unilateral global bifurcation for a class of quasilinear elliptic systems and applications abstract in this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled extending the bifurcation theorem of 1 to this aim we use several results such that fredholm operator of index 0 theory eigenvalues of elliptic operators and the krein rutman theorem lastly we apply this result to some particular systems arising from population dynamics and determine a region of existence of coexistence states

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

the dual variable method for finite element discretizations of navier stokes equations the numerical solution of two dimensional transient incompressible navier stokes problems is considered the dual variable method originally developed in the context of a finite difference discretization is a technique to considerably reduce the size of the linear system to be solved at each time step the steps involved in the method are 1 the determination of the rank of the discrete divergence operator a 2 the determination of a basis for the null space of a n a and 3 the calculation of a particular solution of the discrete continuity equation a finite element implementation of the method is presented using quadrilateral piecewise bilinear velocity constant pressure elements algorithms for the determination of a basis for n a and a particular solution are presented numerical comparisons of primitive versus dual variable formulations on several problems demonstrate the advantage of the dual variable method in terms of both execution speed and memory requirements

iterative solution of navier stokes dual variable difference equations abstract the dual transformation applied to implicit finite difference approximations of the navier stokes equations reduces the number of unknowns by a factor of three removes the pressures from the discrete equations and produces velocities which satisfy the discrete continuity equation exactly new iterative methods for the solution of the unsymmetrical dual variable system are developed and are proven to converge for a large class of problems these iterative methods involve a sequence of discrete laplacian systems whose solutions converge to the solution of the dual variable system they take advantage of the special structure of the dual variable coefficient matrix are very fast compared to the direct methods currently used are less memory intensive and can be more easily vectorized and parallelized

on the indices of fixed points of mappings in cones and applications abstract assume that k is a cone in a banach space and a k k is completely continuous we obtain a formula for the index in k of a fixed point of a under the assumption that a linearization exists and satisfies an invertibility condition we then use this formula to generalize some results of amann on the number of fixed points of a to the case where k has empty interior

unified formulation of the momentum weighted interpolation for collocated variable arrangements abstract momentum weighted interpolation mwi is a widely used discretisation method to prevent pressure velocity decoupling in simulations of incompressible and low mach number flows on meshes with a collocated variable arrangement despite its popularity a unified and consistent formulation of the mwi is not available at present in this work a discretisation procedure is devised following an in depth analysis of the individual terms of the mwi derived from physically consistent arguments based on which a unified formulation of the mwi for flows on structured and unstructured meshes is proposed including extensions for discontinuous source terms in the momentum equations as well as discontinuous changes of density as shown by the presented analysis and numerical results the mwi enforces a low pass filter on the pressure field which suppresses oscillatory solutions furthermore the numerical dissipation of kinetic energy introduced by the mwi is shown to converge with third order in space and is independent of the time step if the mwi is derived consistently from the momentum equations in the presence of source terms the low pass filter on the pressure field can be shaped by a careful choice of the interpolation coefficients to ensure the filter only acts on the driving pressure gradient that is associated with the fluid motion which is shown to be vitally important for the accuracy of the numerical solution to this end a force balanced discretisation of the source terms is proposed that precisely matches the discretisation of the pressure gradients and preserves the force applied to the flow using representative test cases of incompressible and low mach number flows including flows with discontinuous source terms and two phase flows with large density ratios the newly proposed formulation of the mwi is favourably compared against existing formulations and is shown to significantly reduce or even eliminate solution errors with an increased stability for flows with large density ratios

positive steady state solutions of a leslie gower predator prey model with holling type ii functional response and density dependent diffusion abstract in this paper we consider a leslie gower predator prey model with holling type ii functional response and density dependent diffusion under zero dirichlet boundary condition using degree theory bifurcation theory energy estimates and asymptotical behavior analysis the existence and multiplicity of positive steady state solutions were shown under certain conditions on the parameters

conservation properties of unstructured staggered mesh schemes classic cartesian staggered mesh schemes have a number of attractive properties they do not display spurious pressure modes and they have been shown to locally conserve mass momentum kinetic energy and circulation to machine precision recently a number of generalizations of the staggered mesh approach have been proposed for unstructured triangular or tetrahedral meshes these unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic cartesian mesh method this work addresses the momentum kinetic energy and circulation conservation properties of unstructured staggered mesh methods it is shown that with certain choices of the velocity interpolation unstructured staggered mesh discretizations of the divergence form of the navier stokes equations can conserve kinetic energy and momentum both locally and globally in addition it is shown that unstructured staggered mesh discretizations of the rotational form of the navier stokes equations can conserve kinetic energy and circulation both locally and globally the analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation these novel conserving unstructured staggered mesh schemes have not been previously analyzed it is shown that they are first order accurate on nonuniform two dimensional unstructured meshes and second order accurate on uniform unstructured meshes numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented

an application of network theory to the solution of implicit navier stokes difference equations abstract many natural implicit discretizations of the navier stokes equations can with the proper identifications be regarded as systems defining flows on associated networks in this paper we describe a method which through the introduction of a different set of network variables significantly reduces the size of the original system the method avoids the need to compute pressures and produces velocities that are exactly discrete divergence free we illustrate the technique by applying it to the implicit finite difference equations of krzhivitski and ladyzhenskaya and an implicit mac like finite difference system

the null space and non conventional basis functions in the mixed finite element method the two matrix equations obtained from the mixed finite element method are uncoupled through the use of the qr decomposition the procedure obtains the basis for the null space which is composed of solutions to the homogeneous flux equilibrium equation the result is that even for statically indeterminate problems stresses can be obtained without solving for the displacement field furthermore the stress field can be decomposed directly into contributions from applied forces and from prescribed displacements the appearance of kinetic modes can easily be monitored the possibility of utilizing non conventional basis functions for use in conjunction with the qr decomposition is explored briefly simple examples are given to illustrate the theoretical results

uniform holder bounds for a strongly coupled elliptic system with strong competition abstract this article is concerned with a strongly coupled elliptic system modeling the steady state of populations that compete in some region we prove that the solutions are uniformly holder bounded as the competition rate tends to infinity the proof relies on the blow up technique and the monotonicity formula

an unconditionally stable convergent finite difference method for navier stokes problems on curved domains a new finite difference scheme is presented for solving two dimensional transient incompressible navier stokes problems on a bounded simply connected region for which there exists a c 2 invertible mapping onto the unit square the method is proven to be unconditionally stable and convergent and reduces to the well known krzhivitski and ladyzhenskaya scheme for rectangular domains

coexistence and stability of a competition diffusion system in population dynamics abstract the coexistence and stability of the population densities of two competing species in a bounded habitat are investigated in the present paper where the effect of dispersion transportation is taken into consideration the mathematical problem involves a coupled system of lotka volterra type reaction diffusion equations together with some initial and boundary conditions including the dirichlet neumann and third type necessary and sufficient conditions for the coexistence and competitive exclusion are established and the effect of diffusion is explicitly given for the stability problem general criteria for the stability and instability of a steady state solution are established and then applied to various situations depending on the relative magnitude among the physical parameters also given are necessary and sufficient conditions for the existence of multiple steady state solutions and the stability or instability of each of these solutions special attention is given to the neumann boundary condition with respect to which some threshold results for the coexistence and stability or instability of the four uniform steady states are characterized it is shown in this situation that only one of the four constant steady states is asymptotically stable while the remaining three are unstable the stability or instability of these states depends solely on the relative magnitude among the various rate constants and is independent of the diffusion coefficients

an initial boundary value problem for a certain density dependent diffusion system on considere le systeme parabolique quasilineaire u t x c 1 d 1 v v e 1 a 1 v b 1 u u v t x c 2 d 2 u v e 2 a 2 u b 2 o v dans o t u x s 0 v x s sur o t u x o w 1 x v x o w 2 x sur ou r n est un domaine borne a frontiere lisse

convex control systems and convex optimal control problems with constraints this note discusses the concepts of convex control systems and convex optimal control problems we study control systems governed by ordinary differential equations in the presence of state and target constraints our note is devoted to the following main question under which additional assumptions is a sophisticated constrained optimal control problem equivalent to a simple convex minimization problem in a related hilbert space we determine some classes of convex control systems and show that for suitable cost functionals and constraints optimal control problems for these classes of systems correspond to convex optimization problems the latter can be reliably solved using standard numerical algorithms and effective regularization schemes in particular we propose a conceptual computational approach based on gradient type methods and proximal point techniques

analysis and convergence of a covolume method for the generalized stokes problem we introduce a covolume or mac like method for approximating the generalized stokes problem two grids are needed in the discretization a triangular one for the contimlity equation and a quadrilateral one for the momentum equation the velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants error in the l 2 norm for the pressure and error in a mesh dependent h 1 norm as well as in the l 2 norm for the velocity are shown to be of first order provided that the exact velocity is in h 2 and the true pressure in h i we also introduce the concept of a network model into the discretized linear system so that an efficient pressure recovering technique can be used to simplify a great deal the computational work involved in the augmented lagrangian method given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle point problem

coexistence states of a nonlinear lotka volterra type predator prey model with cross diffusion abstract in this paper a lotka volterra type predator prey elliptic system with nonlinear cross diffusion is considered representing the tendency of predators to get closer to prey under robin boundary condition the necessary and sufficient conditions for the existence of coexistence states are given in terms of principal eigenvalues involving semi trivial solutions using the degree theory this paper also discusses the ecological meaning and effect of decreasing cross diffusion induced on predators by prey in a specific model

coexistence of a general elliptic system in population dynamics this paper is concerned with a strongly coupled elliptic system representing a competitiveinteraction between two species we give a sufficient condition for the existence of positive solutions an example is also given to show that there is a coexistence of a steady state if the cross diffusions or the intraspecific competitions are weak

global and blow up solutions for nonlinear degenerate parabolic systems with crosswise diffusion abstract in this paper the nonlinear degenerate parabolic system u t v 1 u xx au v t u 2 v xx bv with dirichlet boundary condition is studied the regularization method and upper lower solutions technique are employed to show the local existence of a solution for the nonlinear degenerate parabolic system the global existence of a solution is discussed the finite time blow up result together with an estimate of the blow up time are found the blow up set with positive measure is analyzed in some detail

on computational procedures for the force method it is known that the matrix force method has certain advantages over the displacement method for a class of structural problems it is also known that the force method when carried out by the conventional gauss jordan procedure tends to fill in the problem data making the method unattractive for large size sparse problems this poor fill in property however is not necessarily inherent to the method and the sparsity may be maintained if one uses what we call the turn back lu procedure the purpose of this paper is two fold first it is shown that there exist some close relationships between the force method and the least squares problem and that many existing algebraic procedures to perform the force method can be regarded as applications extensions of certain well known matrix factorization schemes for the least squares problem secondly it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint included in this unification is the turn back lu procedure which was originally proposed by topcu in his thesis 8 it is explained why this procedure tends to produce sparse and banded self stress and flexibility matrices with small band width some computational results are presented to demonstrate the superiority of the turn back lu procedure over the other schemes considered in this paper

the spatial behavior of a strongly coupled non autonomous elliptic system abstract this article is concerned with the spatial behavior of a strongly coupled non autonomous elliptic system modeling the steady state of populations that compete in some region as the competition rate tends to infinity we obtain the uniform convergence result and prove that non negative solution of the system converges to the positive and negative parts of a solution of a scalar limit problem

cavity flow dynamics at higher reynolds number and higher aspect ratio abstract a primitive variable unsteady viscous incompressible navier stokes flow simulation in a higher aspect ratio a depth width 2 driven cavity at a higher reynold s number re 10 000 exhibits dynamical features not apparent in previous studies of stationary solutions or at lower reynold s number or in the more usual unit cavity investigations the initial dynamics e g between t 0 sec and t 70 sec reveals transient bifurcations between states including two three and four interacting vortices in an anterior separation region following this there is an intermediate interval e g between t 70 sec and t 140 sec characterized by a vertical oscillation of the primary vortex related to the general activities of the secondary features of the flow the long time behavior e g between t 140 seconds and t 360 seconds is one of qualitative smoothing of all secondary features except a persistent oscillation indicating a hopf bifurcation

a lotka volterra symbiotic model with cross diffusion abstract the main goal of this paper is to study the existence and non existence of coexistence states for a lotka volterra symbiotic model with cross diffusion we use mainly bifurcation methods and a priori bounds to give sufficient conditions in terms of the data of the problem for the existence of positive solutions we also analyze the profiles of the positive solutions when the cross diffusion parameter goes to infinity

cross diffusion induced turing instability for a three species food chain model abstract in this paper we study a strongly coupled reaction diffusion system describing three interacting species in a food chain model where the third species preys on the second one and simultaneously the second species preys on the first one we first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ode system the positive equilibrium solution remains linearly stable for the reaction diffusion system without cross diffusion hence it does not belong to the classical turing instability scheme we further proved that the positive equilibrium solution is globally asymptotically stable for the reaction diffusion system without cross diffusion by constructing a lyapunov function but it becomes linearly unstable only when cross diffusion also plays a role in the reaction diffusion system hence the instability is driven solely from the effect of cross diffusion our results also exhibit some interesting combining effects of cross diffusion intra species competitions and inter species interactions

refuge versus dispersion in the logistic equation abstract in this paper we consider a logistic equation with nonlinear diffusion arising in population dynamics in this model there exists a refuge where the species grows following a malthusian law and in addition there exists also a non linear diffusion representing a repulsive dispersion of the species we prove existence and uniqueness of positive solution and study the behavior of this solution with respect to the parameter the growth rate of the species mainly we use bifurcation techniques the sub supersolution method and a construction of appropriate large solutions

accuracy and conservation properties of a three dimensional unstructured staggered mesh scheme for fluid dynamics a three dimensional unstructured mesh discretization of the rotational from of the incompressible navier stokes is presented the method uses novel and highly efficient algorithms for interpolating the velocity vector and constructing the convention term the resulting discretization is shown to conserve mass kinetic energy and vorticity to machine precision both locally and globally the spatial accuracy of the method is analyzed and found to be second order on regular meshes and first order on irregular meshes the numerical efficiency accuracy and conservation properties of the method are tested on three dimensional meshes and found to be in agreement with theory

a central difference scheme for a pure stream function formulation of incompressible viscous flow we present a numerical scheme for incompressible viscous flow formulated as an equation for the stream function the pure stream function formulation obviates the difficulty associated with vorticity boundary conditions the resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver the advection of vorticity is implemented with a high resolution central scheme that remains stable and accurate in the presence of large gradients the accuracy and robustness of the method are demonstrated for high reynolds number flows in a lid driven cavity

parallel algorithm for solving linear systems arising from pde and integral equations in this paper we consider solving matrix systems arising from the linear integral equations and pde by two recent parallel techniques the systems are indefinite due to linear constraints imposed on the fluid velocity the first approach known as the multilevel algorithm employs a hierarchical technique to compute the constrained linear space for the unknowns followed by the iterative solution of a positive definite reduced problem the second approach exploits the banded structure of sparse matrices to obtain a different reduced system which is determined by the unknowns common to adjacent block rows although the reduced system in this approach may still be indefinite the algorithm converges to the solution at an accelerated rate these methods have two desirable characteristics namely robust numerical convergence and efficient parallelizability

the construction of an optimal weakly divergence free macroelement the divergence free finite element method dffem is a method to find an approximate solution of the navier stokes equations in a divergence free space that is the continuity equation is satisfied a priori dffem eliminates the pressure from the calculations and reduces significantly the dimension of the system to be solved at each time step for the standard 8 node velocity and 4 node pressure dffem an optimal basis for the weakly divergence free subspace is constructed such that each basis function has non zero support on at most nine contiguous elements given this basis weakly divergence free macroelements are constructed

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

hopf bifurcation in the driven cavity abstract incompressible two dimensional calculations are reported for the impulsively started lid driven cavity with aspect ratio two the algorithm is based on the time dependent stream function equation with a crank nicolson differencing scheme for the diffusion terms and with an adams bashforth scheme for the convection terms a multigrid method is used to solve the linear implicit equations at each time step periodic asymptotic solutions have been found for re 10000 and for re 5000 the re 5000 results are validated by grid refinement calculations the solutions are shown to be precisely periodic and care is taken to demonstrate that asymptotic states have been reached a discussion is included about the indicators that are used to show that an asymptotic state has been reached and to show that the asymptotic state is indeed periodic

an iterative method for solving finite difference approximations to the stokes equations a new iterative method is presented for solving finite difference equations which approximate the steady stokes equations the method is an extension of successive over relaxation and has two iteration parameters perturbation methods are used to analyze the iteration matrix sufficient conditions for the convergence of the iterative method are obtained and it is shown that many reasonable finite difference schemes for the stokes equations satisfy these conditions computational examples are given to show the efficiency of the method

global solutions to a class of multi species reaction diffusion systems with cross diffusions arising in population dynamics in this paper an n species strongly coupled cooperating diffusive system is considered in a bounded smooth domain subject to homogeneous neumann boundary conditions employing the method of energy estimates we obtain some conditions on the diffusion matrix and inter specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution the globally asymptotical stability of the constant positive steady state is also discussed as a consequence all the results hold true for multi species lotka volterra type competition model and prey predator model

the construction of a null basis for a discrete divergence operator the divergence free finite element method dffem is a method to find an approximate solution of the navier stokes equations in a divergence free space that is the continuity equation is satisfied a priori dffem eliminates the pressure from the calculations and significantly reduces the dimension of the system to be solved at each time step for the standard 9 node velocity and 4 node pressure dffem a basis for the divergence free subspace is constructed such that each basis function has nonzero support on at most 4 contiguous elements given this basis discretely divergence free macro elements can be constructed and used in the implementation of the dffem

coexistence theorem of steady states for nonlinear self cross diffusion systems with competitive dynamics in this paper we discuss the existence of positive solutions to certain nonlinear elliptic systems representing competitive interaction with self cross diffusions between two species the method employed is the fixed point index theory in a positive cone sufficient conditions for the existence of positive solutions are provided

discrete calculus methods for diffusion a general methodology for the solution of partial differential equations is described in which the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations the fact that the calculus is exact gives these methods the ability to capture the physics of pde systems well the construction of both node and cell based methods of first and second order are described for the problem of unsteady heat conduction though the method is applicable to any pde system the performance of these new methods are compared to classic solution methods on unstructured 2d and 3d meshes for a variety of simple and complex test cases

finding a sparse vector in a subspace linear sparsity using alternating directions is it possible to find the sparsest vector direction in a generic subspace mathcal s subseteq mathbb r p with text dim left mathcal s right n this problem can be considered a homogeneous variant of the sparse recovery problem and finds connections to sparse dictionary learning sparse pca and many other problems in signal processing and machine learning in this paper we focus on a planted sparse model for the subspace the target sparse vector is embedded in an otherwise random subspace simple convex heuristics for this planted recovery problem provably break down when the fraction of nonzero entries in the target sparse vector substantially exceeds o 1 sqrt n in contrast we exhibit a relatively simple nonconvex approach based on alternating directions which provably succeeds even when the fraction of nonzero entries is omega 1 to the best of our knowledge this is the first practical algorithm to achieve linear scaling under the planted sparse model empirically our proposed algorithm also succeeds in more challenging data models e g sparse dictionary learning

on positive solutions of some pairs of differential equations this paper is a sequel to the author s paper 6 which studied positive solutions of a pair of differential equations introduced by conway gardner and smoller 4 their equations were equations for population problems in biology their significance is discussed further in 141 the purpose of the present paper is to obtain further results for the conwat gardner smoller equations and to show how the methods in 6 can be modified to decide exactly when the classical predator prey system as in blat and brown 2 has a strictly positive solution we also show how the methods can be applied to a number of other problems in particular our results answer some questions in conway s secondly we show how the asymptotic methods developed in 6 can be used to study the existence and uniqueness of strictly positive solutions of competing species problems our results suggest that this is a much more complicated problem than the predator prey problem we do obtain a condition which is except for a few special cases necessary and sufficient for the existence of a strictly positive solution however the condition is complicated and rather implicit thirdly we prove the uniqueness of the strictly positive solution in the conway smoller system if d is small and n 1 that is the case of ordinary differential equations this partially answers a question in 4 our proof is a local result and can be used in other situations for example it can be used to show that in certain cases of asocial nonlinearities where our notation follows 4 sect 4 there are exactly two strictly positive solutions for small d we also show how our method applies to more general predator prey problems in section 1 we find necessary and sufficient conditions for the existence of strictly positive solutions of predator prey systems in section 2 we study competing species models by iteration and asymptotic methods finally in section 3 we prove our uniqueness results in an appendix we briefly discuss how the methods of section 3 apply to more general models

an elliptic cross diffusion system describing two species models on a bounded domain with different natural conditions this paper is concerned with an elliptic cross diffusion system describing two species models on a bounded domain where consists of a finite number of subdomains i i 1 m separated by interfaces j j 1 m 1 and the natural conditions of the subdomains i are different this system is strongly coupled and the coefficients of the equations are allowed to be discontinuous on interfaces j the main goal is to show the existence of nonnegative solutions for the system by schauder s fixed point theorem furthermore as applications the existence of positive solutions for some lotka volterra models with cross diffusion self diffusion and discontinuous coefficients are also investigated

a discrete vector potential model for unsteady incompressible viscous flows abstract a recent approach to generate a zero divergence velocity field by operating directly on the discretized navier stokes equations is used to obtain the decoupling of the pressure from the velocity field by following the methodology suggested by amit hall and porsching the feasibility of treating three dimensional flows and multiply connected domains is analyzed the present model keeps the main features of the classical vector potential method in that it generates a divergence free velocity field through an algebraic manipulation of the discrete equations at the same time the boundary conditions are still imposed on the discrete values of the primitive variables the accuracy of the method is tested against the exact solution for a recirculating unsteady flow both in simply and doubly connected domains several applications to flow fields in three dimensional enclosures or in multiply connected domains are presented and discussed in terms of accuracy and efficiency of the method

brief paper finite horizon dynamic optimization of nonlinear systems in real time a novel scheme for constructing and tracking the solution trajectories to regular finite horizon deterministic optimal control problems with nonlinear dynamics is devised the optimal control is obtained from the states and costates of hamiltonian odes integrated online in the one dimensional case the initial costate is found by successively solving two first order quasi linear partial differential equations whose independent variables are the time horizon duration t and the final penalty coefficient s these pdes should in general be integrated off line the solution rendering not only the missing initial condition sought in the particular t s situation but additional information on the boundary values of the whole two parameter family of control problems which can be used for designing the definitive objective functional optimal trajectories for the model are then generated in real time and used as references to be followed by the physical system numerical improvements are suggested for accurate integration of naturally unstable hamiltonian dynamics and strategies are proposed for tracking their results in finite time or asymptotically when perturbations in the state of the system appear the whole procedure is tested in models arising in aero navigation optimization

a fast direct solver for the biharmonic problem in a rectangular grid we present a fast direct solver methodology for the dirichlet biharmonic problem in a rectangle the solver is applicable in the case of the second order stephenson scheme j w stephenson j comput phys 55 1984 pp 65 80 as well as in the case of a new fourth order scheme which is discussed in this paper and is based on the capacitance matrix method b l buzbee and f w dorr siam j numer anal 11 1974 pp 1136 1150 p bjorstad siam j numer anal 20 1983 pp 59 71 the discrete biharmonic operator is decomposed into two components the first is a diagonal operator in the eigenfunction basis of the laplacian to which the fft algorithm is applied the second is a low rank perturbation operator given by the capacitance matrix which is due to the deviation of the discrete operators from diagonal form the sherman morrison formula g h golub and c f van loan matrix computations 3rd ed john hopkins university press baltimore md 1996 is applied to obtain a fast solution of the resulting linear system of equations

positive steady state solutions of a competing reaction diffusion system abstract this paper is concerned with positive steady state solutions of a coupled reaction diffusion system which models the coexistence problem of two competing species in ecology the main purpose of the paper is to determine the set of natural growth rate r1 r2 of the two competing species so that the coupled system possesses positive solutions it is shown that is a connected unbounded region in r2 whose boundary consists of two monotone nondecreasing curves r1 h2 r2 and r2 h1 r1 for every r1 r2 inside the coupled system has positive solutions and for r1 r2 outside there exists no positive solution the functions h1 r1 and h2 r2 are constructed in terms of the limit of the corresponding time dependent solution with a specific initial function

global aspect of steady states for competitive diffusive systems with homogeneous dirichlet conditions abstract this paper is concerned with the competitive diffusive systems u t d l u xx a l b l u c l v u v t d 2 v xx a 2 b 2 u c 2 u v x t 0 1 0 with homogeneous dirichlet boundary conditions from a global bifurcation view point the dependency of steady states on the parameters a i b i c i and d i i 1 2 is investigated in particular bifurcation of coexist ence solutions amd their stabilities are our main interests

divergence free bases for finite element schemes in hydrodynamics in a number of finite element schemes satisfying a divergence free condition a difficulty has been the lack of explicit bases or even their dimension in the present paper we employ graph theory to calculate the dimensions of these vector spaces and to obtain bases for them

unsteady solution of incompressible navier stokes equations abstract a new time accurate finite difference numerical method for solving incompressible navier stokes equations is presented with primitive variables as the unknowns the numerical scheme is a crank nicolson implicit treatment of all terms of the equations with central differencing for space derivatives the pressure and the nonlinearities in convection terms are not lagged to obtain the incompressible solution at the advanced time level a continuous auxiliary system is introduced in pseudo time with artificial compressibility time accurate solutions are presented for two dimensional fluid flows in a square cavity with an impulsively starting lid and with an oscillating lid

the derivation of minimal support basis functions for the discrete divergence operator the divergence free finite element method dffem is a method to find an approximate solution of the navier stokes equations in a divergence free space that is the continuity equation is satisfied a priori dffem eliminates the pressure from the calculations and significantly reduces the dimension of the system to be solved at each time step for a low order and nonconforming element we construct a basis for the divergence free subspace such that basis functions have smallest nonzero support

galerkin methods for the time dependent incompressible elasticity equations abstract we consider semidiscrete and single step fully discrete approximations to the solutions of the time dependent incompressible elasticity equations two methods of dealing with the constraint are analysed a a formulation involving a discrete form of incompressibility condition and b a penalty formulation h1 and l2 error estimates are obtained

an absolutely stable weak galerkin finite element method for the darcy stokes problem abstract in this paper we apply the weak galerkin wg finite element method to the darcy stokes equations this method provides accurate approximations for the velocity and the pressure variables general polygonal or polyhedral partitions can be applied in this method the finite element space which is made up of piecewise polynomials is easy to be constructed these advantages make the weak galerkin finite element method efficient and highly flexible optimal rates of convergence for the velocity function u and the pressure function p are established in corresponding norms in addition the convergence rates are independent of the viscosity parameter several numerical experiments are provided to illustrate the robustness flexibility and validity of the weak galerkin finite element method

an algorithm to compute a sparse basis of the null space leta be a realm n matrix with full row rankm in many algorithms in engineering and science such as the force method in structural analysis the dual variable method for the navier stokes equations or more generally null space methods in quadratic programming it is necessary to compute a basis matrixb for the null space ofa hereb isn r r n m of rankr withab 0 in many instancesa is large and sparse and often banded the purpose of this paper is to describe and test a variation of a method originally suggested by topcu and called the turnback algorithm for computing a banded basis matrixb two implementations of the algorithm are given one using gaussian elimination and the other using orthogonal factorization by givens rotations the fortran software was executed on an ibm 3081 computer with an fps 164 attached array processor at the triangle universities computing center and on a cyber 205 vector computer test results on a variety of structural analysis problems including two and three dimensional frames plane stress plate bending and mixed finite element problems are discussed these results indicate that both implementations of the algorithm yielded a well conditioned banded basis matrixb whena is well conditioned however the orthogonal implementation yielded a better conditionedb for large illconditioned problems

limiting structure of steady states to the lotka volterra competition model with large diffusion and advection abstract this paper is concerned with the neumann problem of a stationary lotka volterra competition model with diffusion and advection first we obtain some sufficient conditions of the existence of nonconstant solutions by the leray schauder degree theory next we derive a limiting system as diffusion and advection of one of the species tend to infinity the limiting system can be reduced to a semilinear elliptic equation with nonlocal constraint in the simplified 1d case the global bifurcation structure of nonconstant solutions of the limiting system can be classified depending on the coefficients for example this structure involves a global bifurcation curve which connects two different singularly perturbed states boundary layer solutions and internal layer solutions our proof employs a levelset analysis for the associate integral mapping

positive steady state solutions of a competing reaction diffusion system with large cross diffusion coefficients abstract this paper deals with a coupled system of reaction diffusion equations modeling the competition of two species with cross diffusion we discuss the existence of positive steady state solutions in relation to the intrinsic birth rates a 1 a 2 of species and the cross diffusion coefficients 12 21 our results show that positive steady state solutions exist if a 1 and a 2 lie in certain range or if 12 and 21 are sufficiently large

positive solutions of a lotka volterra competition model with cross diffusion abstract this paper concerns a lotka volterra competition reaction diffusion system with nonlinear diffusion effects we first briefly discuss the stability of trivial solutions by spectrum analysis based on the boundedness of the solutions the existence of the positive steady state solution is also investigated by the monotone iteration method

high accuracy difference schemes for a class of three space dimensional singular parabolic equations with variable coefficients abstract in this article two level implicit difference methods of o k 2 h 4 using 19 spatial grid points for the solution of three space dimensional heat conduction equation and unsteady navier stokes equations in polar coordinates are proposed unconditionally stable adi methods for the solution of the heat conduction equation in polar coordinates are also discussed numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

semi discretization in time and numerical convergence of solutions of a nonlinear cross diffusion population model a positivity preserving numerical scheme for a strongly coupled cross diffusion model for two competing species is presented based on a semi discretization in time the variables are the population densities of the species existence of strictly positive weak solutions to the semidiscrete problem is proved moreover it is shown that the semidiscrete solutions converge to a non negative solution of the continuous system in one space dimension the proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional

a finite difference galerkin formulation for the incompressible navier stokes equations abstract the development of a new computational method for solving the incompressible navier stokes equations in primitive variable form is presented it is found that certain finite difference approximations for these equations can be transformed into an equivalent system which efficiently determines the discrete velocity field and which completely eliminates the pressure two such difference schemes for two dimensional problems are examined and some preliminary numerical results are discussed for the steady driven cavity problem

smooth solutions to a quasi linear system of diffusion equations for a certain population model abstract the system of diffusion equations proposed by kawasaki shigesada and teramoto describes a population model of two competing species with self and cross population pressures the densities of the two species are denoted by u and v in this paper the author studies the initial boundary value problem associated with the neumann boundary condition corresponds to the case where the flux is zero at the boundary many investigators have considered nonlinear diffusion systems arising from various physical and biological problems these equations however have a special structure the highest order derivatives are not coupled or at least the coefficient matrix for the highest order derivatives is positive definite this is not the case for the system and hence some of the techniques which are effective for the system are no longer applicable to our case nevertheless we can still use sobolevski s method see reference 2 to establish the local existence of solutions under the special assumption c1 c2 in we can also prove the global existence of solutions by energy estimates the unusual structure of seems to make it difficult to settle the question of asymptotic stability of solutions

an hybrid finite volume finite element method for variable density incompressible flows this paper is devoted to the numerical simulation of variable density incompressible flows modeled by the navier stokes system we introduce an hybrid scheme which combines a finite volume approach for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint the breakthrough relies on the definition of a suitable footbridge between the two methods through the design of compatibility condition in turn the method is very flexible and allows to deal with unstructured meshes several numerical tests are performed to show the scheme capabilities in particular the viscous rayleigh taylor instability evolution is carefully investigated

positive solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract the aim of this paper is to investigate the existence uniqueness and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions including a system of quasilinear parabolic and ordinary differential equations also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady state solution the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations it is shown that the time dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a porous medium type of problem a heat transfer problem and a two component competition model in ecology these applications illustrate some very interesting distinctive behavior of the time dependent solutions between density independent and density dependent diffusions

construction of null bases for the divergence operator associated with incompressible navier stokes equations the dual variable method dvm and the divergence free finite element method dffem both significantly reduce the dimension of the system to be solved at each time step of discrete transient navier stokes equations we establish an equivalence of these two methods and investigate various approaches to finding sparse bases for the null space of operators associated with each of the methods for the standard eight node velocity and four node pressure dffem a basis for the divergence free subspace is constructed such that each basis function has nonzero support on at most nine contiguous elements

a numerical method for large eddy simulation in complex geometries we discuss the development of a numerical algorithm and solver capable of performing large eddy simulation in the very complex geometries often encountered in industrial applications the algorithm is developed for unstructured hybrid grids is non dissipative yet robust at high reynolds numbers on highly skewed grids simulation results for a variety of flows are presented

an efficient iterative method for the generalized stokes problem the generalized stokes problem which arises frequently in the simulation of time dependent navier stokes equations for incompressible fluid flow gives rise to symmetric linear systems of equations these systems are indefinite due to a set of linear constraints on the velocity causing difficulty for most preconditioners and iterative methods this paper presents a novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method this new method generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite numerical experiments indicating superior convergence compared to existing methods are presented a natural extension of this method to elliptic problems is also proposed along with theoretical bounds on the rate of convergence and results of experiments demonstrating robust and effective preconditioning

minimum norm solutions to linear elastic analysis problems en analyse d elasticite lineaire un probleme consiste a trouver les vecteurs de contraintes et de deformations etant donnes un modele d elements finis d une structure et un systeme de charges externes on montre que ce probleme est un cas particulier du probleme de norme minimum pour des systemes indetermines d equations lineaires

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

positive steady states for a prey predator model with some nonlinear diffusion terms abstract this paper discusses a prey predator system with strongly coupled nonlinear diffusion terms we give a sufficient condition for the existence of positive steady state solutions our proof is based on the bifurcation theory some a priori estimates for steady state solutions will play an important role in the proof

brief paper initial condition of costate in linear optimal control using convex analysis the pontryagin maximum principle is one of the most important results in optimal control and provides necessary conditions for optimality in the form of a mixed initial terminal boundary condition on a pair of differential equations for the system state and its conjugate costate unfortunately this mixed boundary value problem is usually difficult to solve since the pontryagin maximum principle does not give any information on the initial value of the costate in this paper we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality which is often much easier to solve than the original optimal control problem we present that the solution to the dual optimization problem supplements the necessary conditions of the pontryagin maximum principle and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem

coexistence of three species in a strongly coupled elliptic system abstract a strongly coupled elliptic system which describes three interacting species with homogeneous dirichlet boundary conditions is considered it is shown that there is no coexistence state if diffusion rates are strong or if the intrinsic growth rates are slow making use of the schauder fixed point theory we derive some sufficient conditions to have a semi coexistence or a coexistence state for the strongly coupled elliptic problem moreover our results reveal that this problem possesses at least one coexistence state if cross diffusions and intra specific competitions are weak

brief paper optimal control problems with multiple characteristic time points in the objective and constraints in this paper we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points these time points can be either fixed or variable using the control parametrization technique and a time scaling transformation this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems each of these approximate problems can be viewed as a finite dimensional optimization problem new gradient formulae for the cost and constraint functions are derived with these gradient formulae standard gradient based optimization methods can be applied to solve each approximate optimal parameter selection problem for illustration two numerical examples are solved

a new fourth order compact scheme for the navierstokes equations in irregular domains we present a high order finite difference scheme for navierstokes equations in irregular domains the scheme is an extension of a fourth order scheme for navierstokes equations in streamfunction formulation on a rectangular domain ben artzi etal 2010 the discretization offered here contains two types of interior points the first is regular interior points where all eight neighboring points of a grid point are inside the domain and not too close to the boundary the second is interior points where at least one of the closest eight neighbors is outside the computational domain or too close to the boundary in the second case we design discrete operators which approximate spatial derivatives of the streamfunction on irregular meshes using discretizations of pure derivatives in the x y and along the diagonals of the element

time dependent viscous incompressible navier stokes equations the finite difference galerkin formulation and streamfunction algorithms abstract the finite difference galerkin fdg method is extended to time dependent incompressible navier stokes equations two algorithm development examples are given that use a staggered grid and centered differencing scheme for the primitive variables mass balance is used to solve the essential problems associated with applying the fdg method the use of the fdg method with this underlying discretization is shows to be the discrete analog of the continuum manipulations that lead to the fourth order streamfunction equation asymptotic and time evolution results obtained with a crank nicolson adams basforth algorithm are compared with published computations for re 400 1000 and 3200

a robust numerical method for stokes equations based on divergence free h div finite element methods a computational method based on a divergence free h div approach is presented for the stokes equations in this article this method is designed to find velocity approximation in an exact divergence free subspace of the corresponding h div finite element space that is the continuity equation is strongly enforced a priori and the pressure is eliminated from the linear system in calculation a strength of this approach is that the saddle point problem for stokes equations is reduced to a symmetric positive definite problem in a subspace for which basis functions are readily available the resulting discrete system can then be solved by using existing sophisticated solvers the aim of this article is to demonstrate the efficiency and robustness of h div finite element methods for stokes equations the results not only confirm the existing theoretical results but also reveal additional advantages of the method in dealing with discontinuous boundary conditions

coexistence and asymptotic periodicity in a competitor competitor mutualist model in this paper the competitor competitor mutualist three species lotka volterra model is discussed firstly by schauder fixed point theory the coexistence state of the strongly coupled system is given applying the method of upper and lower solutions and its associated monotone iterations the true solutions are constructed our results show that this system possesses at least one coexistence state if cross diffusions and cross reactions are weak secondly the existence and asymptotic behavior of t periodic solutions for the periodic reaction diffusion system under homogeneous dirichlet boundary conditions are investigated sufficient conditions which guarantee the existence of t periodic solution are also obtained

strongly coupled elliptic systems and applications to lotka volterra models with cross diffusion abstract the aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations the existence problem is for nonquasimonotone functions arising in the system while the monotone iterations require some mixed monotone property of these functions applications are given to three lotka volterra model problems with cross diffusion and self diffusion which are some extensions of the classical competition prey predator and cooperating ecological systems the monotone iterative schemes lead to some true positive solutions of the competition system and to quasisolutions of the prey predator and cooperating systems also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems

analysis of a population model with strong cross diffusion in unbounded domains we study a parabolic population model in the full space and prove the global in time existence of a weak solution this model consists of two strongly coupled diffusion equations describing the population densities of two competing species the system features intrinsic growth inter and intra specific competition of the species as well as diffusion cross diffusion and self diffusion and drift terms related to varying environment quality the cross diffusion terms can be large making the system non parabolic for large initial data for the first time solutions in unbounded domains are studied the method of our proof is a combination of a time semi discretization a transformation of the system guaranteeing the positivity of the solution a special entropy symmetrizing the system and compactness arguments 2000 mathematics subject classification 35k55 primary 92d25 35d05 secondary

primitive variable strongly implicit calculation procedure for viscous flows at all speeds a coupled solution procedure is described for solving the compressible form of the time dependent twodimensional navier stokes equations in body fitted curvilinear coordinates this approach employs the strong conservation form of the governing equations but uses primitive variables m v p t rather than the more traditional conservative variables p pw pv et as unknowns a coupled modified strongly implicit procedure cmsip is used to efficiently solve the newton linearized algebraic equations it appears that this procedure is effective for mach numbers ranging from the incompressible limit mx 0 01 to supersonic generally smoothing was not needed to control spatial oscillations in pressure for subsonic flows despite the use of central differences dual time stepping was found to further accelerate convergence for steady flows sample calculations including steady and unsteady low mach number internal and external flows and a steady shock boundarylayer interaction flow illustrate the capability of the present solution algorithm

a covolume method based on rotated bilinears for the generalized stokes problem we introduce a covolume or marker and cell mac method for approximating the generalized stokes problem on an axiparallel domain two grids are needed the primal grid made up of rectangles and the dual grid of quadrilaterals the velocity is approximated by nonconforming rotated bilinear elements with degrees of freedom at midpoints of rectangular elements and the pressure by piecewise constants the error in the velocity in the h 1 h norm and the pressure in the l2 norm are of first order provided that the exact velocity is in h2 and the exact pressure in h1