statistics of shocks in solutions of inviscid burgers equation the purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid burgers equation having a typical realizationb y of the brownian motion as an initial datum this case was proposed and studied numerically in the companion paper by she aurell and frisch the description of the statistics is given in terms of the behavior of the convex hull of the random process w y int limits 0 y b eta eta d eta the hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied

fast and stable explicit operator splitting methods for phase field models numerical simulations of phase field models require long time computations and therefore it is necessary to develop efficient and highly accurate numerical methods in this paper we propose fast and stable explicit operator splitting methods for both one and two dimensional nonlinear diffusion equations for thin film epitaxy with slope selection and the cahn hilliard equation the equations are split into nonlinear and linear parts the nonlinear part is solved using a method of lines together with an efficient large stability domain explicit ode solver the linear part is solved by a pseudo spectral method which is based on the exact solution and thus has no stability restriction on the time step size we demonstrate the performance of the proposed methods on a number of one and two dimensional numerical examples where different stages of coarsening such as the initial preparation alternating rapid structural transition and slow motion can be clearly observed

periodic solutions for a neutral delay hassell varley type predator prey system the main aim of this paper is to discuss the neutral predator prey model with hassell varley type functional response and two time varying delays some new sufficient conditions are obtained for the existence of positive periodic solutions by applying the coincidence degree theorem finally numerical simulations are then carried out as supporting evidences of our analytical results

asymptotic solution of the wang uhlenbeck recurrence time problem a langevin particle is initiated at the origin with positive velocity its trajectory is terminated when it returns to the origin in 1945 wang and uhlenbeck posed the problem of finding the joint probability density function pdf of the recurrence time and velocity naming it the recurrence time problem we show that the short time asymptotics of the recurrence pdf is similar to that of the integrated brownian motion solved in 1963 by mckean we recover the long time t ensuremath 3 2 decay of the first arrival pdf of diffusion by solving asymptotically an appropriate variant of mckean s integral equation

parallel mic 0 preconditioning of 3d elliptic problems discretized by rannacher turek finite elements novel parallel algorithms for the solution of large fem linear systems arising from second order elliptic partial differential equations in 3d are presented the problem is discretized by rotated trilinear nonconforming rannacher turek finite elements the resulting symmetric positive definite system of equations ax f is solved by the preconditioned conjugate gradient algorithm the preconditioners employed are obtained by the modified incomplete cholesky factorization mic 0 of two kinds of auxiliary matrices b that both are constructed as locally optimal approximations of a in the class of m matrices uniform estimates for the condition number k b 1a are derived two parallel algorithms based on the different block structures of the related matrices b are studied the numerical tests confirm theory in that the algorithm scales as o n 7 6 in the matrix order n

stability analysis of hereditary iterative processes by employing the theory of difference inequalities several comparison theorems in the context of vector lyapunov functions are developed these comparison theorems are applied to study various stability properties of the equilibrium state of the hereditary iterative processes examples are given to demonstrate the obtained results the method provides a tool to investigate the fundamental problem of stability versus hereditary effect

numerical methods and applications iterative methods based on linearization for nonlinear elliptic grid systems e g dyakonov how to solve stiff systems of differential equations by explicit methods v i lebedev on numerical methods of solving navier stokes equations in velocity pressurevariables g m kobelkov stiff systems of ordinary differential equations r p fedorenko convergence rate estimates of finite element methods for second order hyperbolic equations a a zlotnik fictitious domain methods and computation of homogenized properties of composites n s bakhvalov and a v knyazev

efficient krylov based exponential time differencing method in application to 3d advection diffusion reaction systems abstract the number of ordinary differential equations generally increases exponentially as the partial differential equation is posed on a domain with more dimensions this is of course the curse of dimensionality for exponential time differencing methods the computational challenge in applying exponential time differencing methods for solving partial differential equations in high spatial dimensions is how to compute the matrix exponential functions for very large matrices accurately and efficiently in this paper our main aim is to design a krylov subspace approximation based locally extrapolated exponential time differencing method and compare its performance in terms of accuracy and efficiency to the already available method in the literature for solving a three dimensional nonlinear advection diffusion reaction systems the fundamental idea of the proposed method is to compute only the action of the matrix exponential on a given state vector instead of computing the matrix exponential itself and then multiplying it with given vector the stability and local truncation error of the proposed method have been examined calculation of local truncation error and empirical convergence analysis indicate the proposed method is second order accurate in time the performance and reliability of this novel method have been investigated by testing it on systems of the three dimensional nonlinear advection diffusion reaction equations and three dimensional viscous nonlinear burgers equation

periodic solutions of periodic delay lotka volterra equations and systems abstract by using the continuation theorem of coincidence degree theory sufficient and realistic conditions are obtained for the existence of positive periodic solutions for both periodic lotka volterra equations and systems with distributed or state dependent delays our results substantially extend and improve existing results

positive periodic solutions for a class of neutral delay gause type predator prey system abstract by using a continuation theorem based on coincidence degree theory we establish easily verifiable criteria for the existence of positive periodic solutions for neutral delay gause type predator prey system x t x t r t a t x t 1 x t 2 t x t y t 1 t y t y t d t h t x t 2 t in addition our results are applicable to neutral delay predator prey systems with different types of functional responses such as holling type ii and ivlev type

canonical euler splitting method for nonlinear composite stiff evolution equations in this paper a new splitting method called canonical euler splitting method ces is constructed and studied which can be used for the efficient numerical solution of general nonlinear composite stiff problems in evolution equations of various type such as ordinary differential equations odes semi discrete unsteady partial differential equations pdes and ordinary or partial volterra functional differential equations vfdes and can significantly improve the computing speed on the basis of ensuring the computing quality stability consistency and convergence theories of this method are established a series of numerical experiments are given which check the efficiency of ces method and confirm our theoretical results

a stabilized explicit lagrange multiplier based domain decomposition method for parabolic problems a fully explicit stabilized domain decomposition method for solving moderately stiff parabolic partial differential equations pdes is presented writing the semi discretized equations as a differential algebraic equation dae system where the interface continuity constraints between subdomains are enforced by lagrange multipliers the method uses the runge kutta chebyshev projection scheme to integrate the dae explicitly and to enforce the constraints by a projection with mass lumping techniques and node to node matching grids the method is fully explicit without solving any linear system a stability analysis is presented to show the extended stability property of the method the method is straightforward to implement and to parallelize numerical results demonstrate that it has excellent performance

a canonical process for estimation of convex functions the invelope of integrated brownian motion t4 a process associated with integrated brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities respectively we call this process the invelope and show that it is an almost surely uniquely defined function of integrated brownian motion its role is comparable to the role of the greatest convex minorant of brownian motion plus a parabolic drift in the problem of estimating monotone functions an iterative cubic spline algorithm is introduced that solves the constrained least squares problem in the limit situation and some results obtained by applying this algorithm are shown to illustrate the theory

second order chebyshev methods based on orthogonal polynomials stabilized methods also called chebyshev methods are explicit runge kutta methods with extended stability domains along the negative real axis these methods are intended for large mildly stiff problems originating mainly from parabolic pdes the aim of this paper is to show that with the use of orthogonal polynomials we can construct nearly optimal stability polynomials of second order with a three term recurrence relation these polynomials can be used to construct a new numerical method which is implemented in a code called rock2 this new numerical method can be seen as a combination of van der houwen sommeijer type methods and lebedev type methods

multiplicity of positive periodic solutions to a generalized delayed predator prey system with stocking abstract in this paper by using the continuation theorem of coincidence degree theory the existence of multiple positive periodic solutions for a generalized delayed predator prey system with stocking is established when our result is applied to a delayed predator prey system with nonmonotonic functional response and stocking we establish the sufficient condition for the existence of multiple positive periodic solutions for the system

an implicit explicit runge kutta chebyshev scheme for diffusion reaction equations an implicit explicit imex extension of the explicit runge kutta chebyshev rkc scheme designed for parabolic pdes is proposed for diffusion reaction problems with severely stiff reaction terms the imex scheme treats these reaction terms implicitly and diffusion terms explicitly within the setting of linear stability theory the new imex scheme is unconditionally stable for reaction terms having a jacobian matrix with a real spectrum for diffusion terms the stability characteristics remain unchanged a numerical comparison for a stiff nonlinear radiation diffusion problem between an rkc solver an imex rkc solver and the popular implicit bdf solver vodpk using the krylov solver gmres illustrates the excellent performance of the new scheme

positive periodic solutions for neutral delay ratio dependent predator prey model with holling type iii functional response abstract by using a continuation theorem based on coincidence degree theory some new and interesting sufficient conditions are obtained for the existence of positive periodic solutions for neutral delay ratio dependent predator prey model with holling type iii functional response x t x t r t a t x t 1 x t 2 c t x 2 t m 2 y 2 t x 2 t y t y t y t d t h t x 2 t t m 2 y 2 t t x 2 t t an example is represented to illustrate the feasibility of our main results

positive periodic solutions for the neutral ratio dependent predator prey model by using a continuation theorem based on coincidence degree theory we obtain some new sufficient conditions for the existence of positive periodic solutions for the neutral ratio dependent predator prey model with holling type ii functional response

amf runge kutta formulas and error estimates for the time integration of advection diffusion reaction pdes the convergence of a family of amf runge kutta methods in short amf rk for the time integration of evolutionary partial differential equations pdes of advection diffusion reaction type semi discretized in space is considered the methods are based on very few inexact newton iterations applied to implicit runge kutta formulas by combining the use of a natural splitting for the underlying jacobians and the approximate matrix factorization amf technique this approach allows a very cheap implementation of the runge kutta formula under consideration particular amf rk methods based on radau iia formulas are considered these methods have given very competitive results when compared with important formulas in the literature for multidimensional systems of non linear parabolic pde problems uniform bounds for the global time space errors on semi linear pdes when simultaneously the time step size and the spatial grid resolution tend to zero are derived numerical illustrations supporting the theory are presented

krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes application to discontinuous galerkin methods abstract integration factor methods are a class of exactly linear part time discretization methods in q nie y t zhang r zhao efficient semi implicit schemes for stiff systems journal of computational physics 214 2006 521 537 a class of efficient implicit integration factor iif methods were developed for solving systems with both stiff linear and nonlinear terms arising from spatial discretization of time dependent partial differential equations pdes with linear high order terms and stiff lower order nonlinear terms the tremendous challenge in applying iif temporal discretization for pdes on high spatial dimensions is how to evaluate the matrix exponential operator efficiently for spatial discretization on unstructured meshes to solve pdes on complex geometrical domains how to efficiently apply the iif temporal discretization was open in this paper we solve this problem by applying the krylov subspace approximations to the matrix exponential operator then we apply this novel time discretization technique to discontinuous galerkin dg methods on unstructured meshes for solving reaction diffusion equations numerical examples are shown to demonstrate the accuracy efficiency and robustness of the method in resolving the stiffness of the dg spatial operator for reaction diffusion pdes application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown

extrapolated stabilized explicit rungekutta methods extrapolated stabilized explicit rungekutta methods eserk are proposed to solve multi dimensional nonlinear partial differential equations pdes in such methods it is necessary to evaluate the function nt times per step but the stability region is o nt2 hence the computational cost is o nt times lower than for a traditional explicit algorithm in that way stiff problems can be integrated by the use of simple explicit evaluations in which case implicit methods usually had to be used therefore they are especially well suited for the method of lines mol discretizations of parabolic nonlinear multi dimensional pdes in this work first s stages first order methods with extended stability along the negative real axis are obtained they have slightly shorter stability regions than other traditional first order stabilized explicit rungekutta algorithms also called rungekuttachebyshev codes later they are used to derive nt stages second and fourth order schemes using richardson extrapolation the stability regions of these fourth order codes include the interval 0 01nt2 0 nt being the number of total functions evaluations which are shorter than stability regions of rock4 methods for example however the new algorithms neither suffer from propagation of errors as other rungekuttachebyshev codes as rock4 or dumka nor internal instabilities additionally many other types of higher order and also lower order methods can be obtained easily in a similar way these methods also allow adaptation of the length step with no extra cost hence the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way i e with minimal number of additional stages we compare the new techniques with other well known algorithms with good results in very stiff diffusion or reactiondiffusion multi dimensional nonlinear equations

discontinuous galerkin hp adaptive methods for multiscale chemical reactors quiescent reactors abstract we present a class of chemical reactor systems modeled numerically using a fractional multistep method between the reacting and diffusing modes of the system subsequently allowing one to utilize algebraic techniques for the resulting reactive subsystems a mixed form discontinuous galerkin method is presented with implicit and explicit imex timestepping strategies coupled to dioristic entropy schemes for h p adaptivity of the solution where the h and p are adapted based on an l 1 stability result finally we provide some numerical studies on the convergence behavior adaptation and asymptotics of the system applied to a pair of equilibrium problems as well as to general three dimensional nonlinear lotka volterra chemical systems

a local anisotropic adaptive algorithm for the solution of low mach transient combustion problems a novel numerical algorithm for the simulation of transient combustion problems at low mach and moderately high reynolds numbers is presented these problems are often characterized by the existence of a large disparity of length and time scales resulting in the development of directional flow features such as slender jets boundary layers mixing layers or flame fronts this makes local anisotropic adaptive techniques quite advantageous computationally in this work we propose a local anisotropic refinement algorithm using for the spatial discretization unstructured triangular elements in a finite element framework for the time integration the problem is formulated in the context of semi lagrangian schemes introducing the semi lagrange galerkin slg technique as a better alternative to the classical semi lagrangian sl interpolation the good performance of the numerical algorithm is illustrated by solving a canonical laminar combustion problem the flame vortex interaction first a premixed methane air flame vortex interaction with simplified transport and chemistry description test i is considered results are found to be in excellent agreement with those in the literature proving the superior performance of the slg scheme when compared with the classical sl technique and the advantage of using anisotropic adaptation instead of uniform meshes or isotropic mesh refinement as a more realistic example we then conduct simulations of non premixed hydrogen air flame vortex interactions test ii using a more complex combustion model which involves state of the art transport and chemical kinetics in addition to the analysis of the numerical features this second example allows us to perform a satisfactory comparison with experimental visualizations taken from the literature

rkc time stepping for advection diffusion reaction problems the original explicit runge kutta chebyshev rkc method is a stabilized second order integration method for pure diffusion problems recently it has been extended in an implicit explicit manner to also incorporate highly stiff reaction terms this implicit explicit rkc method thus treats diffusion terms explicitly and the highly stiff reaction terms implicitly the current paper deals with the incorporation of advection terms for the explicit method thus aiming at the implicit explicit rkc integration of advection diffusion reaction equations in a manner that advection and diffusion terms are treated simultaneously and explicitly and the highly stiff reaction terms implicitly

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

properties of positive definite solutions of the equation x a x 2a i abstract in this paper we discuss some properties of a positive definite solution of the matrix equation x a x 2 a i two effective iterative methods for computing a positive definite solution of this equation are proposed necessary and sufficient conditions for existence of a positive definite solution are derived numerical experiments are executed with these methods

numerical solution of a two dimensional hyperbolic transmission problem an initial boundary value problem for a two dimensional hyperbolic equation in two disjoint rectangles is investigated the existence and uniqueness and a priori estimates for weak solutions in appropriate sobolev like spaces are proved few finite difference schemes approximating this problem are proposed and analyzed

dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field endowed with a hierarchical gaussian process prior computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of markov chain monte carlo and are compounded by high dimensionality of the posterior we address these challenges by introducing truncated karhunen loeve expansions based on the prior distribution to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior we seek a solution of this problem using galerkin projection on a polynomial chaos basis and use the solution to construct a reduced dimensionality surrogate posterior density that is inexpensive to evaluate we demonstrate the formulation on a transient diffusion equation with prescribed source terms inferring the spatially varying diffusivity of the medium from limited and noisy data

quadratic functionals and small ball probabilities for the m fold integrated brownian motion let the gaussian process xm t be the m fold integrated brownian motion for positive integer m the laplace transform of the quadratic functional of xm t is found by using an appropriate self adjoint integral operator the result is then used to show the power of a general connection between small ball probabilities for the gaussian process the connection is discovered by introducing an independent random shift the interplay between our results and the principal eigenvalues for nonuniform elliptic generators on an unbounded domain is also discussed

estimation of a convex function characterizations and asymptotic theory we study nonparametric estimation of convexregression and density functions by methods of least squares in the regression and density cases and maximum likelihood in the density estimation case we provide characterizations of these estimators prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated the asymptotic distribution theory relies on the existence of an invelope function for integrated two sided brownian motion t 4 which is established in a companion paper by groeneboom jongbloed and wellner

on stabilized integration for time dependent pdes an integration method is discussed which has been designed to treat parabolic and hyperbolic terms explicitly and stiff reaction terms implicitly the method is a special two step form of the one step imex implicit explicit rkc runge kutta chebyshev method the special two step form is introduced with the aim of getting a non zero imaginary stability boundary which is zero for the one step method having a non zero imaginary stability boundary allows for example the integration of pure advection equations space discretized with centered schemes the integration of damped or viscous wave equations the integration of coupled sound and heat flow equations etc for our class of methods it also simplifies the choice of temporal step sizes satisfying the von neumann stability criterion by embedding a thin long rectangle inside the stability region embedding rectangles or other tractable domains with this purpose is an idea of wesseling

runge kutta chebyshev projection method in this paper a fully explicit stabilized projection method called the runge kutta chebyshev rkc projection method is presented for the solution of incompressible navier stokes systems this method preserves the extended stability property of the rkc method for solving odes and it requires only one projection per step an additional projection on the time derivative of the velocity is performed whenever a second order approximation for the pressure is desired we demonstrate both by numerical experiments and by order analysis that the method is second order accurate in time for both the velocity and the pressure being explicit the rkc projection method is easy to implement and to parallelize hence it is an attractive candidate for the solution of large scale moderately stiff diffusion like problems

fourth order chebyshev methods with recurrence relation in this paper a new family of fourth order chebyshev methods also called stabilized methods is constructed these methods possess nearly optimal stability regions along the negative real axis and a three term recurrence relation the stability properties and the high order make them suitable for large stiff problems often space discretization of parabolic pdes a new code rock4 is proposed illustrated at several examples and compared to existing programs

absolute monotonicity of polynomials occurring in the numerical solution of initial value problems this paper deals with polynomial approximations o x to the exponential function exp x related to numerical procedures for solving initial value problems motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument denoted by r o at which o is absolutely monotonic for given integersp 1 m 1 we determine the maximum ofr o when o varies over the class of all polynomials of a degree m with phi left x right exp left x right mathcal o left x p 1 right left for x to 0 right forx 0

existence of four periodic solutions for a generalized delayed ratio dependent predator prey system in this paper by using the continuation theorem of coincidence degree theory we obtain the existence condition of at least four periodic solutions for a delayed ratio dependent predator prey system an example is represented to illustrate the feasibility of our main result

rkc an explicit solver for parabolic pdes the fortran program rkc is intended for the time integration of parabolic partial differential equations discretized by the method of lines it is based on a family of runge kutta chebyshev formulas with a stability bound that is quadratic in the number of stages remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size moreover they make it possible to evaluate the explicit formulas in just a few vectors of storage these characteristics of the program make it especially attractive for problems in several spatial variables rkc is compared to the bdf solver vodpk on two test problems in three spatial variables

an essentially non oscillatory semi lagrangian method for tidal flow simulations we develop an essentially non oscillatory semi lagrangian method for solving two dimensional tidal flows the governing equations are derived from the incompressible navier stokes equations with assumptions of shallow water flows including bed frictions eddy viscosity wind shear stresses and coriolis forces the method employs the modified method of characteristics to discretize the convective term in a finite element framework limiters are incorporated in the method to reconstruct an essentially non oscillatory algorithm at minor additional cost the central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized the resulting semi discretized system is then solved by an explicit runge kutta chebyshev scheme with extended stages this scheme adds in a natural way a stabilizing stage to the conventional runge kutta method using the chebyshev polynomials the proposed method is verified for the recirculation tidal flow in a channel with forward facing step we also apply the method for simulation of tidal flows in the strait of gibraltar in both test problems the proposed method demonstrates its ability to handle the interaction between water free surface and bed frictions copyright 2009 john wiley sons ltd

multiple periodic solutions for a non autonomous delayed predator prey model with harvesting terms abstract by using the generalized continuation theorem the existence of at least two positive periodic solutions for a non autonomous delayed periodic predator prey model with harvesting x t x t r t a t x t b t x t m x t y t h 1 t y t y t c t x t t m x t t d t h 2 t is established the method used here is more convenient than mawhin s continuation theorem and the main results show that the harvesting terms consult in the additional periodic solution

integrated brownian motion conditioned to be positive we study the two dimensional process of integrated brownian motion and brownian motion where integrated brownian motion is conditioned to be positive the transition density of this process is derived from the asymptotic behavior of hitting times of the unconditioned process explicit expressions for the transition density in terms of confluent hypergeometric functions are derived and it is shown how our results on the hitting time distributions imply previous results of isozaki watanabe and goldman the conditioned process is characterized by a system of stochastic differential equations sdes for which we prove an existence and unicity result some sample path properties are derived from the sdes and it is shown that t to t 9 10 is a critical curve for the conditioned process in the sense that the expected time that the integral part of the conditioned process spends below any curve t to t alpha is finite for alpha 9 10 and infinite for alpha geq 9 10