spline space and its b splines on an n 1 direction mesh in r n it is well known that splines play an important role in many fields especially their close relationship with wavelets makes them have more widespread applications in numerous scientific and engineering domains univariate and bivariate splines have been well studied and lots of results have been obtained because of the intrinsic difficulty between bivariate case and higher dimension three or more dimensions settings the study of splines on higher dimensions are very limited for example the study of the bivariate splines on a three direction mesh triangulation has obtained many important and excellent results but almost all of those results have no analog generalization to higher dimensions in this paper we will study the higher dimension splines defined on n 1 mesh simplical partitions which is the analog of bivariate splines on three mesh triangulations we have also pointed out many interesting differences between bivariate splines and higher dimensional cases our main results are that similar to bivariate and trivariate cases a necessary and sufficient condition for skr to contain a b spline is k r 1 n 1 for r 1 being odd and k r n 1 1 for r 0 being even

a new formula for box splines on three directional meshes 1 introductionbox splines on three directional meshes and their translates provide a usefultool in the finite element method see e g 4 5 this class of multivariatesplines is contained in a broader class of splines introduced by de boor anddevore 1 some basic recurrence relations obeyed by the latter splines havebeen derived in 2 computations with these recursions could be cumbersomeand time consuming in this note we offer a closed formula for multivariatebox splines on three directional meshes see 3 in 2 we give notation and definitions which will be used in subsequent sec tions the main result is derived in 3 an application is given in 4 2 notation and definitionsby ej 1 j s we will denote the jth coordinate vector in k5 also let

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

algorithms for cardinal interpolation using box splines and radial basis functions we describe an algorithm for bivariate cardinal interpolation which can be applied to translates of basis functions which include box splines or radial basis functions the algorithm is based on a representation of the fourier transform of the fundamental interpolant hence fast fourier transform methods are available in numerical tests the 4 directional box spline transformed to the characteristical submodule of 2 the thin plate spline and the multiquadric case give comparably equal and good results

finite approximations of frobenius perron operators a solution of ulam s conjecture to multi dimensional transformations abstract we prove that ulam s piecewise constant approximation algorithm is convergent for computing an absolutely continuous invariant measure associated with a piecewise c2 expanding transformation or a jablonski transformation s 0 1 n rn 0 1 n this solves an extension of ulam s conjecture to multi dimensions and generalizes the convergence result given by t y li for one dimensional transformations

the spectral analysis of frobenius perron operators abstract it is known that the frobenius perron operator p s l 1 0 1 l 1 0 1 associated with a transformation s from 0 1 to itself with inf s 1 is quasi compact as an operator on the banach space bv 0 1 of functions of bounded variation in l 1 0 1 and thus p s bv 0 1 bv 0 1 possesses only the finite peripheral spectrum and in particular 1 is an isolated eigenvalue of p s in this paper we show that under mild conditions on s the spectrum of p s l 1 x l 1 x is either the closed unit disk c 1 or a cyclic subset of c 1

hex splines a novel spline family for hexagonal lattices this paper proposes a new family of bivariate nonseparable splines called hex splines especially designed for hexagonal lattices the starting point of the construction is the indicator function of the voronoi cell which is used to define in a natural way the first order hex spline higher order hex splines are obtained by successive convolutions a mathematical analysis of this new bivariate spline family is presented in particular we derive a closed form for a hex spline of arbitrary order we also discuss important properties such as their fourier transform and the fact they form a riesz basis we also highlight the approximation order for conventional rectangular lattices hex splines revert to classical separable tensor product b splines finally some prototypical applications and experimental results demonstrate the usefulness of hex splines for handling hexagonally sampled data

a stable three level finite difference scheme for solving the parabolic two step model in a 3d micro sphere heated by ultrashort pulsed lasers ultrashort pulsed lasers with pulse durations of the order of sub picosecond to femtosecond domain possess exclusive capabilities in limiting the undesirable spread of the thermal process zone in the heated sample parabolic two step micro heat transport equations have been widely applied for thermal analysis of thin metal films exposed to picosecond thermal pulses in this study we develop a three level finite difference scheme for solving the heat transport equations in a three dimensional micro sphere heated by ultrashort pulsed lasers it is shown that the scheme is unconditionally stable the method is illustrated by numerical examples

symmetric box splines on root lattices root lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions due to their highly symmetric structure one root lattice the cartesian grid is almost exclusively used since it matches the coordinate grid but it is less efficient than other root lattices box splines on the other hand generalize tensor product b splines by allowing non cartesian directions they provide in any number of dimensions higher order reconstructions of fields often of higher efficiency than tensored b splines but on non cartesian lattices such as the bcc body centered cubic or the fcc face centered cubic lattice only some box splines and then only up to dimension three have been investigated this paper derives and completely characterizes efficient symmetric box spline reconstruction filters on all irreducible root lattices that exist in any number of dimensionsn 2 n 3 for d n and d n lattices in all cases box splines are constructed by convolution using the lattice directions generalizing the known constructions in two and three variables for each box spline we document the basic properties for computational use the polynomial degree the continuity the linear independence of shifts on the lattice and optimal quasi interpolants for fast approximation of fields

a finite element method for the frobenius perron operator equation we present a finite element framework to numerically solve the absolutely continuous invariant measure associated with a non singular transformation both convergence analysis and error estimates of the method are given for a class of mappings in r n

analysis and application of finite volume element methods to a class of partial differential equations abstract the finite volume element fve methods for a class of partial differential equations are discussed and analyzed in this paper the new initial values are introduced in the finite volume element schemes and we obtain optimal error estimates in l p and w 1 p 2 p as well as some superconvergence estimates in w 1 p 2 p the main results in this paper perfect the theory of the finite volume element methods

grid point interpolation on finite regions using c 1 box splines multivariate grid point interpolation on finite regions is considered by translates of c 1 box splines defined on a s 1 direction mesh in mathbb r s in general this problem will give more degrees of freedom than the number of interpolation conditions and hence the problem has no unique solution among all interpolants the one minimizing a smoothing functional is chosen two choices of the smoothing functional are proposed and the related existence and uniqueness problems are studied examples showing constructions of box spline surfaces on nonrectangular regions in lr2 and rectangular regions in mathbb r 3 are presented the method can also be used to construct a smooth interpolant to a set of points given at an almost arbitrary set of grid points in mathbb r s

high order finite difference methods for a second order dual phase lagging models of microscale heat transfer two adi solvers are given for 2d and 3d heat conduct equations at microscale they are uniquely solvable it is shown that they are of order 2 in time and order 4 in space numerical results exhibit the efficiency of the solvers in this paper a compact alternating direction implicit adi method which combines the fourth order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators is firstly presented for solving two dimensional 2d second order dual phase lagging models of microscale heat transfer by the discrete energy method it is shown that it is second order accurate in time and fourth order accurate in space with respect to l2 norms additionally the compact adi method is successfully generalized to solve corresponding three dimensional 3d problem also the convergence result of the solver for 3d case is given rigorously finally numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results

a mixed collocation finite difference method for 3d microscopic heat transport problems three dimensional time dependent initial boundary value problems of a novel microscopic heat equation are solved by the mixed collocation finite difference method in and on the boundaries of a particle when the thickness is much smaller than both the length and width the collocation method on fixed grid size is used to approximate the space operator whereas the finite difference scheme is used for time discretization this new mixed method is applied to a novel heat problem in a particle in order to compute the temperature distribution in and on the particle s surface the second derivatives of the basis functions for the spectral approximation are derived direct substitution of derivatives in the model transforms the differential equation into a linear system of equations that is solved by the specific preconditioned conjugate gradient method the high order accuracy and resolution achieved by the proposed method allows one to obtain engineering accuracy solution on coarse meshes the consistency stability and convergence analysis are provided and numerical results are presented

bivariate interpolation with quadratic box splines etablissement des resultats d existence et d unicite pour l interpolation a deux variables par des fonctions splines quadratiques encapsulees

effortless quasi interpolation in hierarchical spaces we present a general and simple procedure to construct quasi interpolants in hierarchical spaces such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement the proposed hierarchical quasi interpolants are described in terms of the so called truncated hierarchical basis assuming a quasi interpolant is selected for each space associated with a particular level in the hierarchy the hierarchical quasi interpolants are obtained without any additional manipulation the main properties like polynomial reproduction of the quasi interpolants selected at each level are locally preserved in the hierarchical construction we show how to construct hierarchical local projectors and the local approximation order of the underling hierarchical space is also investigated the presentation is detailed for the truncated hierarchical b spline basis and we discuss its extension to a more general framework

the high order multistep adi solver for two dimensional nonlinear delayed reaction diffusion equations with variable coefficients abstract in this paper a high order compact multistep alternating direction implicit adi difference scheme is constructed and analyzed for two dimensional 2d nonlinear delayed reaction diffusion equations ndres with variable coefficients by the discrete energy method it is presented that it achieves the convergence rate of o 2 h x 4 h y 4 with respect to h 1 and l 2 norms in non constrained temporal grids and is unconditionally stable finally numerical results demonstrate the effectiveness of our adi solver and exhibit the correctness of theoretical results

unconditionally stable finite difference scheme and iterative solution of 2d microscale heat transport equation abstract a two dimensional time dependent heat transport equation at the microscale is derived a second order finite difference scheme in both time and space is introduced and the unconditional stability of the finite difference scheme is proved a computational procedure is designed to solve the discretized linear system at each time step by using a preconditioned conjugate gradient method numerical results are presented to validate the accuracy of the finite difference scheme and the efficiency of the proposed computational procedure

high order approximation of the frobenius perron operator we formulate a general convergence theory for the finite dimensional projection approximation of the fixed point of a class of linear operators in particular we explore the relation with the standard framework of stability and consistency of the approximation imply convergence applications include the frobenius perron operator p s corresponding to a nonsingular measurable transformation from 0 1 to itself numerical experiments are also given to support the theoretical analysis

cardinal interpolation by multivariate splines the purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials in particular the notion of a commutator is introduced and its connection with the marsden identity is observed the order of a commutator is shown to be equivalent to the strang and fix conditions that arise in the study of the local approxima tion orders using quasi interpolants we also prove that scaled cardinal interpolants give these local approximation orders introduction cardinal interpolation by bivariate box splines was first studied by de boor hollig and riemenschneider 3 the purpose of our paper is to investigate the cardinal interpolation problem from a different point of view in particular the notion of a commutator is introduced it will be shown that this notion generalizes the marsden identity for univariate splines to the multivariate setting the order of a commutator will be shown to be equivalent to the strang and fix conditions used in the study of the order of controlled approximation by dahmen and micchelli 8 or local approximation by de boor and jia 4 an application to obtain approximation orders through the constructive method of scaled cardinal interpolation will also be studied in this paper 1 preliminaries this section consists of preliminary material for multivariate cardinal interpolation our approach is motivated by the work in 5 and 13 where cardinal interpolation in l2 was connected with certain convolution operators lr let z denote the set of integers and z the nonnegative ones for any given complex sequence by

a basic norm equivalence for the theory of multilevel methods subspace decompositions of finite element spaces based onl 2 like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods recently several authors have proved the equivalence of the associated discrete norms with theh 1 norm the present paper gives an elementary self contained derivation of this result which is based on the use ofk functionals known from the theory of interpolation spaces

a compact difference scheme for numerical solutions of second order dual phase lagging models of microscale heat transfer dual phase lagging dpl models constitute a family of non fourier models of heat conduction that allow for the presence of time lags in the heat flux and the temperature gradient these lags may need to be considered when modeling microscale heat transfer and thus dpl models have found application in the last years in a wide range of theoretical and technical heat transfer problems consequently analytical solutions and methods for computing numerical approximations have been proposed for particular dpl models in different settings in this work a compact difference scheme for second order dpl models is developed providing higher order precision than a previously proposed method the scheme is shown to be unconditionally stable and convergent and its accuracy is illustrated with numerical examples

high accuracy finite difference scheme for three dimensional microscale heat equation a compact finite difference scheme is developed to the three dimensional microscale heat transport equation this new scheme is fourth order in space and second order in time it is proved to be unconditionally stable with respect to initial values numerical results are provided for comparison testing purpose

h1 galerkin mixed finite element methods for pseudo hyperbolic equations h 1 galerkin mixed finite element methods are discussed for a class of second order pseudo hyperbolic equations depending on the physical quantities of interest two methods are discussed optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension an extension to problems in two and three space variables is also discussed the existence and uniqueness are derived and it is showed that the h 1 galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the lbb consistency condition

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

simplicial grid refinement on freudenthal s algorithm and the optimal number of congruence classes in the present paper we investigate freudenthal s simplex refinement algorithm which can be considered to be the canonical generalization of bank s well known red refinement strategy for triangles freudenthal s algorithm subdivides any given n simplex into 2 n subsimplices in such a way that recursive application results in a stable hierarchy of consistent triangulations our investigations concentrate in particular on the number of congruence classes generated by recursive refinements after presentation of the method and the basic ideas behind it we will show that freudenthal s algorithm produces at most n 2 congruence classes for any initial n simplex no matter how many subsequent refinements are performed moreover we will show that this number is optimal in the sense that recursive application of any affine invariant refinement strategy with 2 n sons per element results in at least n 2 congruence classes for almost all n simplices

optimal lattices for sampling the generalization of the sampling theorem to multidimensional signals is considered with or without bandwidth constraints the signal is modeled as a stationary random process and sampled on a lattice exact expressions for the mean square error of the best linear interpolator are given in the frequency domain moreover asymptotic expansions are derived for the average mean square error when the sampling rate tends to zero and infinity respectively this makes it possible to determine the optimal lattices for sampling in the low rate sampling case or equivalently for rough processes the optimal lattice is the one which solves the packing problem whereas in the high rate sampling case or equivalently for smooth processes the optimal lattice is the one which solves the dual packing problem in addition the best linear interpolation is compared with ideal low pass filtering cardinal interpolation

a spacetime discontinuous galerkin method for hyperbolic heat conduction non fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation for applications involving very short length or time scales hyperbolic conduction models better represent the physical thermal transport processes this paper reviews the maxwell cattaneo vernotte modification of the fourier conduction law and describes its implementation within a spacetime discontinuous galerkin sdg finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time a causal advancing front meshing procedure enables a patch wise solution procedure with linear complexity in the number of spacetime elements an h adaptive scheme and a special sdg shock capturing operator accurately resolve sharp solution features in both space and time numerical results for one spatial dimension demonstrate the convergence properties of the sdg method as well as the effectiveness of the shock capturing method simulations in two spatial dimensions demonstrate the proposed method s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place

on the approximation of invariant measures given a discrete dynamical system defined by the map x x the density of the absolutely continuous a c invariant measure if it exists is the fixed point of the frobenius perron operator defined on l1 x ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator t y li proved the convergence of the scheme for expanding maps of the interval g keller and m blank extended this result to piecewise expanding maps of the cube in n we show convergence of a variation of ulam s scheme for maps of the cube for which the frobenius perron operator is quasicompact we also give sufficient conditions on for the existence of a unique fixed point of the matrix approximation and if the fixed point of the operator is a function of bounded variation we estimate the convergence rate

an alternative discretization and solution procedure for the dual phase lag equation we describe an alternative numerical treatment of the dual phase lag equation often used to account for microscale short time heat transport the approach consists of an undecomposed formulation of the partial differential equation resulting from taylor expansion with respect to lag times of the original delay partial differential equation trapezoidal integration in time and centered differencing in space provide an accurate discretization as demonstrated by comparisons with analytical and experimental results in one dimension and via grid function convergence tests in three dimensions for relatively fine 3 d grids the approach is approximately six times faster than a standard explicit scheme and nearly three times faster than an implicit method employing conjugate gradient iteration at each time step

numerical methods for nonlinear second order hyperbolic partial differential equations i time linearized finite difference methods for 1 d problems abstract a three parameter family of finite difference methods for one dimensional hyperbolic equations with damping and nonlinearities based on the introduction of a new dependent variable and three implicitness parameters for the time discretization and the diffusion and reaction terms is presented and shown to result in two time level semi discrete equations second order accurate spatial discretizations and time linearization of the nonlinear source terms are shown to result in three time level linear finite difference equations which are second order accurate in time whenever the reaction and diffusion terms are allocated in the proportion 1 2 1 to the past current and future time levels second order accurate in time time linearized three point compact operator methods for the spatial derivatives which only involve three time levels and three grid points and are fourth order accurate in space are also presented and their linear stability analyzed an exhaustive comparison between analytical and numerical results indicates that the accuracy of time linearized methods increases as the time step and grid spacing are decreased and is nearly independent of the order of the spatial discretization when there is a sufficient number of grid points to resolve the wave structure and the best accuracy is achieved when the three implicitness parameters are equal to 1 2 it is also shown that the accuracy of time linearized methods is a strong function of the wave speed as the wave speed is increased and the wave equation tends to a parabolic one the methods are still three time level techniques which may not preserve the positivity of the solution whereas a crank nicolson discretization of the corresponding parabolic equation does preserve this property provided that there are no mathematical incompatibilities between the initial and the boundary conditions it is also shown that for one dimensional wave equations with monotonic kinetics and time dependent forcing there are very few differences between second and first order time accurate discretizations of the diffusion terms provided that the time step and grid size are sufficiently small and the discretization of the reaction terms plays a less important rule than that of the diffusion terms

multigrid method and fourth order compact scheme for 2d poisson equation with unequal mesh size discretization abstract a fourth order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two dimensional poisson equation in a rectangular domain multigrid methods using a partial semicoarsening strategy and line gauss seidel relaxation are designed to solve the resulting sparse linear systems numerical experiments are conducted to test the accuracy of the fourth order compact difference scheme and to compare it with the standard second order difference scheme convergence behavior of the partial semicoarsening and line gauss seidel relaxation multigrid methods is examined experimentally

voronoi splines we introduce a framework for construction of non separable multivariate splines that are geometrically tailored for general sampling lattices voronoi splines are b spline like elements that inherit the geometry of a sampling lattice from its voronoi cell and generate a lattice shift invariant spline space for approximation in rd the spline spaces associated with voronoi splines have guaranteed approximation order and degree of continuity by exploiting the geometric properties of voronoi polytopes and zonotopes we establish the relationship between voronoi splines and box splines which are used for a closed form characterization of the former for cartesian lattices voronoi splines coincide with tensor product b splines and for the 2 d hexagonal lattice the proposed approach offers a reformulation of hex splines in terms of multi box splines while the construction is for general multidimensional lattices we particularly characterize bivariate and trivariate voronoi splines for all 2 d and 3 d lattices and specifically study them for body centered cubic and face centered cubic lattices

a new two level implicit discretization of o k 2 kh 2 h 4 for the solution of singularly perturbed two space dimensional non linear parabolic equations we propose a new two level implicit difference method of o k2 kh2 h4 for the solution of singularly perturbed non linear parabolic differential equation e uxx uyy f x y t u ux u y ut 0 0 subject to appropriate initial and dirichlet boundary conditions where k 0 and h 0 are grid sizes in time and space directions respectively and e 0 is a small parameter we also develop new methods of o kh2 h4 for the estimates of u x and u y in all cases we use 9 spatial grid points and a single computational cell the proposed methods are directly applicable to singular problems we do not require any special scheme to solve singular problems we also discuss alternating direction implicit adi method for solving diffusion equation in polar cylindrical coordinates this method permits multiple use of the one dimensional tri diagonal algorithm with a considerable saving in computing time and produces a very efficient solver it is shown that the adi method is unconditionally stable numerical experiments are conducted to test the high accuracy of the proposed methods and compared with the exact solutions

a new high accuracy two level implicit off step discretization for the system of three space dimensional quasi linear parabolic partial differential equations we present a new two level compact implicit numerical scheme based on off step discretization for the solution of the system of three space dimensional quasi linear parabolic partial differential equations subject to suitable initial and boundary conditions further we derive the estimates of first order space derivatives of the solution the proposed methods are fourth order accurate in space and second order accurate in time and involve 19 spatial grid points of a single compact cell also we develop the alternating direction implicit adi scheme for a linear parabolic equation with variable coefficients which is shown to be unconditionally stable for the heat equation in polar coordinates the methods so proposed do not require any modification when applied to the singular problems at the point of singularity unlike the numerical scheme proposed earlier in mohanty and jain 1994 and mohanty 1997 2003 the proposed methods are directly applicable to parabolic equations with singular coefficients this is the main highlight of our work the method successfully works for the navier stokes equations of motion in polar coordinates many physical problems are solved to illustrate the accuracy of the proposed methods

an unconditionally stable finite difference scheme for solving a 3d heat transport equation in a sub microscale thin film heat transport at the microscale is of vital importance in microtechnology applications the heat transport equation is different from the traditional heat diffusion equation since a second order derivative of temperature with respect to time and a third order mixed derivative of temperature with respect to space and time are introduced in this study we develop a finite difference scheme with two levels in time for the 3d heat transport equation in a sub microscale thin film it is shown by the discrete energy method that the scheme is unconditionally stable the 3d implicit scheme is then solved by using a preconditioned richardson iteration so that only a tridiagonal linear system is solved for each iteration the numerical procedure is employed to obtain the temperature rise in a gold sub microscale thin film

computing the invariant measure and the lyapunov exponent for one dimensional maps using a measure preserving polynomial basis we consider a generalisation of ulam s method for approximating invariant densities of one dimensional maps rather than use piecewise constant polynomials to approximate the density we use polynomials of degree n which are defined by the requirement that they preserve the measure on n 1 neighbouring subintervals over the whole interval this results in a discontinuous piecewise polynomial approximation to the density we prove error results where this approach is used to approximate smooth densities we also consider the computation of the lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself together with using cubic polynomials in the density approximation this yields a very efficient method for computing highly accurate estimates of the lyapunov exponent we illustrate the theoretical findings with some examples

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

the finite element numerical analysis for a class nonlinear evolution equations a complete stability and convergence analysis is given for two level scheme for the approximate solution of a class generalized nerve conduction equation defining a kind of truncation error and using a kind of technique optimal order estimates in l 2 is constructively obtained

error estimates of the markov finite approximation of the frobenius perron operator in the practical application of the modern ergodic theory one of the main problems is related to the computation of the absolutely continuous invariant measures for nonsingular transformations on measure spaces see lasota and mackey l a simple example is constructed in li 2 to show that the round off error can completely dominate the calculation if one is to compute the invariant measure directly using a sequence of cesaro sums an alternative is to compute the invariant density function for the frobenius perron operator p l 0 1 l 0 1 associated with the nonsingular measurable transformations s 0 11 0 11 the frobenius perron operator p is defined by

a compact finite difference scheme for solving a one dimensional heat transport equation at the microscale 431 heat transport at the microscale is of vital importance in microtechnology applications the heat transport equation is different from the traditional heat diffusion equation since a second order derivative of temperature with respect to time and a third order mixed derivative of temperature with respect to space and time are introduced in this study we develop a high order compact finite difference scheme for the heat transport equation at the microscale it is shown by the discrete fourier analysis method that the scheme is unconditionally stable numerical results show that the solution is accurate

a local lagrange interpolation method based on c 1 cubic splines on freudenthal partitions a trivariate lagrange interpolation method based on c 1 cubic splines is described the splines are defined over a special refinement of the freudenthal partition of a cube partition the interpolating splines are uniquely determined by data values but no derivatives are needed the interpolation method is local and stable provides optimal order approximation and has linear complexity

a finite difference scheme for solving a three dimensional heat transport equation in a thin film with microscale thickness heat transport at the microscale is of vital importance in microtechnology applications the heat transport equation is different from the traditional heat diffusion equation since a second order derivative of temperature with respect to time and a third order mixed derivative of temperature with respect to space and time are introduced in this study we develop a finite difference scheme with two levels in time for the three dimensional heat transport equation it is shown by the discrete energy method that the scheme is unconditionally stable the three dimensional implicit scheme is then solved by using a preconditioned richardson iteration so that only a tridiagonal linear system is solved each iteration numerical results show that the solution is accurate copyright 2001 john wiley sons ltd

a general meshsize fourth order compact difference discretization scheme for 3d poisson equation a fourth order compact difference scheme with unrestricted general meshsizes in different coordinate directions is derived to discretize three dimensional poisson equation on a regular cubic domain the difference scheme derivation procedure makes use of the symbolic representation of the finite difference schemes and is easier to understand in such complex three dimensional manipulations we use a preconditioned conjugate gradient method to solve the resulting sparse linear systems and verify the formal order of convergence of the derived fourth order finite difference scheme

semi cardinal interpolation and difference equations from cubic b splines to a three direction box spline construction this paper considers the problem of interpolation on a semi plane grid from a space of box splines on the three direction mesh building on a new treatment of univariate semi cardinal interpolation for natural cubic splines the solution is obtained as a lagrange series with suitable localization and polynomial reproduction properties it is proved that the extension of the natural boundary conditions to box spline semi cardinal interpolation attains half of the approximation order of the cardinal case

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

an alternating direction implicit method for a second order hyperbolic diffusion equation with convection a numerical method is presented to solve a two dimensional hyperbolic diffusion problem where is assumed that both convection and diffusion are responsible for flow motion since direct solutions based on implicit schemes for multidimensional problems are computationally inefficient we apply an alternating direction method which is second order accurate in time and space the stability of the alternating direction method is analyzed using the energy method numerical results are presented to illustrate the performance in different cases

an unconditionally stable three level explicit difference scheme for the schro dinger equation with a variable coefficient a three level explicit difference scheme for the schrodinger equation with a variable coefficient is established it is proved by the discrete energy method that the scheme is unconditionally stable

convergence of classes of high order semi lagrangian schemes for the vlasov poisson system abstract in this paper we present some classes of high order semi lagran gian schemes for solving the periodic one dimensional vlasov poisson system in phase space on uniform grids we prove that the distribution function f t x v and the electric field e t x converge in the l 2 norm with a rate of displaystyle mathcal o left delta t 2 h m 1 frac h m 1 delta t right where m is the degree of the polynomial reconstruction and delta t and h are respectively the time and the phase space discretization parameters

a new high accuracy two level implicit off step discretization for the system of two space dimensional quasi linear parabolic partial differential equations abstract in this article we propose a two level compact implicit off step discretization for the solution of the system of 2d quasi linear parabolic partial differential equations subject to suitable initial and boundary conditions new methods for the estimates of first order space derivatives of the solution are also derived these methods are fourth order accurate in space and second order accurate in time and involve only nine spatial grid points of a single compact cell we further develop the alternating direction implicit adi scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates the proposed methods are directly applicable to singular problems without the need of any special technique which is the main advantage of this work the method is effectively applied to the time dependent navier stokes model equations in polar coordinates numerical examples are provided to illustrate the accuracy of the methods

a domain decomposition method for solving thin film elliptic interface problems with variable coefficients a domain decomposition method is developed for solving thin film elliptic interface problems with variable coefficients in this study the elliptic equation with variable coefficients is discretized using second order finite differences while a discrete interface equation is obtained using the immersed interface method in order to obtain a second order global accuracy the obtained linear system is solved using a preconditioned richardson iteration which is shown to converge fast when the grid size in the thickness direction is much smaller than the grid sizes in both the length and width directions to simplify the computation a domain decomposition algorithm is obtained based on a parallel gaussian elimination procedure the method is illustrated by a numerical example copyright 1999 john wiley sons ltd

numerical inversion of a laplace transform solution of a diffusion equation with a mixed derivative term we consider a diffusion equation with a mixed derivative term modelling the transient flow of a second grade fluid the same diffusion equation with a mixed derivative term also comes up in the modelling of heat conduction in which the history of the temperature gradient is included the equations differ in the sign of the coefficient of the mixed derivative term we show that the method of lines solution is stable for the transient flow of a second grade fluid and unstable for heat conduction with the temperature gradient history included a laplace transform solution for the heat conduction case is then investigated the laplace transform solution is inverted numerically using a fourier based method a stable solution is obtained the solutions are compared and discussed

quasi interpolation projectors for box splines we consider box spline quasi interpolants based on local linear functionals of point evaluator and integral type the approximations are easy to compute and reproduce the whole spline space in question

the projection method for computing multidimensional absolutely continuous invariant measures we present an algorithm for numerically computing an absolutely continuous invariant measure associated with a piecewisec2 expanding mappings on a bounded region rn the method is based on the galerkin projection principle for solving an operator equation in a banach space with the help of the modern notion of functions of bounded variation in multidimension we prove the convergence of the algorithm

stability in terms of two measures of impulsive integro differential equations via variation of the lyapunov method this paper establishes several criteria on stability in terms of two measures for impulsive integro differential equations with fixed moments of impulse effects by using the generalized variation of parameters technique unified with the lyapunov second method

bivariate box splines and smooth pp functions on a three direction mesh abstract let s denote the space of bivariate piecewise polynomial functions of degree k and smoothness on the regular mesh generated by the three directions 1 0 0 1 1 1 we construct a basis for s in terms of box splines and truncated powers this allows us to determine the polynomials which are locally contained in s and to give upper and lower bounds for the degree of approximation for 2k 2 3 k 2 3 the case of minimal degree k for given smoothness we identify the elements of minimal support in s and give a basis for s loc f s supp f with a convex subset of r 2

runge kutta methods and viscous wave equations we study the numerical time integration of a class of viscous wave equations by means of runge kutta methods the viscous wave equation is an extension of the standard second order wave equation including advection diffusion terms differentiated in time the viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient a stable runge kutta methods are then very good candidates for time integration in particular diagonally implicit ones special attention is paid to the question how the a stability property can be translated to this non standard class of viscous wave equations

markov finite approximation of frobenius perron operator the problem of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory 3 lasota and yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations s from 0 l to itself 4 later in 6 li and yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping the classical birkhoff s individual ergodic theorem says that if u is an ergodic invariant probability measure under s then for any measurable set a c 0 11 the time average lim 1 n c ax s x which measures the average time spent in a under iteration of s exists to be i for p almost all x where xa is the characteristic function of a 1 on a and 0 off a this seems to suggest that we may use the cesaro sum of the above form to calculate the invariant measure a simple and important example in 5 however shows that the computer round off error can completely dominate the calculation and make the implementation difficult in order to overcome this difficulty li 5 proposed a rigorous numerical procedure which can be implemented on a computer with negligible round off error problem a mapping s 0 l 0 l is called piecewise c2 if there is a partition 0 a a a 1 of 0 l such that for k 1 r the restriction sk of s on uk a is a c2 function which can be extended to the closed interval ak i ak as a c2 function s need not be continuous at the point ak for a c 0 11 we write s a for x s x e a throughout the paper we assume s is a nonsingular measurable transformation i e s is measurable and for any measurable subset a c 0 l with m a 0 m s a 0 here m denotes the lebesgue measure on 0 11 the operator ps l 0 1 l 0 1 defined by