coarse stability and bifurcation analysis using time steppers a reaction diffusion example evolutionary pattern forming partial differential equations pdes are often derived as limiting descriptions of microscopic kinetic theory based models of molecular processes e g reaction and diffusion the pde dynamic behavior can be probed through direct simulation time integration or more systematically through stability bifurcation calculations time stepper based approaches like the recursive projection method shroff g m keller h b 1993 siam j numer anal 30 1099 1120 provide an attractive framework for the latter we demonstrate an adaptation of this approach that allows for a direct effective coarse bifurcation analysis of microscopic kinetic based models this is illustrated through a comparative study of the fitzhugh nagumo pde and of a corresponding lattice boltzmann model

jet schemes of locally complete intersection canonical singularities let x be a variety defined over an algebraically closed field k of characteristic zero the mth jet scheme xm of x is a scheme whose closed points over x x are morphisms ox x k t tm 1 when x is a smooth variety this is an affine bundle over x of dimension m 1 dim x the space of arcs x of x is the projective limit x proj limm xm our main result is a proof of the following theorem which was conjectured by eisenbud and frenkel

high order exactly well balanced numerical methods for shallow water systems abstract this paper is concerned with the development of high order methods for the shallow water system that preserve every stationary solution of the system and not only the water at rest ones the pursued strategy is based on the generalized hydrostatic reconstruction introduced in 10 and on the technique proposed in 8 to obtain high order well balanced reconstruction operators the more difficult step in the reconstruction procedure is the first one given a cell value a stationary solution of the system has to be found whose average at the cell equals the given value the implementation of this step is discussed in detail as well as the similarities and differences with some previous approaches to obtain well balanced methods for the shallow water system finally the well balanced properties of the developed numerical methods are checked in some test cases

finite difference heterogeneous multi scale method for homogenization problems in this paper we propose a numerical method the finite difference heterogeneous multi scale method fd hmm for solving multi scale parabolic problems based on the framework introduced in commun math sci 1 1 87 the numerical method relies on the use of two different schemes for the original equation at different grid level which allows to give numerical results at a much lower cost than solving the original equations we describe the strategy for constructing such a method discuss generalization for cases with time dependency random correlated coefficients nonconservative form and implementation issues finally the new method is illustrated with several test examples

computing motivic zeta functions on log smooth models we give an explicit formula for the motivic zeta function in terms of a log smooth model it generalizes the classical formulas for snc models but it gives rise to much fewer candidate poles in general this formula plays an essential role in recent work on motivic zeta functions of degenerating calabi yau varieties by the second named author and his collaborators as a further illustration we explain how the formula for newton non degenerate polynomials can be viewed as a special case of our results

vanishing of principal value integrals on surfaces principal value integrals are associated to multi valued rational differential forms with normal crossings support on a non singular algebraic variety we prove their vanishing on rational surfaces in the context of a conjecture of denef jacobs as an appli cation we obtain a strong vanishing result for candidate poles of p adic and motivic igusa zeta functions

wavelet based numerical homogenization a numerical homogenization procedure for elliptic differential equations is presented it is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions if the operator is in divergence form this is preserved by the homogenization procedure for periodic problems results similar to classical effective coefficient theory are proved the procedure can be applied to problems that are not cell periodic

coarse grained numerical bifurcation analysis of lattice boltzmann models abstract in this paper we study the coarse grained bifurcation analysis approach proposed by i g kevrekidis and collaborators in pnas c theodoropoulos y h qian i g kevrekidis coarse stability and bifurcation analysis using time steppers a reaction diffusion example proc natl acad sci 97 18 2000 9840 9843 we extend the results obtained in that paper for a one dimensional fitzhugh nagumo lattice boltzmann lb model in several ways first we extend the coarse grained time stepper concept to enable the computation of periodic solutions and we use the more versatile newton picard method rather than the recursive projection method rpm for the numerical bifurcation analysis second we compare the obtained bifurcation diagram with the bifurcation diagrams of the corresponding macroscopic pde and of the lattice boltzmann model most importantly we perform an extensive study of the influence of the lifting or reconstruction step on the minimal successful time step of the coarse grained time stepper and the accuracy of the results it is shown experimentally that this time step must often be much larger than the time it takes for the higher order moments to become slaved by the lowest order moment which somewhat contradicts earlier claims

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

random homogenization analysis in linear elasticity based on analytical bounds and estimates in this work random homogenization analysis of heterogeneous materials is addressed in the context of elasticity where the randomness and correlation of components properties are fully considered and random effective properties together with their correlation for the two phase heterogeneous material are then sought based on the analytical results of homogenization in linear elasticity when the randomness of bulk and shear moduli the volume fraction of each constituent material and correlation among random variables are considered simultaneously formulas of random mean values and mean square deviations of analytical bounds and estimates are derived from random factor method results from the random factor method and the monte carlo method are compared with each other through numerical examples and impacts of randomness and correlation of random variables on the random homogenization results are inspected by two methods moreover the correlation coefficients of random effective properties are obtained by the monte carlo method the random factor method is found to deliver rapid results with comparable accuracy to the monte carlo approach

on the smallest poles of igusa s p adic zeta functions let k be a p adic field we explore igusa s p adic zeta function which is associated to a k analytic function on an open and compact subset of k n first we deduce a formula for an important coefficient in the laurent series of this meromorphic function at a candidate pole afterwards we use this formula to determine all values less than 1 2 for n 2 and less than 1 for n 3 which occur as the real part of a pole

high order nonlinear interpolatory reconstruction operators and associated multiresolution schemes this paper is devoted to the construction and analysis of a new family of nonlinear reconstruction operators and associated interpolating subdivision schemes they are based on centered piecewise nonlinear polynomial interpolations adapted to discontinuities we analyze both subdivision schemes and multiresolution schemes associated to these reconstructions some practical properties of these schemes are demonstrated in various numerical examples

generalized p fem in homogenization summary a new finite element method for elliptic problems with locally periodic microstructure of length varepsilon 0 is developed and analyzed it is shown that the method converges as varepsilon rightarrow 0 to the solution of the homogenized problem with optimal order in varepsilon and exponentially in the number of degrees of freedom independent of varepsilon 0 the computational work of the method is bounded independently of varepsilon numerical experiments demonstrate the feasibility and confirm the theoretical results

the motivic zeta function and its smallest poles abstract let f be a regular function on a nonsingular complex algebraic variety of dimension d we prove a formula for the motivic zeta function of f in terms of an embedded resolution this formula is over the grothendieck ring itself and specializes to the formula of denef and loeser over a certain localization we also show that the space of n jets satisfying f 0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension d n 2 finally we look at the consequences for the poles of the motivic zeta function

large scale statistics of the kuramoto sivashinsky equation a wavelet based approach we show that the statistical properties of the large scales of the kuramoto sivashinsky equation in the extended system limit can be understood in terms of the dynamical behavior of the same equation in a small finite domain our method relies on the description of the solutions of this equation in terms of wavelets and allows us to model the energy transfer between small and large scales we show that the effective equation obtained in this way can be consistently approximated by a forced burgers equation only for scales far from the cutoff between small and large wavelengths copyright ital 1996 the american physical society

back in the saddle again a computer assisted study of the kuramoto sivashinsky equation a numerical and analytical study of the kuramoto sivashinsky partial differential equation pde in one spatial dimension with periodic boundary conditions is presented the structure stability and bifurcation characteristics of steady state and time dependent solutions of the pde for values of the parameter alpha less than 40 are examined the numerically observed primary and secondary bifurcations of steady states as well as bifurcations to constant speed traveling waves limit cycles are analytically verified persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of o 2 their effect on the system dynamics is discussed and several tertiary bifurcations observed numerically are presented

a finite volume method with a modified eno scheme using a hermite interpolation to solve advection diffusion equations in this paper we have developed a finite volume eno scheme third order accurate based on cell averages and a tvd runge kutta time discretization to solve advection diffusion equations in a two dimensional spatial domain we have designed a special interpolating polynomial based on a modified eno scheme and a hermite procedure which avoids the excessive smearing in regions with sharpconcentration fronts and the overcompression effects produced by the modified eno technique thesemodifications do not affect the non oscillatory philosophy since we compare divided differences inthe modified eno scheme and in the evaluation of the hermite polynomial derivatives numericalresults compare favourably with their respective analytical solutions copyright 2001 john wiley sons ltd

deciding the nature of the coarse equation through microscopic simulations the baby bathwater scheme recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form the closure is obtained on demand through appropriately initialized bursts of microscopic simulation the effective coupling of microscopic simulators with macroscopic behavior requires certain decisions about the nature of the unavailable coarse equation such decisions include a the highest spatial derivative active in the coarse equation b whether the equation satisfies certain conservation laws or c whether the coarse dynamics is hamiltonian or dissipative these decisions affect the number and type of boundary conditions as well as the algorithms employed in the absence of an explicit formula for the temporal derivative we propose implement and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation in the same spirit we show how to determine whether a certain coarse conservation law exists or not and we discuss plausibility tests for the existence of a coarse hamiltonian or integrability we believe that such schemes constitute an important part of the equation free approach to multiscale computation

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

a class of extended limiters applied to piecewise hyperbolic methods piecewise hyperbolic methods phms have been widely used for the approximation of hyperbolic conservation laws in this paper we have studied the power limiters in order to analyze their suitable use in the design of phms we formulate a new phm based on these limiters improving the behavior of the phm in siam j sci comput 15 1994 pp 892 915 at local extrema jump discontinuities and discontinuities in derivative we show that this new hyperbolic reconstruction we call it powerphm is local total variation bounded and it behaves stably with a reduced numerical viscosity compared to phm we show numerical evidence of the above mentioned features by means of one and two dimensional numerical tests with the euler equations of gas dynamics

patch dynamics with buffers for homogenization problems an important class of problems exhibits smooth behaviour on macroscopic space and time scales while only a microscopic evolution law is known for such time dependent multi scale problems an equation free framework has been proposed of which patch dynamics is an essential component patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales it uses appropriately initialized simulations of the available microscopic model in a number of small boxes patches which cover only a fraction of the space time domain we show that it is possible to use arbitrary boundary conditions for these patches provided that suitably large buffer regions shield the boundary artefacts from the interior of the patches we analyze the accuracy of this scheme for a diffusion homogenization problem with periodic heterogeneity and illustrate the approach with a set of numerical examples which include a non linear reaction diffusion equation and the kuramoto sivashinsky equation

sensitivity and randomness in homogenization of periodic fiber reinforced composites via the response function method the main issue this paper addresses is the derivation and implementation of a general homogenization method including the simultaneous determination of sensitivity gradients and probabilistic moments of the effective elasticity tensor this is possible with an application of the perturbation method based on taylor expansion and with the effective modules method the computational procedure is implemented using plane strain analysis carried out with the finite element method program mcceff and the symbolic computations system maple the sensitivity gradients and probabilistic moments are commonly determined on the basis of partial derivatives for the homogenized elasticity tensor calculated using the response function method with respect to some composite parameters they are subjected separately to a normalization procedure in deterministic analysis and the relevant algebraic combinations for the stochastic case this enriched homogenization procedure is tested on a periodic fiber reinforced two component composite where the material parameters are taken as design variables and then the input random quantities the results of computational analysis are compared against the results of the central finite difference approach in the case of sensitivity gradients determination as well as the direct monte carlo simulation approach this numerical methodology may be further applied not only in the context of the homogenization method but also to extend various discrete computational techniques such as boundary finite element and finite difference together with various meshless methods

motivic zeta functions of abelian varieties and the monodromy conjecture we prove for abelian varieties a global form of denef and loeser s motivic monodromy conjecture in arbitrary characteristic more precisely we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field its motivic zeta function has a unique pole at chai s base change conductor c a of a and that the order of this pole equals one plus the potential toric rank of a moreover we show that for every embedding of ql in c the value exp 2 ic a is an l adic tame monodromy eigenvalue of a the main tool in the paper is edixhoven s filtration on the special fiber of the neron model of a which measures the behavior of the neron model under tame base change

evaluation of kriging based surrogate models constructed from mesoscale computations of shock interaction with particles abstract macro scale computations of shocked particulate flows require closure laws that model the exchange of momentum energy between the fluid and particle phases closure laws are constructed in this work in the form of surrogate models derived from highly resolved mesoscale computations of shock particle interactions the mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of mach number and particle volume fraction two kriging based methods viz the dynamic kriging method dkg and the modified bayesian kriging method mbkg are evaluated for their ability to construct surrogate models with sparse data i e using the least number of mesoscale simulations it is shown that if the input data is noise free the dkg method converges monotonically convergence is less robust in the presence of noise the mbkg method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments this work is the first step towards a full multiscale modeling of interaction of shocked particle laden flows

projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum we show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters one containing the stiff or fast components and one containing the slow components these methods have large average step sizes relative to the fast components conventional implicit methods involve the solution of nonlinear equations at each step which for large problems requires significant communication between processors on a multiprocessor machine for such problems the methods proposed here have significant potential for speed improvement

lower bound for the poles of igusa s p adic zeta functions let k be a p adic field r the valuation ring of k p the maximal ideal of r and q the cardinality of the residue field r p let f be a polynomial over r in n 1 variables and let be a character of r times let mi u be the number of solutions of f u in r pi n for i in mathbb z geq 0 and u in r p i these numbers are related with igusa s p adic zeta function zf s of f we explain the connection between the mi u and the smallest real part of a pole of zf s we also prove that mi u is divisible by q ulcorner n 2 i 1 urcorner where the corners indicate that we have to round up this will imply our main result zf s has no poles with real part less than n 2 we will also consider arbitrary k analytic functions f

local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws this paper constructs a local third order accurate shock capturing method for hyperbolic scalar conservation laws based on numerical fluxes with a total variation diminishing tvd runge kutta evolution in time using the idea recently introduced by c w shu and s j osher for essentially nonoscillatory eno methods the constructed method is an upwind conservative scheme that is local in the sense that numerical fluxes are reconstructed without using extrapolation from the data of the smoothest neighboring cell to design the method a new concept of local smoothing is introduced to prevent the increasing of total variation of the solution near discontinuities and to achieve third order accuracy the method becomes third order accurate in smooth regions of the solution except at local extrema where it may degenerate to o h 3 2 thus giving better accuracy than tvd methods at local extrema the main advantage of this method lies on the property that is more local than that of eno and tvd upwind

periodically forced finite networks of heterogeneous globally coupled oscillators a low dimensional approach we study a network of 500 globally coupled modified van der pol oscillators the value of a parameter associated with each oscillator is drawn from a normal distribution giving a heterogeneous network for strong enough coupling the oscillators all have the same period and we consider periodic forcing of the network when it is in this state by exploiting the correlations that quickly develop between the state of an oscillator and the value of its parameter we obtain an approximate low dimensional description of the system in terms of the first few coefficients in a polynomial chaos expansion standard bifurcation analysis can then be performed on the low dimensional system which results from this computational coarse graining and the results obtained from this predict very well the behaviour of the high dimensional system for any set of realisations of the random parameter situations in which the method begins to fail are also discussed

discontinuous galerkin bgk method for viscous flow equations one dimensional systems this paper is about the construction of a bgk navier stokes bgk ns solver in the discontinuous galerkin dg framework since in the dg formulation the conservative variables and their slopes can be updated simultaneously the flow evolution in each element involves only the flow variables in the nearest neighboring cells instead of using the semidiscrete approach in the runge kutta discontinuous galerkin rkdg method the current dg bgk method integrates the governing equations in time as well due to the coupling of advection and dissipative terms in the gas kinetic formulation the dg bgk method solves the viscous governing equations directly numerical examples for the one dimensional compressible navier stokes solutions will be presented

on a nonlinear 4 point ternary and non interpolatory subdivision scheme eliminating the gibbs phenomenon abstract a nonlinear ternary 4 point non interpolatory subdivision scheme is presented it is based on a nonlinear perturbation of the 4 point subdivision scheme studied in 16 the convergence of the scheme and the regularity of the limit function are analyzed it is shown that the gibbs phenomenon that is classical in linear schemes is eliminated we also establish the stability of the subdivision scheme that is not a consequence of its convergence due to its non linearity to the best of our knowledge this is the first ternary non interpolatory subdivision scheme that can be found in the literature

a flux split algorithm applied to relativistic flows the equations of rfd can be written as a hyperbolic system of conservation laws by choosing an appropriate vector of unknowns we give an explicit formulation of the full spectral decomposition of the jacobian matrices associated with the fluxes in each spatial direction which is the essential ingredient of the techniques we propose in this paper these techniques are based on the recently derived flux formula of marquina a new way to compute the numerical flux at a cell interface which leads to a conservative upwind numerical scheme using the spectral decompositions in a fundamental way we construct high order versions of the basic first order scheme described by r donat and a marquina in j comput phys 125 42 1996 and test their performance in several standard simulations in one dimension two dimensional simulations include a wind tunnel with a flat faced step and a supersonic jet stream both of them in strongly ultrarelativistic regimes

poles of the topological zeta function associated to an ideal in dimension two to an ideal in mathbb c x y one can associate a topological zeta function this is an extension of the topological zeta function associated to one polynomial but in this case we use a principalization of the ideal instead of an embedded resolution of the curve in this paper we will study two questions about the poles of this zeta function first we will give a criterion to determine whether or not a candidate pole is a pole it turns out that we can know this immediately by looking at the intersection diagram of the principalization together with the numerical data of the exceptional curves afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two the same results are valid for related zeta functions as for instance the motivic zeta function

relations between numerical data of an embedded resolution introduction let k be an algebraically closed field of characteristic zero and f x y e k x y let x h be an embedded resolution of f 0 in the affine plane a2 constructed by successive blowing ups and denote by ei i e i the irreducible components of h f 0 we associate to each ei i e i a pair of numerical data n vi where ni and vi 1 are the multiplicities of ei in the divisor of respectively f o h and h dx a dy on x fix one exceptional curve e with numerical data n v and say e intersects k times another irreducible component denote these components by e1 ek then we have

diffusion front capturing schemes for a class of fokker planck equations application to the relativistic heat equation in this research work we introduce and analyze an explicit conservative finite difference scheme to approximate the solution of initial boundary value problems for a class of limited diffusion fokker planck equations under homogeneous neumann boundary conditions we show stability and positivity preserving property under a courant friedrichs lewy parabolic time step restriction we focus on the relativistic heat equation as a model problem of the mentioned limited diffusion fokker planck equations we analyze its dynamics and observe the presence of a singular flux and an implicit combination of nonlinear effects that include anisotropic diffusion and hyperbolic transport we present numerical approximations of the solution of the relativistic heat equation for a set of examples in one and two dimensions including continuous initial data that develops jump discontinuities in finite time we perform the numerical experiments through a class of explicit high order accurate conservative and stable numerical schemes and a semi implicit nonlinear crank nicolson type scheme