hermite birkhoff interpolation on scattered data on the sphere and other manifolds abstract the hermite birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered each proposed interpolant is expressed as a linear combination of basis functions the combination coefficients being incomplete taylor expansions of the interpolated function at the interpolation points the basis functions depend on the geodesic distance are orthonormal with respect to the point evaluation functionals and have all derivatives equal zero up to a certain order at the interpolation points a remarkable feature of such interpolants which belong to the class of partition of unity methods is that their construction does not require solving linear systems numerical tests are given to show the interpolation performance

positive varieties of tree languages pin s variety theorem for positive varieties of string languages and varieties of finite ordered semigroups is proved for trees i e a bijective correspondence between positive varieties of tree languages and varieties of finite ordered algebras is established this in turn is extended to generalized varieties of finite ordered algebras which corresponds to steinby s generalized variety theorem also families of tree languages and classes of ordered algebras that are definable by ordered syntactic or translation monoids are characterized

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

on the riemann problem for a prototype of a mixed type conservation law we study the system of conservation laws given by with initial value the system is elliptic when u2 v2 2 and hyperbolic when u2 v2 2 following liu s construction it is found that the system always has a weak solution which however is not necessarily unique

an unconditionally stable method for the euler equations we discuss how to combine a front tracking method with dimensional splitting to solve systems of conservation laws numerically in two space dimensions in addition we present an adaptive grid refinement strategy the method is unconditionally stable and allows for moderately highcflnumbers typically 1 4 and thus it is highly efficient the method is applied to the euler equations of gas dynamics in particular it is tested on an expanding circular gas front a wind tunnel with a step a double mach reflection and a shock bubble interaction the method shows very sharp resolution of shocks

global solutions for the ultra relativistic euler equations abstract a front tracking scheme for the ultra relativistic euler equations is introduced this scheme is based on piecewise constant approximations to the front tracking riemann solutions where continuous rarefaction waves are approximated by finite collections of discontinuities so called non entropy shocks we study the interaction estimates of the generalized shocks entropy and nonentropy shocks of the ultra relativistic euler equations and the outcoming asymptotic riemann solution moreover we use a new function to measure the strengths of the waves of the ultra relativistic euler equations this function has the important implication that the strength is non increasing for the interactions of the generalized shocks this enables us to define a new kind of total variation of a solution the main application of this scheme is proving the global existence of weak solutions for the ultra relativistic euler equations

reconstruction of a function from hermite birkhoff data abstract birkhoff or lacunary interpolation is an extension of polynomial interpolation that appears when observation gives irregular information about some function and its derivatives a birkhoff interpolation problem is not always solvable even in the appropriate polynomial or rational space in this paper we split up the initial problem in subproblems having a unique polynomial solution and use multinode rational basis functions in order to obtain a global interpolant

on the convergence of operator splitting applied to conservation laws with source terms the order of convergence for operator splitting applied to conservation laws with source terms is studied the operator splitting procedure is based on local solutions of the associated homogeneous conservation law and an ordinary differential equation we prove that for scalar problems the error introduced by the splitting is linear with respect to the time step the theoretical results are illustrated by numerical examples

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

asymptotic expansions for interpolating periodic splines a detailed study is made of the structure of smooth periodic interpolating splines of odd degree over a uniform mesh for splines of degree 2r 1 all terms of the asymptotic expansion of the error at the mesh points of the first 2r 2 derivatives are given this result is used to derive a number of other high order spline approximations

minimum sobolev norm interpolation of scattered derivative data abstract we study the problem of reconstructing a function on a manifold satisfying some mild conditions given data of the values and some derivatives of the function at arbitrary points on the manifold while the problem of finding a polynomial of two variables with total degree n given the values of the polynomial and some of its derivatives at exactly the same number of points as the dimension of the polynomial space is sometimes impossible we show that such a problem always has a solution in a very general situation if the degree of the polynomials is sufficiently large we give estimates on how large the degree should be and give explicit constructions for such a polynomial even in a far more general case as the number of sampling points at which the data is available increases our polynomials converge to the target function on the set where the sampling points are dense numerical examples in single and double precision show that this method is stable efficient and of high order

scattered data interpolation based upon bivariate recursive polynomials abstract in this paper firstly based on new recursive algorithms of non tensor product typed bivariate divided differences scattered data interpolation schemes are constructed in the cases of odd and even interpolating nodes respectively moreover the corresponding error estimation is worked out and equivalent formulae are obtained between bivariate high order non tensor product typed divided differences and high order partial derivatives furthermore the operation count for the addition subtractions multiplication and divisions approximates o n 2 in the computation of the interpolating polynomials presented while the operation count approximates o n 3 in the case of radial basis functions for sufficiently large n finally several numerical examples show that it is valid for the recursive interpolating polynomial schemes and these interpolating polynomials change as the order of the interpolating nodes although the node collection is the same

varieties of recognizable tree series over fields we introduce varieties of recognizable s tree series k s vts for short over a field k and a ranked alphabet s our variety theorem establishes a bijective correspondence between these k s vtss and the varieties of finite dimensional k s algebras k s vfda for short a k s algebra is a k vector space equipped with multilinear s operations the link between k s vtss and k s vfdas is provided by the syntactic k s algebras of tree series the most immediate predecessors of this study are berstel s and reutenauer s 1982 2 work on tree series over fields reutenauer s 1980 27 theory of varieties of string series bozapalidis and his associates 1983 1989 1991 8 5 4 work on syntactic k s algebras steinby s 1979 1992 30 31 theory of varieties of tree languages and our previous work 2009 on series of general algebras and their syntactic k s algebras

an improved front tracking method for the euler equations an improved front tracking method for hyperbolic conservation laws is presented the improved method accurately resolves discontinuities as well as continuous phenomena the method is based on an improved front interaction model for a physically more accurate modeling of the euler equations as compared to standard front tracking methods the resulting algorithm is also more efficient than existing front tracking methods the improved front tracking method is applied to the euler equations for one dimensional unsteady flow and two dimensional steady supersonic flow the results are compared to results of a standard front tracking method and a finite volume method

shuffle on positive varieties of languages we show there is a unique maximal positive variety of languages which does not contain the language ab this variety is the unique maximal positive variety satisfying the two following conditions it is strictly included in the class of rational languages and is closed under the shuffle operation it is also the largest proper positive variety closed under length preserving morphisms the ordered monoids of the corresponding variety of ordered monoids are characterized as follows for every pair a b of mutually inverse elements and for every element z of the minimal ideal of the submonoid generated by a and b abzab ab in particular this variety is decidable

multivariate hermite birkhoff interpolation by a class of cardinal basis functions abstract a class of cardinal basis functions for hermite birkhoff interpolation to multivariate real functions on scattered data is constructed the argument is developed first recalling some classical approaches to the multivariate hermite interpolation problem and then introducing suitable cardinal basis functions satisfying a vanishing property on the derivatives a noteworthy special case involving shepard s functions is finally discussed including some numerical examples

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

recognisability for algebras of infinite trees we develop an algebraic language theory for languages of infinite trees we define a class of algebras called w hyperclones and we show that a language of infinite trees is regular if and only if it is recognised by a finitary path continuous w hyperclone

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed

large time step shock capturing techniques for scalar conservation laws for a scalar conservation law u t f u x with f of constant sign the first order upwind difference scheme is a special case of godonov s method the method is equivalent to solving a sequence of riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level the difference scheme is stable and the solutions to the associated sequence of riemann problems do not interact provided the courant number nu is less than 1 by allowing and explicitly handling such interactions it is possible to obtain a generalized method which is stable for nu much larger than 1 in many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods in particular shocks can be correctly computed with virtually no smearing the generalized method is rather unorthodox and still has some problems associated with it nonetheless preliminary results are quite encouraging

general varieties of tree languages we consider varieties of tree languages which are not restricted to a fixed ranked alphabet varieties of finite algebras that contain algebras of all finite types and a matching notion of varieties of congruences a variety theorem that yields isomorphisms between the lattices formed by these three types of varieties is proved to achieve this some basic universal algebra is suitably generalized and we define syntactic algebras so that two tree languages over any alphabets belong to the same varieties exactly in case their syntactic algebras are isomorphic many families of regular tree languages are shown to be varieties in our sense in particular we prove that every family of regular tree languages which can be characterized by syntactic monoids is such a variety but that the converse does not hold

a front tracking method for conservation laws in one dimension abstract we present a front tracking technique for conservation laws in one dimension the method is based on approximations to the solution of riemann problems where the solution is represented by piecewise constant states separated by discontinuities the discontinuities are tracked until they interact at this point a new riemann problem is solved and so on no finite differences are used this method is tested on the system of nonstationary gas dynamics defined by the euler equations and three test cases are presented

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

a new front tracking scheme for the ultra relativistic euler equations the ultra relativistic euler equations for an ideal gas are described in terms of the pressure p the spatial part email protected r 3 of the dimensionless four velocity and the particle density n two schemes for these equations are presented in one space dimension namely a front tracking and a cone grid scheme a new front tracking technique for the ultra relativistic euler equations is introduced which gives weak solutions the front tracking method is based on piecewise constant approximations to riemann solutions called front tracking riemann solutions where continuous rarefaction waves are approximated by finite collections of discontinuities so called non entropy shocks this method can be used for analytical as well as for numerical purposes a new unconditionally stable cone grid scheme is also derived in this paper which is based on the riemann solution for the ultra relativistic euler equations both schemes are compared by two numerical examples where explicit solutions are known

a numerical method for first order nonlinear scalar conservation laws in one dimension abstract a numerical method for first order nonlinear scalar hyperbolic conservation laws in one dimension is presented using an idea by dafermos in this paper it is proved that it may be used as a numerical method for a general flux function and a general initial value it is possible to give explicit error estimates for the numerical method the error in the method is far smaller than in any other method the numerical method is illustrated in an example

second order front tracking for the euler equations a second order front tracking method is developed for solving the hyperbolic system of euler equations of inviscid fluid dynamics numerically meshless front tracking methods are usually limited to first order accuracy since they are based on a piecewise constant approximation of the solution here second order convergence is achieved by deriving a piecewise linear reconstruction of the piecewise constant front tracking solution the linearization is performed by decomposing the front tracking solution into its wave components and by linearizing the wave solutions separately in order to construct a physically correct linearization the physical phenomena of the front are taken into account in terms of the front types of the previously developed improved front interaction model this front interaction model is also extended to include front numbers used in the wave decomposition it is illustrated numerically for sod s riemann problem the two interacting blast waves problem and a two dimensional supersonic airfoil flow validation study that the proposed front tracking method achieves second order convergence also in the presence of strong discontinuities and their interactions