stabilized finite element methods with shock capturing for advection diffusion problems abstract stabilized fem of streamline diffusion type for advection diffusion problems may exhibit local oscillations in crosswind direction s as a remedy a shock capturing variant of such stabilized schemes is considered as an additional consistent but nonlinear stabilization we prove existence of discrete solutions then we present some a priori and a posteriori estimates finally we address the efficient solution of the arising nonlinear discrete problems

a monolithic multi time step computational framework for first order transient systems with disparate scales abstract developing robust simulation tools for problems involving multiple mathematical scales has been a subject of great interest in computational mathematics and engineering a desirable feature to have in a numerical formulation for multiscale transient problems is to be able to employ different time steps multi time step coupling and different time integrators and different numerical formulations mixed methods in different regions of the computational domain to this end we present two new monolithic multi time step mixed coupling methods for first order transient systems we shall employ unsteady advection diffusion reaction equation with linear decay as the model problem which offers several unique challenges in terms of non self adjoint spatial operator and rich features in the solutions we shall employ the dual schur domain decomposition technique to split the computational domain into an arbitrary number of subdomains it will be shown that the governing equations of the decomposed problem after spatial discretization will be differential algebraic equations this is a crucial observation to obtain stable numerical results two different methods of enforcing compatibility along the subdomain interface will be used in the time discrete setting a systematic theoretical analysis which includes numerical stability influence of perturbations bounds on drift along the subdomain interface will be performed the first coupling method ensures that there is no drift along the subdomain interface but does not facilitate explicit implicit coupling the second coupling method allows explicit implicit coupling with controlled but non zero drift in the solution along the subdomain interface several canonical problems will be solved to numerically verify the theoretical predictions and to illustrate the overall performance of the proposed coupling methods finally we shall illustrate the robustness of the proposed coupling methods using a multi time step transient simulation of a fast bimolecular advective diffusive reactive system

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

algebraic and analytical tools for the study of the period function abstract in this paper we consider analytic planar differential systems having a first integral of the form h x y a x b x y c x y 2 and an integrating factor x not depending on y our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results theorem a gives a characterization of isochronicity a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone theorem b is intended for being applied in combination with theorem a in an algebraic setting that we shall specify finally theorem c is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers four different applications are given to illustrate these results

a parameter free dynamic diffusion method for advection diffusion reaction problems abstract in this paper we present a two scale finite element formulation named dynamic diffusion dd for advection diffusion reaction problems by decomposing the velocity field in coarse and subgrid scales the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse scale kinetic energy this is done locally and dynamically by imposing some constraints on the resolved scale solution yielding a parameter free consistent method the subgrid scale space is defined by using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales convergence tests on a two dimensional example are reported yielding optimal rates in addition numerical experiments show that dd method is robust for a wide scope of application problems

an adaptive stabilized method for advection diffusion reaction equation abstract we introduce and analyze a residual local projection relp adaptive finite element scheme for the advection reaction diffusion equation this scheme is based on franca et al 2009 and combined with a residual a posteriori error estimator see verfurth 1998 the a posteriori estimator is proved to be consistent reliable and efficient more specifically we proved global upper and local lower error estimates with constants which are independent of the physical parameters and the mesh size several numerical experiments illustrate the effectiveness of this method

a study of solvers for nonlinear afc discretizations of convection diffusion equations abstract nonlinear discretizations are necessary for convection diffusion equations for obtaining accurate solutions that satisfy the discrete maximum principle the numerical solution of the arising nonlinear problems is often difficult this paper presents several approaches for solving the nonlinear problems of algebraic flux correction afc schemes for the kuzmin limiter and the bjk limiter comprehensive numerical studies are performed at examples that model the transport of energy from a body in a flow field in two and three dimensions it turns out that the most efficient approach from the point of view of computing times is a simple fixed point iteration because the iteration matrix possesses properties that can be exploited by the solvers of the arising linear systems of equations

a class of hamiltonian systems with increasing periods on considere un systeme hamiltonien de la forme x g y y f x ou h f et h g satisfont certaines hypotheses on montre qu il y a un intervalle maximal 0 a a 0 tel que chaque solution x t y t avec x 0 a 0 a y 0 0 est periodique avec son orbite enfermant 0 0 soit p a la plus petite periode de cette solution alors p 0 a r est differentiable avec p a 0

on the stable solution of transient convection diffusion equations a transient convection diffusion equation is considered which particularly arises in optimization problems for the asymmetric flow field flow fractionation af4 process a time dependent generalization of the monotone edge averaged finite element eafe scheme is used to obtain a stable discretization in the convection dominated regime new error estimates are proved for this scheme and a comparison with the popular supg method is presented numerical experiments demonstrate that the eafe method is more suitable for problems where boundary layers are formed we consider transient convection diffusion equation with dominating convection the model has an application in optimal control of the asymmetric flow field flow fractionation process for a stable discretization we propose a monotone edge averaged finite element eafe scheme eafe is generalized to a time dependent case and a new error estimate is proved we present numerical results for comparison with a popular supg method

pseudo isochronicity in a class of septic differential systems this paper deals with the pseudo isochronicity for a class of septic differential systems in this paper we transform infinity into the origin so that the properties of infinity can be investigated with the methods developed for finite critical points by calculating the singular point quantities and period constants of the origin the problem of infinity being a pseudo isochronous center has been solved in this case

an a posteriori error estimator for an unsteady advectiondiffusionreaction problem in this work we introduce an a posteriori error estimator of the residual type for the unsteady advectiondiffusionreaction problem for the discretization in time we use an implicit euler scheme and a continuous piecewise linear triangular finite elements for the space together with a stabilized scheme we prove that the approximation error is bounded by above and below by the error estimator using that an adaptive algorithm is proposed analyzed and tested numerically to prove the efficiency of our approach

optimal adaptivity for the supg finite element method abstract for convection dominated problems the streamline upwind petrov galerkin method supg also named streamline diffusion finite element method sdfem ensures a stable finite element solution even on coarse meshes i e in the preasymptotic range based on some a posteriori error estimators from the literature we formulate an adaptive mesh refining algorithm for supg we prove that the adaptively generated supg solutions converge at asymptotically optimal rates towards the exact solution the main focus thus is on the mathematical proof that the supg stabilization does not spoil the asymptotic convergence behavior of the adaptive scheme

ising like dynamics in large scale functional brain networks brain rest is defined more or less unsuccessfully as the state in which there is no explicit brain input or output this work focuses on the question of whether such state can be comparable to any known dynamical state for that purpose correlation networks from human brain functional magnetic resonance imaging are contrasted with correlation networks extracted from numerical simulations of the ising model in two dimensions at different temperatures for the critical temperature tc striking similarities appear in the most relevant statistical properties making the two networks indistinguishable from each other these results are interpreted here as lending support to the conjecture that the dynamics of the functioning brain is near a critical point

stabilized galerkin approximation of convection diffusion reaction equations discrete maximum principle and convergence we analyze a nonlinear shock capturing scheme for h1 conforming piecewise affine finite element approximations of linear elliptic problems the meshes are assumed to satisfy two standard conditions a local quasi uniformity property and the xu zikatanov condition ensuring that the stiffness matrix associated with the poisson equation is an m matrix a discrete maximum principle is rigorously established in any space dimension for convection diffusion reaction problems we prove that the shock capturing finite element solution converges to that without shock capturing if the cell peclet numbers are sufficiently small moreover in the diffusion dominated regime the difference between the two finite element solutions super converges with respect to the actual approximation error numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

a simple subgrid scale stabilized method for the advection diffusion reaction equation abstract a subgrid scale method has been developed for the advection diffusion reaction equation with a simple intrinsic time scale parameter the proposed formulation combines very good accuracy and stability in all the physical regimes of the equation that is in the exponential regime for both negative and positive source terms and in the propagation regime numerical examples confirm the robustness and accuracy of the method

nonconforming streamline diffusion finite element methods for convection diffusion problems we analyze nonconforming finite element approximations of streamline diffusion type for solving convection diffusion problems both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same o h k 1 2 order of convergence in l 2 as in the conforming case for convection dominated problems a rigorous error analysis supported by numerical experiments is given

linearizability conditions for lotka volterra planar complex quartic systems having homogeneous nonlinearities in this paper we investigate the linearizability problem for the two dimensional lotka volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials i e we consider systems of the form email protected x 1 a 3 0x 3 a 2 1x 2y a 1 2xy 2 a 0 3y 3 email protected y 1 b 3 0x 3 b 2 1x 2y b 1 2xy 2 b 0 3y 3 the necessary and sufficient conditions for the linearizability of this system are found from them the conditions for isochronicity of the corresponding real system can be derived

a unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty we discuss stabilized galerkin approximations in a new framework widening the scope from the usual dichotomy of the discontinuous galerkin method on the one hand and petrov galerkin methods such as the supg method on the other the idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation thus circumventing the need of a petrov galerkin type choice of spaces this is made possible by adding a higher order penalty term giving l2 control of the jumps in the gradients between adjacent elements we consider convection diffusion reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces the standard h1 conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides the crouzeix raviart element moreover we show how the formulation extends to discontinuous galerkin interior penalty methods in a natural way by domain decomposition using nitsche s method

ale supg finite element method for convection diffusion problems in time dependent domains conservative form a streamline upwind petrov galerkin supg finite element method for a convection dominated transient convection diffusion reaction equation in time dependent domains is proposed the time dependent domain is handled by the arbitrary lagrangian eulerian ale approach whereas the supg method is used for the spatial discretization further the first order modified backward euler and the second order modified crank nicolson methods are used for the temporal discretization it is shown that the stability of the semi discrete continuous in time conservative ale supg equation is independent of the mesh velocity whereas the stability of the fully discrete scheme with the implicit euler time discretization is unconditionally stable and is only conditionally stable time step depends on mesh velocity with the crank nicolson method numerical results are presented to support the stability estimates and to show the influence of the supg stabilization parameter in a time dependent domain further the proposed numerical scheme is applied to a boundary layer problem in a time dependent domain c 2017 elsevier inc all rights reserved

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

on the upper bound of the criticality of potential systems at the outer boundary using the roussarie ecalle compensator abstract this paper is concerned with the study of the criticality of families of planar centers more precisely we study sufficient conditions to bound the number of critical periodic orbits that bifurcate from the outer boundary of the period annulus of potential centers in the recent years the new approach of embedding the derivative of the period function into a collection of functions that form a chebyshev system near the outer boundary has shown to be fruitful in this issue in this work we tackle with a remaining case that was not taken into account in the previous studies in which the roussarie ecalle compensator plays an essential role the theoretical results we develop are applied to study the bifurcation diagram of the period function of two different families of centers the power like family x x q x p p q r with p q and the family of dehomogenized loud s centers

supg reduced order models for convection dominated convection diffusion reaction equations abstract this paper presents a streamline upwind petrov galerkin supg reduced order model rom based on proper orthogonal decomposition pod this rom is investigated theoretically and numerically for convection dominated convection diffusion reaction problems the supg finite element method was used on realistic meshes for computing the snapshots leading to some noise in the pod data numerical analysis is used to propose the scaling of the stabilization parameter for the supg rom two approaches are used one based on the underlying finite element discretization and the other one based on the pod truncation the resulting supg roms and the standard galerkin rom g rom are studied numerically for many settings the results obtained with the supg roms are more accurate finally one of the choices for the stabilization parameter is recommended

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

on the period function in a class of generalized lotka volterra systems in this note motivated by the recent results of wang et al wang et al local bifurcations of critical periods in a generalized 2d lv system appl math comput 214 2009 17 25 we study the behaviour of the period function of the center at the point 1 1 of the planar differential systemu u p 1 v q v email protected q u p 1 where p q email protected r with pq 0 and m 0 our aim is twofold firstly we determine regions in the parameter space for which the corresponding system has a center with a monotonic period function secondly by taking advantage of the results of wang et al we show some properties of the bifurcation diagram of the period function and we make some comments for further research the differential system under consideration is a generalization proposed by farkas and noszticzius of the lotka volterra model

asymptotic expansion of the period function abstract let p be a not necessarily bounded polycycle of an analytic vector field on an open set of the plane suppose that the singularities which appear after desingularization of the vertices of p are formally linearizable consider the function t defined by the return time near p it is shown that the function t and its derivative t have asymptotic expansions in x r and x log x r it is also shown that under some other conditions imposed on the polycycle vertices the asymptotic expansions of t and t converge absolutely and uniformly to these functions respectively these results are applied to the polycycles of the analytic vector fields which have a darboux first integral in particular it is obtained that if p is a polycycle of a hamiltonian vector field with an analytic polynomial if p is unbounded hamiltonian function t is a nonoscillating function another application concerns the nilpotent centers or focus since the singularities which appear after desingularization of such a singularity have analytic first integrals

anderson acceleration for fixed point iterations this paper concerns an acceleration method for fixed point iterations that originated in work of d g anderson j assoc comput mach 12 1965 pp 547 560 which we accordingly call anderson acceleration here this method has enjoyed considerable success and wide usage in electronic structure computations where it is known as anderson mixing however it seems to have been untried or underexploited in many other important applications moreover while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities this method has received relatively little attention from these communities over the years a recent paper by h fang and y saad numer linear algebra appl 16 2009 pp 197 221 has clarified a remarkable relationship of anderson acceleration to quasi newton secant updating methods and extended it to define a broader anderson family of acceleration methods in this paper our goals are to shed additional light on anderson acceleration and to draw further attention to its usefulness as a general tool we first show that on linear problems anderson acceleration without truncation is essentially equivalent in a certain sense to the generalized minimal residual gmres method we also show that the type 1 variant in the fang saad anderson family is similarly essentially equivalent to the arnoldi full orthogonalization method we then discuss practical considerations for implementing anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

scale free brain functional networks functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites analysis of the resulting networks in different tasks shows that a the distribution of functional connections and the probability of finding a link versus distance are both scale free b the characteristic path length is small and comparable with those of equivalent random networks and c the clustering coefficient is orders of magnitude larger than those of equivalent random networks all these properties typical of scale free small world networks reflect important functional information about brain states

a hybrid multi time step framework for pore scale and continuum scale modeling of solute transport in porous media abstract in this paper we propose a computational framework which is based on a domain decomposition technique to employ both finite element method which is a popular continuum modeling approach and lattice boltzmann method which is a popular pore scale modeling approach in the same computational domain to bridge the gap across the disparate length and time scales we first propose a new method to enforce continuum scale boundary conditions i e dirichlet and neumann boundary conditions onto the numerical solution from the lattice boltzmann method this method is based on maximization of entropy and preserve the non negativity of discrete distributions under the lattice boltzmann method the proposed computational framework allows different grid sizes orders of interpolation and time steps in different subdomains this allows for different desired resolutions in the numerical solution in different subdomains through numerical experiments the effect of grid and time step refinement disparity of time steps in different subdomains domain partitioning and the number of iteration steps on the accuracy and rate of convergence of the proposed methodology are studied finally to showcase the performance of this framework in porous media applications we use it to simulate the dissolution of calcium carbonate in a porous structure

localized axial green s function method for the convection diffusion equations in arbitrary domains a localized axial green s function method lagm is proposed for the convection diffusion equation the axial green s function method agm enables us to calculate the numerical solution of a multi dimensional problem using only one dimensional green s functions for the axially split differential operators this agm has been developed not only for the elliptic boundary value problems but also for the steady stokes flows however this paper is concerned with the localization of the agm this localization of the method is needed for practical purpose when computing the axial green s function specifically for the convection diffusion equation on a line segment that we call the local axial line although our focus is mainly on the convection dominated cases in arbitrary domains this method can solve other cases in a unified way numerical results show that despite irregular types of discretization on an arbitrary domain we can calculate the numerical solutions using the lagm without loss of accuracy even in cases of large convection in particular it is also shown that randomly distributed axial lines are available in our lagm and complicated domains are not a burden

lie symmetries for the orbital linearization of smooth planar vector fields around singular points abstract this work is a generalization of the method proposed in i a garcia s maza linearization of analytic isochronous centers from a given commutator j math anal appl 339 1 2008 740 745 of linearization of analytic isochronous centers from a given commutator in this paper we propose a constructive procedure to get the change of variables that orbitally linearizes a smooth planar vector field on c 2 around an elementary singular point i e a singular point with associated eigenvalues c satisfying 0 or a nilpotent singular point from a given infinitesimal generator of a lie symmetry

isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials in this work we study isochronous centers of two dimensional autonomous system in the plane with linear part of center type and nonlinear part given by homogeneous polynomials of fifth degree a complete classification of the necessary conditions for the time reversible systems of this class is given in order to have an isochronous center at the origin an open problem is stated for the sufficient conditions moreover we find two nonreversible isochronous families from the center cases known all the computations in order to obtain necessary conditions for such isochronous centers are given in polar coordinates and we give a proof of the isochronicity of these systems by using different methods

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part ii analysis for p1 and q1 finite elements abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers a review and a comparison of the most methods which have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers are given in part i v john p knobloch on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review comput methods appl mech engrg 196 2007 2197 2215 in the present paper the most promising of these sold methods are investigated in more detail for p 1 and q 1 finite elements in particular the dependence of the results on the mesh the data of the problems and parameters of the methods are studied analytically and numerically furthermore the numerical solution of the nonlinear discrete problems is discussed and the capability of adaptively refined grids for reducing spurious oscillations is examined our conclusion is that also for simple problems any of the sold methods generally provides solutions with non negligible spurious oscillations

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

a new projection based stabilized method for steady convection dominated convection diffusion equations abstract we propose a new projection based stabilized method for the steady convection dominated convection diffusion equations this method directly increases the l 2 stability instead of the h 1 stability which the variational multiscale vms methods do numerical analysis and numerical experiments illustrate and confirm that our new method has effective numerical performances on convection dominated convection diffusion problems

isochronicity into a family of time reversible cubic vector fields in this work we study necessary and sufficient conditions for the existence of isochronous centers into a family of cubic time reversible systems this class of reversible systems is characterized by the existence of an inverse integrating factor which is a certain power of an invariant straight line

analytic integrability around a nilpotent singularity abstract in this work it is characterized the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions

properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation abstract in this work we study the period function t of solutions to the conservative equation x t f x t 0 we present conditions on f that imply the monotonicity and convexity of t as a consequence we obtain the criterium established by c chicone and find conditions easier to apply we also get a condition obtained by cima gasull and manosas about monotonicity and following some of their calculations present results on the period function of hamiltonian systems where h x y f x n 1 y n using the monotonicity of t we count the homogeneous solutions to the semilinear elliptic equation u u 1 in two dimensions

a stabilised finite element method for the convection diffusion reaction equation in mixed form abstract this paper is devoted to the approximation of the convection diffusion reaction equation using a mixed first order formulation we propose and analyse a stabilised finite element method that allows equal order interpolations for the primal and dual variables this formulation reminiscent of the galerkin least squares method is proven stable and convergent in addition a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out and the different methods are compared in terms of accuracy stability and sharpness of the layers for two different classical test problems

linearization of analytic isochronous centers from a given commutator in this paper we propose an algorithmic way to get the change of variables that linearizes an analytic isochronous center from a given commutator moreover we use this method in order to obtain the linearization of some isochronous centers of the existent literature

linearizability and integrability of vector fields via commutation abstract in this paper we consider complex smooth and analytic vector fields x in a neighborhood of a nondegenerate singular point it is proved the equivalence between linearizability and commutation i e the existence of a commuting vector field y such that the lie brackets x y 0 for complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point it is also proved the equivalence between integrability and the existence of a smooth vector field y such that y is a normalizer of x i e x y x

isochronicity for several classes of hamiltonian systems abstract in this paper we study isochronous centers of analytic hamiltonian systems giving special attention to the polynomial case we first revisit the potential systems and we show the connection between isochronicity and involutions we then study a more general system namely the ones associated to hamiltonians of the form h x y a x b x y c x y 2 as an application we classify the cubic polynomial hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial hamiltonian isochronous centers

darboux linearization and isochronous centers with a rational first integral abstract in this paper we study isochronous centers of polynomial systems it is known that a center is isochronous if and only if it is linearizable we introduce the notion of darboux linearizability of a center and give an effective criterion for verifying darboux linearizability if a center is darboux linearizable the method produces a linearizing change of coordinates most of the known polynomial isochronous centers are darboux linearizable moreover using this criterion we find a new two parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family we also determine all hamiltonian cubic systems which are darboux linearizable in the second part of this work we restrict to the study of isochronous centers having a rational first integral we prove that under certain conditions the cycle vanishing at the isochronous center is either zero homologous in the closure of a generic fiber or the function obtained from the first integral by eliminating the indeterminacy points has several critical points in the singular fiber passing through the isochronous center

linearizability conditions for lotka volterra planar complex cubic systems in this paper we investigate the linearizability problem for the two dimensional planar complex system the necessary and sufficient conditions for the linearizability of this system are found from them the conditions for isochronicity of the corresponding real system can be derived

the period function of the generalized lotka volterra centers abstract the present paper deals with the period function of the quadratic centers in the literature different terminologies are used to classify these centers but essentially there are four families hamiltonian reversible q 3 r codimension four q 4 and generalized lotka volterra systems q 3 lv chicone c chicone review in mathscinet ref 94h 58072 conjectured that the reversible centers have at most two critical periods and that the centers of the three other families have a monotonic period function with regard to the second part of this conjecture only the monotonicity of the hamiltonian and q 4 families w a coppel l gavrilov the period function of a hamiltonian quadratic system differential integral equations 6 1993 1357 1365 y zhao the monotonicity of period function for codimension four quadratic system q 4 j differential equations 185 2002 370 387 has been proved concerning the q 3 lv family no substantial progress has been made since the middle 80s when several authors showed independently the monotonicity of the classical lotka volterra centers f rothe the periods of the volterra lokta system j reine angew math 355 1985 129 138 r schaaf global behaviour of solution branches for some neumann problems depending on one or several parameters j reine angew math 346 1984 1 31 j waldvogel the period in the lotka volterra system is monotonic j math anal appl 114 1986 178 184 by means of the first period constant one can easily conclude that the period function of the centers in the q 3 lv family is monotone increasing near the inner boundary of its period annulus i e the center itself thus according to chicone s conjecture it should be also monotone increasing near the outer boundary which in the poincare disc is a polycycle in this paper we show that this is true in addition we prove that except for a zero measure subset of the parameter plane there is no bifurcation of critical periods from the outer boundary finally we show that the period function is globally i e in the whole period annulus monotone increasing in two other cases different from the classical one

a posteriori optimization of parameters in stabilized methods for convection diffusion problems part i abstract stabilized finite element methods for convection dominated problems require the choice of appropriate stabilization parameters from numerical analysis often only their asymptotic values are known this paper presents a general framework for optimizing stabilization parameters with respect to the minimization of a target functional exemplarily this framework is applied to the supg finite element method and the minimization of a residual based error estimator an error indicator and a functional including the crosswind derivative of the computed solution benefits of the basic approach are demonstrated by means of numerical results

a singularly perturbed model problem for numerical computation abstract in this note we introduce a model problem for the numerical solution of a two dimensional singular perturbation problem to combine a number of typical difficulties in a relatively simple problem we propose to solve the linear convection diffusion problem in the domain exterior of a circle we describe the analytical solution of the problem and we comment on its numerical evaluation for small values of the parameter asymptotic approximations of the solution are given based on expansions by friedlander 1958 and waechter 1968 this information gives insight into the behaviour of the solution and allows the computation of a reference solution for small values of the parameter

appropriate stabilized galerkin approaches for solving two dimensional coupled burgers equations at high reynolds numbers abstract this paper aims to seek proper stabilized galerkin methods for solving the two dimensional coupled burgers equations at high reynolds numbers the stabilization techniques employed here include the streamline upwind petrov galerkin supg method the spurious oscillations at layers diminishing sold method and the characteristic galerkin cg method the first two methods are combined with the crank nicolson scheme for time discretization and the last one is applied in its semi implicit version different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low reynolds numbers this paper mainly focuses on keeping the stability of the solution at high reynolds numbers which is significant in practical applications and also challenging in numerical computation we study two problems equipped with mixed boundary conditions and only dirichlet boundary conditions respectively numerical experiments reveal that the supg method is optimal for the former problem and the sold method is more appropriate for the latter one in addition the performances of these methods demonstrate the difference between the two problems which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high reynolds numbers and last since sold methods have rarely been utilized to solve nonlinear unsteady problems before this study also indicates the potential of this class of methods to solve nonlinear unsteady convection dominated problems

the period function for quadratic integrable systems with cubic orbits this paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves it is proved that each such system has a period function with at most one critical point

shishkin mesh simulation a new stabilization technique for convection diffusion problems abstract a new stabilization procedure is presented it is based on a simulation of the interaction between the coarse and fine parts of a shishkin mesh but can be applied on coarse and irregular meshes and on domains with nontrivial geometries the technique which does not require adjusting any parameter can be applied to different stabilized and non stabilized methods numerical experiments show it to obtain oscillation free approximations on problems with boundary and internal layers on uniform and nonuniform meshes and on domains with curved boundaries

the criticality of centers of potential systems at the outer boundary abstract the number of critical periodic orbits that bifurcate from the outer boundary of a potential center is studied we call this number the criticality at the outer boundary our main results provide sufficient conditions in order to ensure that this number is exactly 0 and 1 we apply them to study the bifurcation diagram of the period function of x y x x 1 p x 1 q y with q p this family was previously studied for q 1 by y miyamoto and k yagasaki

analysis of algebraic flux correction schemes a family of algebraic flux correction afc schemes for linear boundary value problems in any space dimension is studied these methods main feature is that they limit the fluxes along each one of the edges of the triangulation and we suppose that the limiters used are symmetric for an abstract problem the existence of a solution existence and uniqueness of the solution of a linearized problem and an a priori error estimate are proved under rather general assumptions on the limiters for a particular but standard in practice choice of the limiters it is shown that a local discrete maximum principle holds the theory developed for the abstract problem is applied to convection diffusion reaction equations where in particular an error estimate is derived numerical studies show its sharpness

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

a class of reversible cubic systems with an isochronous center abstract we study cubic polynomial differential systems having an isochronous center and an inverse integrating factor formed by two different parallel invariant straight lines such systems are time reversible we find nine subclasses of such cubic systems see theorem 8 we also prove that time reversible polynomial differential systems with a nondegenerate center have half of the isochronous constants equal to zero see theorem 3 we present two open problems

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

an approach to solving systems of polynomials via modular arithmetics with applications the objective of this paper is twofold first we describe a method to solve large systems of polynomial equations using modular arithmetics then we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations

study of the period function of a two parameter family of centers abstract in this paper we study the period function of x 1 x p 1 x q with p q r and p q we prove three independent results the first one establishes some regions in the parameter space where the corresponding center has a monotonous period function this result extends the previous ones by miyamoto and yagasaki for the case q 1 the second one deals with the bifurcation of critical periodic orbits from the center the third one is addressed to the critical periodic orbits that bifurcate from the period annulus of each one of the three isochronous centers in the family when perturbed by means of a one parameter deformation these three results together with the ones that we obtained previously on the issue lead us to propose a conjectural bifurcation diagram for the global behaviour of the period function of the family

stochastic optimal robin boundary control problems of advection dominated elliptic equations in this work we deal with a stochastic optimal robin boundary control problem constrained by an advection diffusion reaction elliptic equation with advection dominated term we assume that the uncertainty comes from the advection field and consider a stochastic robin boundary condition as control function a stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system a global error estimate in the product of physical space and stochastic space for the numerical approximation is derived illustrative numerical experiments are provided

first derivative of the period function with applications abstract given a centre of a planar differential system we extend the use of the lie bracket to the determination of the monotonicity character of the period function as far as we know there are no general methods to study this function and the use of commutators and lie bracket was restricted to prove isochronicity we give several examples and a special method which simplifies the computations when a first integral is known

a parameter uniform numerical method for a singularly perturbed convection diffusion problem posed on an annulus abstract a finite difference method is constructed for a singularly perturbed convection diffusion problem posed on an annulus the method involves combining polar coordinates an upwind finite difference operator and a piecewise uniform shishkin mesh in the radial direction compatibility constraints are imposed on the data in the vicinity of certain characteristic points to ensure that interior layers do not form within the annulus a theoretical parameter uniform error bound is established and numerical results are presented to illustrate the performance of the numerical method applied to particular test problems

linearization of isochronous centers abstract in this paper we study isochronous centers of polynomial systems we first discuss isochronous centers of quadratic systems cubic symmetric systems and reduced kukles system all these systems have rational first integrals we give a unified proof of the isochronicity of these systems by constructing algebraic linearizing changes of coordinates we then study two other classes of systems with isochronous centers namely the class of complex systems z ip z and the class of cubic systems symmetric with respect to a line and satisfying 1 both classes consist of darboux integrable systems we discuss their geometric properties and construct the linearizing changes of coordinates we show that the class of polynomial isochronous systems carries a very rich geometry finally we discuss the geometry of the linearizing changes of coordinates in the complex plane

an assessment of discretizations for convection dominated convection diffusion equations the performance of several numerical schemes for discretizing convection dominated convection diffusion equations will be investigated with respect to accuracy and efficiency accuracy is considered in measures which are of interest in applications the study includes an exponentially fitted finite volume scheme the streamline upwind petrov galerkin supg finite element method a spurious oscillations at layers diminishing sold finite element method a finite element method with continuous interior penalty cip stabilization a discontinuous galerkin dg finite element method and a total variation diminishing finite element method femtvd a detailed assessment of the schemes based on the hemker example will be presented

a robust supg norm a posteriori error estimator for stationary convection diffusion equations a robust residual based a posteriori estimator is proposed for the supg finite element method applied to stationary convection diffusion reaction equations the error in the natural supg norm is estimated the main concern of this paper is the consideration of the convection dominated regime a global upper bound and a local lower bound for the error are derived where the global upper estimate relies on some hypotheses numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses a comparison to other residual based estimators with respect to the adaptive grid refinement is also provided

isochronicity and linearizability of planar polynomial hamiltonian systems in this paper we study isochronicity and linearizability of planar polynomial hamiltonian systems first we prove a theorem which supports a negative answer to the following open question stated by jarque and villadelprat in 15 do there exist planar polynomial hamiltonian systems of even degree having an isochronous center additionally we obtain some conditions for linearizability of complex cubic hamiltonian systems

estimating the clustering coefficient in scale free networks on lattices with local spatial correlation structure abstract when we study the architecture of networks of spatially extended systems the nodes in the network are subject to local correlation structures in this case we show that for scale free networks the traditional way to estimate the clustering coefficient may not be meaningful here we explain why and propose an approach that corrects this problem

on an improved unusual stabilized finite element method for theadvective reactive diffusive equation an improved unusual finite element method is studied herein for a second order linear scalar differential equation including a zero order term the method consists in subtracting from the standard galerkin method a mesh dependent term suggested by static condensation of the bubbles based on this idea a new stabilized parameter is constructed which improves accuracy of the solution several numerical tests attest the efficiency of the present method and an error analysis is perfomed

a finite volume scheme for nonlinear parabolic equations derived from one dimensional local dirichlet problems in this paper we propose a new method to compute the numerical flux of a finite volume scheme used for the approximation of the solution of the nonlinear partial differential equation u t div q f u u 0 in a 1d 2d or 3d domain the function is supposed to be strictly increasing but some values s such that s 0 can exist the method is based on the solution at each interface between two control volumes of the nonlinear elliptic two point boundary value problem q f 0 with dirichlet boundary conditions given by the values of the discrete approximation in both control volumes we prove the existence of a solution to this two point boundary value problem we show that the expression for the numerical flux can be yielded without referring to this solution furthermore we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem numerical results show the increase of accuracy due to the use of this scheme compared to some other schemes

centers with degenerate infinity and their commutators abstract we study polynomial systems with degeneracy at infinity and a center focus equilibrium at the origin we give some general properties related to the existence of polynomial commutators and use these properties in order to characterize uniformly isochronous polynomial centers with polynomial commutator and also we show that the commutator of the centers of the analytic systems whose angular speed is constant can be chosen of radial form finally we characterize the systems y ps j kn 1xhj x qs j kn 1yhj t with polynomial commutator with pj qj hj and kj homogeneous polynomials

completely integrable hamiltonian systems with weak lyapunov instability or isochrony the aim of this paper is to introduce a class of hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details they have an equilibrium point which is stable for some rare elements of the class and unstable in most cases anyhow it is linearly stable all orbits of the linearized system are bounded and no motion is asymptotic in the past namely no non constant solution has the equilibrium as limit point as time goes to minus infinity in the unstable cases there is a sequence of initial data which converges to the equilibrium point whose corresponding solutions are unbounded and the motion is slow so instability is quite weak and perhaps no such explicit examples of instability are known in the literature the stable cases are also interesting since the level sets of the 2 first integrals independent and in involution keep being non compact and stability is related to the isochronous periodicity of all orbits near the equilibrium point and the existence of a further first integral hopefully these superintegrable hamiltonian systems will deserve further research

on the classification of lienard systems with amplitude independent periods abstract we consider conditions under which the second order differential equation x f x x g x 0 has a family of periodic orbits with constant period this condition is equivalent to seeking conditions under which the two dimensional autonomous system x y y g x f x y has a center with constant period i e an isochronous center in turn this is equivalent to the latter system being locally linearizable we give a simple necessary and sufficient condition for this when f and g are analytic functions in the case where when f and g are polynomials we show that this reduces to a finitely determinable system of equations a complete classification is given of all such systems of degree 34 or less

linearizability and local bifurcation of critical periods in a cubic kolmogorov system since chicone and jacobs investigated local bifurcation of critical periods for quadratic systems and newtonian systems in 1989 great attention has been paid to some particular forms of cubic systems having special practical significance but less difficulties in computation this paper is devoted to the linearizability and local bifurcation of critical periods for a cubic kolmogorov system we use the darboux method to give explicit linearizing transformations for isochronous centers investigating the finite generation for the ideal of all period constants which are of the polynomial form in six parameters we prove that at most two critical periods can be bifurcated from the interior equilibrium if it is an isochronous center moreover we prove that the maximum number of critical periods is reachable

nonexistence of isochronous centers in planar polynomial hamiltonian systems of degree four abstract in this paper we study centers of planar polynomial hamiltonian systems and we are interested in the isochronous ones we prove that every center of a polynomial hamiltonian system of degree four that is with its homogeneous part of degree four not identically zero is nonisochronous the proof uses the geometric properties of the period annulus and it requires the study of the hamiltonian systems associated to a hamiltonian function of the form h x y a x b x y c x y 2 d x y 3

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100