity by introducing a wavelet packet decomposition that is invariant to frequency shifts of a signal a frequency shifted wavelet packet library is presented and an efficient best basis algorithm is developed to determine the best signal representation among all frequency shifts of a signal the algorithm computes all frequency shifted coefficients in o n sup 2 operations followed by an efficient depth first tree search of the same complexity

a multiscale restriction smoothed basis method for high contrast porous media represented on unstructured grids a wide variety of multiscale methods have been proposed in the literature to reduce runtime and provide better scaling for the solution of poisson type equations modeling flow in porous media we present a new multiscale restricted smoothed basis msrsb method that is designed to be applicable to both rectilinear grids and unstructured grids like many other multiscale methods msrsb relies on a coarse partition of the underlying fine grid and a set of local prolongation operators multiscale basis functions that map unknowns associated with the fine grid cells to unknowns associated with blocks in the coarse partition these mappings are constructed by restricted smoothing starting from a constant a localized iterative scheme is applied directly to the fine scale discretization to compute prolongation operators that are consistent with the local properties of the differential operators the resulting method has three main advantages first of all both the coarse and the fine grid can have general polyhedral geometry and unstructured topology this means that partitions and good prolongation operators can easily be constructed for complex models involving high media contrasts and unstructured cell connections introduced by faults pinch outs erosion local grid refinement etc in particular the coarse partition can be adapted to geological or flow field properties represented on cells or faces to improve accuracy secondly the method is accurate and robust when compared to existing multiscale methods and does not need expensive recomputation of local basis functions to account for transient behavior dynamic mobility changes are incorporated by continuing to iterate a few extra steps on existing basis functions this way the cost of updating the prolongation operators becomes proportional to the amount of change in fluid mobility and one reduces the need for expensive tolerance based updates finally since the msrsb method is formulated on top of a cell centered conservative finite volume method it is applicable to any flow model in which one can isolate a pressure equation herein we only discuss single and two phase incompressible models compressible flow e g as modeled by the black oil equations is discussed in a separate paper

fast reliable simulations of secondary settling tanks in wastewater treatment with semi implicit time discretization the bio kinetic and sedimentation processes of wastewater treatment plants can be modelled by a large system of coupled nonlinear ordinary and partial differential equations odes and pdes the subprocess of continuous sedimentation which contains concentration discontinuities is modelled by a degenerate parabolic conservation pde with spatially discontinuous coefficients a spatial discretization of this pde described in brger etal 2013 results in a large system of method of lines odes for the entire plant and simulation can be performed by integration in time in practice standard time integration methods available in commercial simulators are often used shortages of such methods are here shown such as the smearing of shock waves by rungekutta methods and long execution times a semi implicit time discretization which is described in detail provides substantially shorter computational times and is more efficient than standard methods

a priori error estimates for numerical methods for scalar conservation laws part i the general approach we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original kuznetsov approximation theory as a first application of this general technique we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution thus we obtain optimal error estimates for the engquist osher scheme without using the fact i that the solution is uniformly bounded ii that the scheme is total variation diminishing and iii that the discrete semigroup associated with the scheme has the l 1 contraction property which guarantees an upper bound for the modulus of continuity in time of the approximate solution

vertex approximate gradient discretization preserving positivity for two phase darcy flows in heterogeneous porous media abstract in this article a new nodal discretization is proposed for two phase darcy flows in heterogeneous porous media the scheme combines the vertex approximate gradient vag scheme for the approximation of the gradient fluxes with an hybrid upwind hu approximation of the mobility terms in the saturation equation the discretization in space incorporates naturally nodal interface degrees of freedom d o f allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves it is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3d large discrete fracture matrix dfm models

convergence of a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper a new explicit finite element method for numerically solving the drift diffusion semiconductor device equations is introduced and analyzed the method uses a mixed finite element method for the approximation of the electric field a finite element method using discontinuous finite elements is used to approximate the concentrations which may display strong gradients the use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable in this paper we carry out the analysis of the model method which employs a continuous piecewise linear approximation to the electric field and a piecewise constant approximation to the electron concentration in a model problem namely the so called unipolar case with the diffusion terms neglected the resulting system of equations is equivalent to a conservation law whose flux the electric field depends globally on the solution the concentration of electrons by exploiting the similarities of this system with classical scalar conservation laws the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme the scheme considered as a scheme for the electron concentration is shown to satisfy a maximum principle and to be total variation bounded its convergence to the unique weak solution is proven numerical experiments displaying the performance of the scheme are shown

error estimates for a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper new error estimates for an explicit finite element method for numerically solving the so called zero diffusion unipolar model a one dimensional simplified version of the drift diffusion semiconductor device equations are obtained the method studied in a previous paper combines a mixed finite element method using a continuous piecewise linear approxima tion of the electric field with an explicit upwinding finite element method using a piecewise constant approximation of the electron concentration by using a suitable extension of kuznetsov approximation theory for scalar conservation laws it is proved that under proper hypotheses on the data the l j error between the approximate and exact electron concentrations of the zero diffusion unipolar model is of order ax1 2 these estimates are sharp

implicit hybrid upwind scheme for coupled multiphase flow and transport with buoyancy abstract numerical simulation of coupled multiphase flow and transport in porous media is used in many scientific and industrial applications such as groundwater management and oil recovery to solve the governing partial differential equations we focus on the fully implicit backward euler finite volume method fim in this numerical scheme large algebraic systems must be solved at each time step using newton s method this can be quite expensive especially for highly nonlinear problems with buoyancy in this work we present a new first order approximation of the flux in the fim leading to improved nonlinear convergence rate and radius this approximation is based on implicit hybrid upwinding and relies on separate treatments of the viscous and buoyancy parts of the numerical flux moreover in the viscous part the total velocity discretization is key as it controls the coupling between the elliptic flow problem and the hyperbolic multiphase transport problem to achieve robustness and efficiency we propose a differentiable total velocity discretization that adapts to the balance of forces at each interface we analyze the theoretical properties of the resulting numerical scheme and prove that saturations remain between physical bounds challenging two and three phase test cases illustrate that our numerical scheme brings a significant reduction in the number of nonlinear iterations compared to a finite volume discretization relying on the widely used phase per phase upstream weighting our numerical scheme therefore reduces the simulation cost while providing similar accuracy

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed