time discontinuous stabilized space time finite elements for timoshenko beams stabilized space time e nite element methods for the transient computational analysis of the elastodynamics of timoshenko beams have been developed the underlying time discontinuous galerkin formulation is implicit unconditionally stable higher order accurate and robust the employed interpolations are continuous in space and time inside time slabs but discontinuous in time between adjacent time slabs to suppress spurious numerical oscillations near discontinuities or high gradients galerkin least squares stabilization has been applied further improvement of the numerical representation of the transient elastic wave propagation phenomena has been achieved by specially designed galerkin gradient least squares operators because of the reduced phase and amplitude errors of these e nite elements much coarser spatial meshes may be used without loss of accuracy the resulting reduction of the number of unknowns leads to a signie cant decrease of computer time furthermore a globaladaptivetime stepping strategy has beenemployed basedon thetemporal jump residual of thetimee niteelement method as error estimator numerical examples of transientelastic wave propagation in timoshenko beams involving a wide spectrum of wave numbers and frequencies demonstrate the good performance of the developed methods

dynamics of a predator prey model we describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator prey model proposed by r m may stability and complexity in model ecosystems princeton university press princeton nj 1974 in particular we give a qualitative description of the bifurcation curve when two limit cycles collapse on a semistable limit cycle and disappear moreover we show that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator prey models

positive steady states of the holling tanner prey predator model with diffusion this paper is concerned with the holling tanner prey predator model with diffusion subject to the homogeneous neumann boundary condition we obtain the existence and non existence of positive non constant steady states

bi discontinuous time step integration algorithms part 2 second order equations abstract in this paper bi discontinuous time step integration algorithms for linear second order equations are investigated the second order equations are manipulated directly both the initial and final displacement and velocity are only weakly enforced at the beginning and at the end of the time interval as a result there may be discontinuous jumps for both the displacement and velocity at the two ends of the time interval if the displacement is assumed to be a polynomial of degree n 1 within the time interval there are n 4 unknowns in the formulation it is found that the order of accuracy would be at least n 1 for displacement and n for velocity in general it is shown that the relevant algorithmic parameters for the second order equations can be related to the algorithmic parameters presented in part 1 of this paper for the first order equations unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation equivalent to the generalized pade approximations with an order of accuracy 2 n 3 can then be constructed systematically the weighting parameters and the corresponding weighting functions are given explicitly it is also shown that the accuracy of the particular solutions is compatible with the accuracy of the homogeneous solutions if the proposed weighting functions are employed

third order complex time step methods for transient analysis abstract using complex time steps and the method for the first order differential equations third order accurate a stable step by step time integration algorithms with controllable numerical dissipation are constructed for transient analysis the results at two sub step locations are evaluated and combined linearly with the corresponding weighting factors to give higher order accurate results at the end of a time step the sub step locations and the weighting factors are complex numbers the amplification factors at infinity are controlled by another algorithmic parameter a fourth order accurate algorithm is obtained when 1 the algorithmic characteristics and the computational procedure for the present third order algorithms are discussed equivalent algorithms and comparisons with other algorithms are given the present algorithms are found to be competitive with the other third and fourth order accurate algorithms

higher order accurate time step integration algorithms by post integration techniques in this paper a framework to construct higher order accurate time step integration algorithms based on the post integration techniques is presented the prescribed initial conditions are naturally incorporated in the formulations and can be strongly or weakly enforced the algorithmic parameters are chosen such that unconditionally a stable higher order accurate time step integration algorithms with controllable numerical dissipation can be constructed for linear problems besides it is shown that the order of accuracy for non linear problems is maintained through the relationship between the present formulation and the runge kutta method the second order differential equations are also considered numerical examples are given to illustrate the validity of the present formulation copyright 2001 john wiley sons ltd

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

mixed finite element formulation for the solution of parabolic problems a mixed formulation of the finite element method is used to establish a higher order incremental method for the solution of parabolic equations the displacement and velocity fields in the terminology of solid mechanics are approximated independently in time using a hierarchical adaptive basis three time bases are tested namely polynomial radial and wavelet bases the time approximation criterion ensures stability and preserves parabolicity the resulting discretization in the time domain leads to an uncoupled governing system in the space dimension this system can be subsequently solved using the alternative formulations for the finite element and boundary element methods the application of the conventional conform displacement formulation of the finite element method is illustrated particular attention is given to the extension of pian and trefftz type hybrid stress finite elements for transient analysis in the time domain the performance of the coupled time space integration method is assessed and the results obtained are compared with the results reported in the literature for alternative time integration procedures

a comprehensive unified set of single step algorithms with controllable dissipation for dynamics part ii algorithms and analysis in part i a comprehensive unified set of single step algorithms with controllable dissipation based on the generalized galerkin formulation was presented in this paper the details of four lower order algorithms using parametrized temporal finite element are presented the analyses of the numerical stability dissipation and dispersion are also provided in comparison with the established algorithms the present algorithms facilitate controllable dissipation and yield results of higher order accuracy examples are also included to illustrate the validity of the algorithms

pattern formation of a spatial predator prey system abstract there are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors to investigate how these migration directed movement and diffusion random movement affect predator prey systems we have studied the spatiotemporal complexity in a predator prey system with holling tanner form a theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated at the same time we present the properties of pattern solutions the results of numerical simulations show that migration has prominent effect on the pattern formation of the population i e changing turing pattern into traveling pattern this study suggests that modelling by migration and diffusion in predator prey systems can account for the dynamical complexity of ecosystems

improving the variational path integral approach to the quantum double well potential an improved variational path integral approach is developed and applied to the quantum double well potential in which part of the quartic term of the potential is included in the trial action the expression of the effective classical potential ecp under a non gaussian expectation is obtained here the frequency and fourth order derivative of the potential are treated as two variational parameters determined by the minimization of the ecp at each point we calculate the ecp the free energy and the level splitting of a symmetrical double well potential it is shown that the present results are better than those of the feynman kleinert gaussian variational method

explicit predictor multicorrector time discontinuous galerkin methods for non linear dynamics abstract explicit predictor multicorrector time discontinuous galerkin tdg methods developed for linear structural dynamics are formulated and implemented in a form suitable for arbitrary non linear analysis of structural dynamics problems the formulation is intended to inherit the accuracy properties of the exact parent implicit tdg methods to this end suitable predictors and correctors are designed to achieve third order accuracy large stability limits and controllable numerical dissipation by means of an algorithmic parameter as the study of a general non linear case is rather complex the analysis of the convergence properties of the resulting algorithms are restricted to conservative duffing oscillators for which closed form solutions are available it is shown that the main properties of the underlying parent scheme can be retained finally results of representative numerical simulations relevant to duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical schemes and confirm the analytical estimates

an efficient time discontinuous galerkin procedure for non linear structural dynamics this paper presents an effective time discontinuous galerkin procedure for non linear dynamics the procedure is based on a cheap iterative algorithm which is rather different from previous efforts in fact the algorithm is designed so that the corrected solutions inherit the desired stability and dissipative properties and the iterations serve only to improve accuracy some numerical tests illustrate the good performance of the present procedure which appears to be competitive with the available implementations of time discontinuous galerkin methods

time discontinuous galerkin methods with energy decaying correction for non linear elastodynamics in this paper a new time discontinuous galerkin tdg formulation for non linear elastodynamics is presented the new formulation embeds an energy correction which ensures truly energy decaying thus allowing to achieve unconditional stability that as shown in the paper is not guaranteed by the classical tdg formulation the resulting method is simple and easily implementable into existing finite element codes moreover it inherits the desirable higher order accuracy and high frequency dissipation properties of the classical formulation numerical results illustrate the very good performance of the proposed formulation copyright 2010 john wiley sons ltd

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

a strongly coupled predator prey system with modified holling tanner functional response in this paper a strongly coupled system of partial differential equations in a bounded domain with the homogeneous neumann boundary condition which models a predator prey system with modified holling tanner functional response is considered first the authors study the stability of the positive constant solution sufficient conditions are derived for the global stability of the positive equilibrium by constructing a suitable lyapunov function by using the leray schauder theorem the authors prove a number of existence and non existence results about the non constant steady states of the system

unconditionally stable higher order newmark methods by sub stepping procedure abstract in this paper a procedure to construct unconditionally stable higher order step by step time integration algorithms based on the newmark method is presented the second order accurate newmark method is extended to third fourth and even higher order by linearly combining the numerical results evaluated at some specific sub step locations the weighting factors and the sub step locations are algorithmic parameters they are chosen to eliminate the leading truncation error terms and to control the stability characteristics the ranges of parameters for unconditionally stable third and fourth order algorithms are given besides by using complex time steps a third order l stable and a fourth order a stable non dissipative algorithms are derived furthermore the accuracy of the excitation responses is shown to be improved by the present procedure as well to maintain high order accuracy the excitation may need to be modified the computational procedure is discussed it is found that there are no changes in the implementation of the underlying newmark method and no extra computations of high order gradients as the numerical results at the sub step locations are obtained independent of one another the evaluations can be computed in parallel comparisons with other higher order algorithms are presented

dynamics and patterns of a diffusive leslie gower prey predator model with strong allee effect in prey abstract this paper is devoted to study the dynamical properties and stationary patterns of a diffusive leslie gower prey predator model with strong allee effect in the prey population we first analyze the nonnegative constant equilibrium solutions and their stabilities and then study the dynamical properties of time dependent solutions moreover we investigate the stationary patterns induced by diffusions turing pattern our results show that the impact of the strong allee effect essentially increases the system spatiotemporal complexity

strategy and stationary pattern in a three species predator prey model abstract in this paper we study a strongly coupled system of partial differential equations which models the dynamics of a two predator one prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey s strategy we demonstrate the emergence of stationary patterns for this system and show that it is due to the cross diffusion that arises naturally in the model as far as the authors are aware this is the first example of stationary patterns in a predator prey system arising solely from the effect of cross diffusion

multiple coexistence states for a prey predator system with cross diffusion abstract we study the multiple existence of positive solutions for the following strongly coupled elliptic system 1 v u u a u cv 0 in 1 u v v b du v 0 in u v 0 on where a b c d are positive constants and is a bounded domain in r n this is the steady state problem associated with a prey predator model with cross diffusion effects and u resp v denotes the population density of preys resp predators in particular the presence of represents the tendency of predators to move away from a large group of preys assuming that is small and that is large we show that the system admits a branch of positive solutions which is s or shaped with respect to a bifurcation parameter so that the system has two or three positive solutions for suitable range of parameters our method of analysis uses the idea developed by du lou j differential equations 144 1998 390 and is based on the bifurcation theory and the lyapunov schmidt procedure

weighting parameters for time step integration algorithms with predetermined coefficients in this paper the effect of using the predetermined coefficients in constructing time step integration algorithms is investigated both first and second order equations are considered the approximate solution is assumed to be in a form of polynomial in the time domain it can be related to the truncated taylor s series expansion of the exact solution therefore some of the coefficients can be predetermined from the known initial conditions if there are m predetermined coefficients and r unknown coefficients in the approximate solution the unknowns can be solved by the weighted residual method the weighting parameter method is used to investigate the resultant algorithm characteristics it is shown that the formulation is consistent with a minimum order of accuracy m r the maximum order of accuracy achievable is m 2r unconditionally stable algorithms exist if m r for first order equations and m 1 r for second order equations hence the dahlquist s theorem is generalized algorithms equivalent to the pade approximations and unconditionally stable algorithms equivalent to the generalized pade approximations are constructed the corresponding weighting parameters and weighting functions for the pade and generalized pade approximations are given explicitly copyright 2000 john wiley sons ltd

stationary patterns of the holling tanner prey predator model with diffusion and cross diffusion this paper is concerned with the holling tanner prey predator model subject to the homogeneous neumann boundary condition we will show that under certain hypotheses even though the unique positive constant steady state is uniformly asymptotically stable for the dynamics with diffusion non constant positive steady states can exist due to the emergence of cross diffusion the biological implication of cross diffusion means that the prey species exercise a self defense mechanism to protect themselves from the attack of the predator our result also demonstrates that cross diffusion can create stationary patterns

a novel time marching scheme using numerical green s functions a comparative study for the scalar wave equation abstract the present paper presents the formulation of a novel time marching method based on the explicit green s approach exga to solve scalar wave propagation problems by means of the weighted residual method in both time and space the time integral expression concerning the exga is readily established the arising exga time integral expression is spatially discretized in a finite element sense and a recursive scheme that employs time domain numerical green s function matrices is adopted to evaluate the displacement and the velocity vectors these green s matrices are computed by the time discontinuous galerkin finite element method only at the first time step the system of coupled equations originated from the time discontinuous galerkin method is then solved by an iterative predictor multicorrector algorithm once the green s matrices are computed no iterative process is required to obtain the displacement and the velocity vectors at any time level at the end of the paper numerical examples are presented in order to compare the proposed approach with other approaches

a particular integral bem time discontinuous fem methodology for solving 2 d elastodynamic problems abstract this study proposes a time discontinuous galerkin finite element method fem for solving second order ordinary differential equations in the time domain the equations are formulated using a particular integral boundary element method bem in the space domain for elastodynamic problems the particular integral bem technique depends only on elastostatic displacement and traction fundamental solutions without resorting to commonly used complex fundamental solutions for elastodynamic problems based on the time discontinuous galerkin fem the unknown displacements and velocities are approximated as piecewise linear functions in the time domain and are permitted to be discontinuous at the discrete time levels this leads to stable and third order accurate solution algorithms for ordinary differential equations numerical results using the time discontinuous galerkin fem are compared with results using a conventional finite difference method the houbolt method both methods are employed for a particular integral bem analysis in elastodynamics this comparison reveals that the time discontinuous galerkin fem is more stable and more accurate than the traditional finite difference methods

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

hierarchical higher order dissipative methods for transient analysis this work focuses on devising an efficient hierarchy of higher order methods for linear transient analysis equipped with an effective dissipative action on the spurious high modes of the response the proposed strategy stems from the norsett idea and is based on a multi stage algorithm designed to hierarchically improve accuracy while retaining the desired dissipative behaviour computational efficiency is pursued by requiring that each stage should involve just one set of implicit equations of the size of the problem to be solved as standard time integration methods and in addition all the stages should share the same coefficient matrix this target is achieved by rationally formulating the methods based on the discontinuous collocation approach the resultant procedure is shown to be well suited for adaptive solution strategies in particular it embeds two natural tools to effectively control the error propagation one estimates the local error through the next stage solution which is one order more accurate the other through the solution discontinuity at the beginning of the current time step which is permitted by the present formulation the performance of the procedure and the quality of the two error estimators are experimentally verified on different classes of problems some typical numerical tests in transient heat conduction and elasto dynamics are presented copyright 2006 john wiley sons ltd

conservation properties of a time fe method part ii time stepping schemes for non linear elastodynamics in the present paper one step implicit integration algorithms for non linear elastodynamics are developed the discretization process rests on galerkin methods in space and time in particular the continuous galerkin method applied to the hamiltonian formulation of semidiscrete non linear elastodynamics lies at the heart of the time stepping schemes algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals we newly introduce the assumed strain method in time which enables the design of energy momentum conserving schemes and which can be interpreted as temporal counterpart of the well established assumed strain method for finite elements in space the numerical examples deal with quasi rigid motion as well as large strain motion copyright 2001 john wiley sons ltd

an efficient hybrid time laplace domain method for elastodynamic analysis based on the explicit green s approach abstract this work aims at presenting a novel hybrid method applied to the numerical solution of elastodynamic problems the formulation is based on the so called explicit green s approach exga recently developed by mansur et al mansur w j loureiro f s soares jr d dors c 2007 explicit time domain approaches based on numerical green s functions computed by finite differences the exga family journal of computational physics 227 851 870 in which a recursive time integration procedure based on green s function is adopted here the finite element method is employed to calculate the green s function in the laplace domain afterwards the zakian s laplace inversion algorithm is employed to compute numerically the green s function in the time domain only at time t t time integration is then carried out by employing the standard exga procedure and another where the convolution integral is replaced by a particular solution that is directly related to the applied load finally some numerical problems are solved to demonstrate the computational efficiency and reliability of the proposed hybrid method

the embedded projection method a general index reduction procedure for constrained system dynamics abstract the embedded projection method epm is an attempt to eliminate most of the difficulties related to the solution of high index differential algebraic equations daes as those typical of the dynamics of constrained systems the intimate coupling between the algebraic and the differential parts lies at the origin of these difficulties therefore we uncouple these components introduce an unconstrained modified state variable and complement the resulting ordinary differential equation ode by additional algebraic variables as a result a general consistent index reduction from arbitrarily high values to one is obtained allowing the use of any suitable ode integration algorithm without the need for a specialized dae solver the epm solution is intrinsically endowed with greater accuracy and stability than conventional counterparts thus balancing the higher complexity of the procedure we illustrate the epm including some applications of holonomic and non holonomic mechanical systems

diffusion has no influence on the global asymptotical stability of the lotka volterra prey predator model incorporating a constant number of prey refuges by constructing a suitable lyapunov function we obtain the global asymptotical stability of the lotka volterra prey predator model incorporating a constant number of prey refuges and diffusion it is shown that diffusion has no influence on the global asymptotical stability of the system with homogeneous neumann boundary condition

the bifurcation structure of the holling tanner model for predator prey interactions using two timing the holling tanner model for predator prey systems has two hopf bifurcation points for certain parameters the dependence of the environmental parameters on the underlying bifurcation structure is uncovered using two timing emphasis is on how the bifurcation diagram changes as the hopf bifurcation points separate two degenerate cases require a modification of conventional two timing when the two hopf bifurcation points nearly coalesce the two stable periodic solution branches are shown to be connected as a ratio of linear growth rates varies the hopf bifurcation points separate further and one limit cycle becomes unstable this situation can correspond to an outbreak in populations the modified two timing analysis analytically captures the unstable and stable limit cycles of the new branch

stationary patterns created by cross diffusion for the competitor competitor mutualist model abstract in this paper we consider a competitor competitor mutualist model with cross diffusion we prove some existence and non existence results concerning non constant positive steady states patterns in particular we demonstrate that the cross diffusion can create patterns when the corresponding model without cross diffusion fails

complex time step newmark methods with controllable numerical dissipation in this paper unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation are presented the algorithms are based on the newmark method with complex time steps the ultimate spectral radius the sub step locations j and the weighting factors j are the algorithmic parameters for an algorithm that is 2n 1 th order accurate the sub step locations which may be complex are shown to be the roots of an nth degree polynomial the polynomial is given explicitly in terms of n and the weighting factors are then obtained by solving a system of n simultaneous equations it is further shown that the order of accuracy is increased by one for the non dissipative algorithms with 1 the stability properties of the present algorithms are studied it is shown that if the ultimate spectral radius is set between 1 and 1 the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity the algorithms are therefore unconditionally c stable when the ultimate spectral radius is set to 0 or 1 the algorithms are found to be equivalent to the first sub diagonal and diagonal pade approximations respectively the present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers the accuracy of the excitation responses is found to be enhanced by the present complex time step procedure to maintain high order accuracy the excitation may need some modifications 1998 john wiley sons ltd

fish hook shaped global bifurcation branch of a spatially heterogeneous cooperative system with cross diffusion abstract in this paper we will consider the following strongly coupled cooperative system in a spatially heterogeneous environment with neumann boundary condition u u u b x v 0 x 1 k x u v v v d x u 0 x u v 0 x where is a bounded domain in r n n 1 with smooth boundary k is a positive constant and are real constants which may be non positive b x 0 and d x 0 are continuous functions in x is a smooth positive function in with x 0 is the outward unit normal vector on and for the case 0 we show that if is small and k is large a spatial segregation of x and b x can cause the positive solution curve to form an unbounded fish hook shaped curve with parameter for the case 0 if is small and k is large and the cooperative effect is strong for species u and very weak for species v then the positive solution set also forms an unbounded fish hook shaped continuum these results are quite different from those of predator prey systems and the cooperative system under dirichlet boundary condition both of which can form a bounded continuum our results deduce that the spatial heterogeneity of environments can produce multiple coexistence states our method of analysis is based on the bifurcation theory and the lyapunov schmidt procedure

collocation methods with controllable dissipation for linear dynamics abstract this paper introduces a unified set of collocation methods for linear dynamics in which a parameter is used to control the position of the collocation points in this manner both a stable algorithms of order 2 p and l stable algorithms of order 2 p 1 are derived from collocation polynomials of degree p in addition algorithms with intermediate accuracy and stability properties are made available the effects of varying the algorithmic parameter are investigated with particular reference to the numerical dissipation of spurious high frequency modes numerical tests are reported which support the theoretical analysis and demonstrate the performance of the proposed algorithms

a unified set of single step asymptotic annihilation algorithms for structural dynamics when solving the equations of structural dynamics using direct time integration meth6ds algorithmic damping is useful in controlling spurious high frequency oscillations ideally an algorithm should possess asymptotic annihilation of the highfrequency response i e spurious oscillations in the high frequencies are eliminated after one time step in this paper a new class of asymptotic annihilation algorithms for structural dynamics is presented that possesses little numerical dissipation in the low frequency regime the algorithms are based upon using finite elements in time the displacement and velocity fields are interpolated independently using time discontinuous functions the equations of motion displacement velocity compatibility and the time continuity of displacement and velocity are satisfied weakly asymptotic annihilation is achieved by choosing the displacement and velocity fields to be of equal order algorithms of any desired order of temporal accuracy can be obtained by appropriate choice of the finite element interpolations in time an analysis of the proposed class of algorithms is given proving the asymptotic annihilation property and the spectral equivalence of the algorithms to the upper diagonal of the pad6 approximation table results from finite difference analyses are presented showing the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low frequency regime

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability

pattern formation of a diffusive predator prey model with strong allee effect and nonconstant death rate abstract this paper is concerned with the pattern formation of a diffusive predator prey model with strong allee effect in the prey population and nonconstant death rate for the predator the necessary and sufficient conditions of turing instability are explicitly obtained by regarding the diffusion coefficient d 2 of the predator as bifurcation parameter we demonstrate that at the critical value of the bifurcation parameter d 2 a turing bifurcation occurs i e a pattern arises the conditions for the stability of the pattern are also derived in detail our results show that strong allee effect in our model plays a crucial role in the formation of stationary pattern which is a strong contrast to the case without strong allee effect

a comprehensive unified set of single step algorithms with controllable dissipation for dynamics part i formulation a new class of time integration algorithms for dynamics is presented the algorithms are based on finite element in time the equations of motion are expressed in first order form and the variables are interpolated via time polynomials the unknown nodal variables are solved by the generalized galerkin method algorithms of any desired order of time accuracy can be achieved by proper choice of the finite element interpolation in time the salient feature of the algorithms is that it can be either asymptotic annihilating or non dissipative besides any desired algorithmic damping between these two extreme cases could be obtained the algorithms are analysed in detail the spectral characteristics equivalent to the upper diagonal diagonal of the pade approximations and other rational approximations are given