on the directional dynamics of additive cellular automata we continue the study of cellular automata ca directional dynamics i e the behavior of the joint action of ca and shift maps this notion has been investigated for general ca in the case of expansive dynamics by boyle and lind and by sablik for sensitivity and equicontinuity in this paper we give a detailed classification for the class of additive ca providing non trivial examples for some classes of sablik s classification moreover we extend the directional dynamics studies by considering also factor languages and attractors

high convergence order finite elements with lumped mass matrix a method for deriving hexahedral finite elements with lumped mass matrices for 3 d problems is presented these elements meet the theoretical conditions for high order convergence and two numerical examples based on the 3 d scalar wave equation show that this is also the case in practice and that their accuracy is comparable to elements with consistent mass matrices

conservation of some dynamical properties for operations on cellular automata we consider the family of all the cellular automata ca sharing the same local rule but having different memories this family contains also all ca with memory m 0 one sided ca which can act both on a z and on a n we study several set theoretical and topological properties for these classes in particular we investigate whether the properties of a given ca are preserved when considering the ca obtained by changing the memory of the original one shifting operation furthermore we focus our attention on the one sided ca acting on a z starting from the one sided ca acting on a n and having the same local rule lifting operation as a particular consequence of these investigations we prove that the long standing conjecture surjectivity dense periodic orbits dpo can be restated in several different but equivalent ways furthermore we give some results on properties conserved under the iteration of the ca global map

the solution of elastostatic and elastodynamic problems with chebyshev spectral finite elements chebyshev spectral finite elements are applied to the solution of elastostatic and elastodynamic problems in two dimensions the accuracy of the spectral approach is judged by examining the dispersion of the solutions it is shown that the spectral approach can achieve nearly zero dispersion for a wide range of spatial and temporal discretizations even coarse mesh solutions using the explicit lumped capacitance approach demonstrate exceptional accuracy a brief description of the development of the chebyshev spectral elements is included

non uniform cellular automata classes dynamics and decidability the dynamical behavior of non uniform cellular automata is compared with the one of classical cellular automata several differences and similarities are pointed out by a series of examples decidability of basic properties like surjectivity and injectivity is also established the final part studies a strong form of equicontinuity property specially suited for non uniform cellular automata

on the undecidability of the limit behavior of cellular automata cellular automata ca are discrete dynamical systems and an abstract model of parallel computation the limit set of a cellular automaton is its maximal topological attractor a well known result due to kari says that all nontrivial properties of limit sets are undecidable in this paper we consider the properties of limit set dynamics i e properties of the dynamics of ca restricted to their limit sets there can be no equivalent of kari s theorem for limit set dynamics anyway we show that there is a large class of undecidable properties of limit set dynamics namely all properties of limit set dynamics which imply stability or the existence of a unique subshift attractor as a consequence we have that it is undecidable whether the cellular automaton map restricted to the limit set is the identity map and whether it is closing injective expansive positively expansive and transitive

highly convergent dynamic models obtained by modal synthesis with application to short wave pulse propagation a general approach for obtaining the matrices of a substructure ensuring minimum modal frequency errors of the whole structure is presented the mass and stiffness matrices of a small component domain of selected dimension are obtained by applying the modal synthesis of a limited number of close to exact modes such that after assembling a larger joined domain model the modal convergence rate of the latter should be as high as possible the goal is achieved by formulating the minimization problem for the penalty type target function representing the cumulative relative modal frequency error of the joined sample domain and by applying the gradient descent minimization method after the optimum matrices of a component domain are obtained they can be used in any structure as higher order elements or super elements the well known generalized mass matrices obtained as a weighted sum of lumped and consistent components can be treated as a special case of the presented approach the obtained dynamic models are used for modelling short transient waves and wave pulses propagating in elastic or acoustic environments by using a only a few nodal points per pulse length copyright 2004 john wiley sons ltd

a convergent scheme for hamilton jacobi equations on a junction application to traffic in this paper we consider first order hamilton jacobi hj equations posed on a junction that is to say the union of a finite number of half lines with a unique common point for this continuous hj problem we propose a finite difference scheme and prove two main results as a first result we show bounds on the discrete gradient and time derivative of the numerical solution our second result is the convergence for a subsequence of the numerical solution towards a viscosity solution of the continuous hj problem as the mesh size goes to zero when the solution of the continuous hj problem is unique we recover the full convergence of the numerical solution we apply this scheme to compute the densities of cars for a traffic model we recover the well known godunov scheme outside the junction point and we give a numerical illustration

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

error estimates for a finite difference scheme associated with hamilton jacobi equations on a junction this paper is concerned with monotone time explicit finite difference schemes associated with first order hamilton jacobi equations posed on a junction they extend the schemes recently introduced by costeseque lebacque and monneau 2013 to general junction conditions on the one hand we prove the convergence of the numerical solution towards the viscosity solution of the hamilton jacobi equation as the mesh size tends to zero for general junction conditions on the other hand we derive optimal error estimates of order delta x frac 1 2 in l loc infty for junction conditions of optimal control type at least if the flux is strictly limited

sand automata as cellular automata in this paper we exhibit a strong relation between the sand automata configuration space and the cellular automata configuration space this relation induces a compact topology for sand automata and a new context in which sand automata are homeomorphic to cellular automata acting on a specific subshift we show that the existing topological results for sand automata including the hedlund like representation theorem still hold in this context we give a characterization of cellular automata which are sand automata and study some dynamical behaviors such as equicontinuity furthermore we deal with simple sand automata we show that the classical definition of nilpotency is not meaningful for sand automata then we introduce the suitable new notion of flattening sand automata finally we prove that this simple dynamical behavior is undecidable

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

dynamical behavior of additive cellular automata over finite abelian groups abstract we study the dynamical behavior of additive d dimensional d 1 cellular automata where the alphabet is any finite abelian group this class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list 38 44 41 among our major contributions there is the proof that topologically transitive additive d dimensional cellular automata over a finite abelian group are ergodic this result represents a solid bridge between the world of measure theory and that of topology and greatly extends previous results obtained in 12 44 for linear ca over z m z i e additive ca in which the alphabet is the cyclic group z m z and the local rules are linear combinations with coefficients in z m z in our scenario the alphabet is any finite abelian group and the global rule is any additive map this class of ca strictly contains the class of linear ca over z m z n i e with the local rule defined by n n matrices with elements in z m z which in turn strictly contains the class of linear ca over z m z in order to further emphasize that finite abelian groups are more expressive than z m z we prove that contrary to what happens in z m z there exist additive ca over suitable finite abelian groups which are roots with arbitrarily large indices of the shift map as a relevant consequence of our results we have that for additive d dimensional ca over a finite abelian group ergodic mixing weak ergodic mixing ergodicity topological mixing weak topological mixing topological total transitivity and topological transitivity are all equivalent properties as a corollary we see that invertible transitive additive ca are isomorphic to bernoulli shifts furthermore we prove that surjectivity implies openness for additive d dimensional ca over a finite abelian group hence we get that topological transitivity is equivalent to the well known devaney notion of chaos when d 1 moreover we provide a first characterization of strong transitivity for additive ca which we suspect to be true also for the general case

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a theory of l 1 dissipative solvers for scalar conservation laws with discontinuous flux we propose a general framework for the study of l1 contractive semigroups of solutions to conservation laws with discontinuous flux u t mathfrak f x u x 0 qquad mathfrak f x u left begin array ll f l u x 0 end array right quad quad quad rm cl where the fluxes fl fr are mainly assumed to be continuous developing the ideas of a number of preceding works baiti and jenssen in j differ equ 140 1 161 185 1997 towers in siam j numer anal 38 2 681 698 2000 towers in siam j numer anal 39 4 1197 1218 2001 towers et al in skr k nor vidensk selsk 3 1 49 2003 adimurthi et al in j math kyoto university 43 1 27 70 2003 adimurthi et al in j hyperbolic differ equ 2 4 783 837 2005 audusse and perthame in proc roy soc edinburgh a 135 2 253 265 2005 garavello et al in netw heterog media 2 159 179 2007 burger et al in siam j numer anal 47 1684 1712 2009 we claim that the whole admissibility issue is reduced to the selection of a family of elementary solutions which are piecewise constant weak solutions of the form c x c l11 left x 0 right we refer to such a family as a germ it is well known that cl admits many different l1 contractive semigroups some of which reflect different physical applications we revisit a number of the existing admissibility or entropy conditions and identify the germs that underly these conditions we devote specific attention to the vanishing viscosity germ which is a way of expressing the condition of diehl j hyperbolic differ equ 6 1 127 159 2009 for any given germ we formulate germ based admissibility conditions in the form of a trace condition on the flux discontinuity line x 0 in the spirit of vol pert math ussr sbornik 2 2 225 267 1967 and in the form of a family of global entropy inequalities following kruzhkov math ussr sbornik 10 2 217 243 1970 and carrillo arch ration mech anal 147 4 269 361 1999 we characterize those germs that lead to the l1 contraction property for the associated admissible solutions our approach offers a streamlined and unifying perspective on many of the known entropy conditions making it possible to recover earlier uniqueness results under weaker conditions than before and to provide new results for other less studied problems several strategies for proving the existence of admissible solutions are discussed and existence results are given for fluxes satisfying some additional conditions these are based on convergence results either for the vanishing viscosity method with standard viscosity or with specific viscosities adapted to the choice of a germ or for specific germ adapted finite volume schemes

a generalized mass lumping technique for vector finite element solutions of the time dependent maxwell equations time domain finite element solutions of maxwell s equations require the solution of a sparse linear system involving the mass matrix at every time step this process represents the bulk of the computational effort in time dependent simulations as such mass lumping techniques in which the mass matrix is reduced to a diagonal or block diagonal matrix are very desirable in this paper we present a special set of high order 1 form also known as curl conforming basis functions and reduced order integration rules that together allow for a dramatic reduction in the number of nonzero entries in a vector finite element mass matrix the method is derived from the nedelec curl conforming polynomial spaces and is valid for arbitrary order hexahedral basis functions for finite element solutions to the second order wave equation for the electric or magnetic field intensity we present a numerical eigenvalue convergence analysis of the method and quantify its accuracy and performance via a series of computational experiments

multidimensional cellular automata closing property quasi expansivity and un decidability issues in this paper we study the dynamics of d dimensional cellular automata ca by considering them as one dimensional 1d ca along some direction slicing constructions these constructions allow to give the d dimensional version of important notions as 1d closing property and lift well known one dimensional results to the d dimensional settings indeed like in one dimensional case closing d dimensional ca have jointly dense periodic orbits and biclosing d dimensional ca are open by the slicing constructions we further prove that for the class of closing d dimensional ca surjectivity implies surjectivity on spatially periodic configurations old standing open problem we also deal with the decidability problem of the d dimensional closing by extending the kari s construction from 31 based on tilings we prove that the two dimensional and then d dimensional closing property is undecidable in such a way we add one further item to the class of dimension sensitive properties i e properties that are decidable in dimension 1 and are undecidable in higher dimensions it is well known that there are not positively expansive ca in dimension 2 and higher as a meaningful replacement we introduce the notion of quasi expansivity for d dimensional ca which shares many global properties in the d dimensional settings with the 1d positive expansivity we also prove that for quasi expansive d dimensional ca the topological entropy which is an uncomputable property for general ca has infinite value in a similar way as quasi expansivity the notions of quasi sensitivity and quasi almost equicontinuity are introduced and studied

superconvergence of discontinuous galerkin and local discontinuous galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension in this paper we study the superconvergence property for the discontinuous galerkin dg and the local discontinuous galerkin ldg methods for solving one dimensional time dependent linear conservation laws and convection diffusion equations we prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection diffusion equations the order of superconvergence for both cases is proved to be k frac 3 2 when piecewise p k polynomials with k geq1 are used the proof is valid for arbitrary nonuniform regular meshes and for piecewise p k polynomials with arbitrary k geq1 improving upon the results in y cheng and c w shu j comput phys 227 2008 pp 9612 9627 y cheng and c w shu computers and structures 87 2009 pp 630 641 in which the proof based on fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise p 1 polynomials

effective nonlinear neumann boundary conditions for 1d nonconvex hamilton jacobi equations abstract we study hamilton jacobi equations in 0 of evolution type with nonlinear neumann boundary conditions in the case where the hamiltonian is not necessarily convex with respect to the gradient variable in this paper we give two main results first we prove for a nonconvex and coercive hamiltonian that general boundary conditions in a relaxed sense are equivalent to effective ones in a strong sense here we exhibit the effective boundary conditions while for a quasi convex hamiltonian we already know them imbert and monneau 2016 second we give a comparison principle for a nonconvex and nonnecessarily coercive hamiltonian where the boundary condition can have constant parts

discrete time high order schemes for viscosity solutions of hamilton jacobi bellman equations a general method for constructing high order approximation schemes for hamilton jacobi bellman equations is given the method is based on a discrete version of the dynamic programming principle we prove a general convergence result for this class of approximation schemes also obtaining under more restrictive assumptions an estimate in l infty of the order of convergence and of the local truncation error the schemes can be applied in particular to the stationary linear first order equation in bbb r n we present several examples of schemes belonging to this class and with fast convergence to the solution

dispersive and dissipative behavior of the spectral element method if the nodes for the spectral element method are chosen to be the gauss legendre lobatto points and a lagrange basis is used then the resulting mass matrix is diagonal and the method is sometimes then described as the gauss point mass lumped finite element scheme we study the dispersive behavior of the scheme in detail and provide both a qualitative description of the nature of the dispersive and dissipative behavior of the scheme along with precise quantitative statements of the accuracy in terms of the mesh size and the order of the scheme we prove that a the gauss point mass lumped scheme i e spectral element method tends to exhibit phase lag whereas the consistent finite element scheme tends to exhibit phase lead b the absolute accuracy of the spectral element scheme is 1 p times better than that of the finite element scheme despite the use of numerical integration c when the order p the mesh size h and the frequency of the wave omega satisfy 2p 1 approx omega h the true wave is essentially fully resolved as a consequence one obtains a proof of the general rule of thumb sometimes quoted in the context of spectral element methods pi modes per wavelength are needed to resolve a wave

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

realization problems for nonuniform cellular automata given a finite set of local rules the sequences built from them give a set of nonuniform cellular automata it is known 1 that for any such set of local rules the subset of nonuniform cellular automata built from them which give a surjective global map is in fact a sofic subshift for the injective sequences this set is only automatic accepted by a buchi automaton and it is not necessarily open or closed we prove the first corresponding realization result we provide for any sft a set of local rules such that the surjective injective nonuniform cellular automata over them are precisely the sequences in the sft up to a renaming of symbols in fact we prove that sfts are exactly the subshifts which are the set of injective sequences of nonuniform cellular automata for some set of local rules we also consider surjectivity of subclasses of nonuniform cellular automata and realizability questions for other properties in particular number conservation and chain transitivity

the vanishing viscosity limit for hamilton jacobi equations on networks abstract for a hamilton jacobi equation defined on a network we introduce its vanishing viscosity approximation the elliptic equation is given on the edges and coupled with kirchhoff type conditions at the transition vertices we prove that there exists exactly one solution of this elliptic approximation and mainly that as the viscosity vanishes it converges to the unique solution of the original problem

a semi lagrangian scheme for hamilton jacobi bellman equations on networks we present a semi lagrangian scheme for the approximation of a class of hamilton jacobi bellman hjb equations on networks the scheme is explicit consistent and stable for large time steps w

optimal blended spectral element operators for acoustic wave modeling spectral element methods based on high order polynomials are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media however to retain computational efficiency very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided in the present work we devise an approach for reducing such errors by considering modified discrete wave operators we analyze consistent and lumped operators together with blended operators weighted averages of consistent and lumped operators furthermore using the operator blending approach and a novel dispersion analysis method we develop optimal spectral element operators that have increased numerical accuracy without resorting to very high order operators the new operators are faster and computationally more efficient than consistent operators our approach is based on the tensor product decomposition of the element matrices into 1d f

numerical algorithms for simulations of a traffic model on road networks we introduce a simulation algorithm based on a fluid dynamic model for traffic flows on road networks which are considered as graphs composed by arcs that meet at some junctions the approximation of scalar conservation laws along arcs is made by three velocities kinetic schemes with suitable boundary conditions at junctions here we describe the algorithm and we give an example

chaos and ergodicity are decidable for linear cellular automata over z mz n abstract we prove that important properties describing complex behaviours as ergodicity chaos topological transitivity and topological mixing are decidable for one dimensional linear cellular automata lca over z m z n theorem 6 and corollary 7 a large and important class of cellular automata ca which are able to exhibit the complex behaviours of general ca and are used in applications in particular we provide a decidable characterization of ergodicity which is known to be equivalent to all the above mentioned properties in terms of the characteristic polynomial of the matrix associated with lca we stress that the setting of lca over z m z n with n 1 is more expressive gives rise to much more complex dynamics and is more difficult to deal with than the already investigated case n 1 the proof techniques from 23 25 used when n 1 for obtaining decidable characterizations of dynamical and ergodic properties can no longer be exploited when n 1 for achieving the same goal indeed in order to get the decision algorithm algorithm 1 we need to prove a non trivial result of abstract algebra theorem 5 which is also of interest in its own we also illustrate the impact of our results in real world applications concerning the important and growing domain of cryptosystems which are often based on one dimensional lca over z m z n with n 1 as a matter of facts since cryptosystems have to satisfy the so called confusion and diffusion properties ensured by ergodicity and chaos respectively of the involved lca algorithm 1 turns out to be an important tool for building chaotic ergodic one dimensional linear ca over z m z n and hence for improving the existing methods based on them

local rule distributions language complexity and non uniform cellular automata this paper investigates a variant of cellular automata namely n ca indeed n ca are cellular automata which can have different local rules at each site of their lattice the assignment of local rules to sites of the lattice completely characterizes n ca in this paper sets of assignments sharing some interesting properties are associated with languages of bi infinite words the complexity classes of these languages are investigated providing an initial rough classification of n ca

mass lumping techniques in the spectral element method on the equivalence of the row sum nodal quadrature and diagonal scaling methods abstract in the context of wave propagation analysis the spectral element method sem in conjunction with a diagonal mass matrix is often the method of choice therefore it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results to this end we compare the performance of three established methods including the row sum method the nodal quadrature method and the diagonal scaling method the theoretical analysis of these methods reveals a close connection between them under certain conditions that are discussed in detail in this article we are able to show a direct equivalence between these three approaches in this regard the attainable accuracy of the numerical integration of the mass matrix plays an important role by means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix sem in comparison to a formulation based on the consistent mass matrix