the lem exponential integrator for advection diffusion reaction equations we implement a second order exponential integrator for semidiscretized advection diffusion reaction equations obtained by coupling exponential like euler and midpoint integrators and computing the relevant matrix exponentials by polynomial interpolation at leja points numerical tests on 2d models discretized in space by finite differences or finite elements show that the leja euler midpoint lem exponential integrator can be up to 5 times faster than a classical second order implicit solver

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

a massively parallel exponential integrator for advection diffusion models this work considers the real leja points method relpm m caliari m vianello l bergamaschi interpolating discrete advection diffusion propagators at spectral leja sequences j comput appl math 172 2004 79 99 for the exponential integration of large scale sparse systems of odes generated by finite element or finite difference discretizations of 3 d advection diffusion models we present an efficient parallel implementation of relpm for polynomial interpolation of the matrix exponential propagators exp dta v and f dta v f z exp z 1 z a scalability analysis of the most important computational kernel inside the code the parallel sparse matrix vector product has been performed as well as an experimental study of the communication overhead as a result of this study an optimized parallel sparse matrix vector product routine has been implemented the resulting code shows good scaling behavior even when using more than one thousand processors the numerical results presented on a number of very large test cases gives experimental evidence that relpm is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures

time stepping methods for the simulation of the self assembly of nano crystals in matlab on a gpu partial differential equations describing the patterning of thin crystalline films are typically of fourth or sixth order they are quasi or semilinear and they are mostly defined on simple geometries such as rectangular domains for the numerical simulation of these kinds of problems spectral methods are an efficient approach we apply several implicit explicit schemes to one recently derived pde that we express in terms of coefficients of trigonometric interpolants while the simplest imex scheme turns out to have the mildest step size restriction higher order sbdf schemes tend to be more unstable and exponential time integrators are fastest for the calculation of very accurate solutions we implemented a reduced model in the expint package syntax 3 and compared various exponential schemes a convexity splitting approach was employed to stabilize the sbdf1 scheme we show that accuracy control is crucial when using this idea therefore we present a time adaptive sbdf1 sbdf1 2 step method that yields convincing results reflecting the change in timescales during topological changes of the nanostructures the implementation of all presented methods is carried out in matlab we used the open source gpumat package to gain up to 5 fold runtime benefits by carrying out calculations on a low cost gpu without having to prescribe any knowledge in low level programming or cuda implementations and found comparable speedups as with matlab s pct or with gpumat run on octave

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

a determining form for the two dimensional navier stokes equations the fourier modes case the determining modes for the two dimensional incompressible navier stokes equations nse are shown to satisfy an ordinary differential equation ode of the form dv dt f v in the banach space x of all bounded continuous functions of the variable s r with values in certain finite dimensional linear space this new evolution ode named determining form induces an infinite dimensional dynamical system in the space x which is noteworthy for two reasons one is that f is globally lipschitz from x into itself the other is that the long term dynamics of the determining form contains that of the nse the traveling wave solutions of the determining form i e those of the form v t s v0 t s correspond exactly to initial data v0 that are projections of solutions of the global attractor of the nse onto the determining modes the determining form is also shown to be dissipative an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the grashof num

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

multi step tangential versus transversal linearizations in non linear dynamics a family of multi step tangential linearization mtnl techniques is developed for numeric analytic and geometric integration of non linear oscillators under deterministic loading a family of multi step transversal linearization mtrl schemes is also explored and a comparison between the two linearization families is provided with a view of bringing out their relative advantages and deficiencies the mtnl methodology essentially works through an application of the concept of mtrl to non linear rate equations different variants of the mtnl are obtained and it turns out that they are specific cases of a generalized mtnl method local error estimates are also provided moreover higher order mtrl and mtnl methods are successfully implemented through chebyshev nodes instead of equidistant nodes this helps to improve accuracy even with larger step sizes a comparison of relative errors through the mtrl and mtnl techniques consistently indicates a superior quality of approximations via the mtrl route whilst the usage of the tangential linearization is very common in non linear computational mechanics it is demonstrated that the transversal version offers in addition to higher numerical accuracy an easier implementation and avoids computation of jacobian matrices this allows the mtrl family to be applicable even to non linear oscillators having c0 vector fields copyright 2007 john wiley sons ltd

on the construction of families of approximate inertial manifolds abstract the concept of approximate inertial manifold is related to the study of the longtime behaviour of partial differential equations in this paper we present a general method in order to construct sequences of such manifolds providing better and better approximations to the attractor le concept de variete inertielle approximative intervient dans l etude du comportement pour les grands temps des equations aux derivees partielles nous presentons dans cet article une methode generale pour construire des suites de telles varietes qui approximent de mieux en mieux l attracteur

rigorous estimates of small scales in turbulent flows we derive rigorous bounds on the length scale of determining local averages volume elements for the 3 d navier stokes equations these length scale estimates are related to kolmogorov s notion of a dissipation length scale in turbulent flows

bifurcation computations on an approximate inertial manifold for the 2d navier stokes equations abstract we compare the minimum number of modes needed to capture certain bifurcations of the 2d navier stokes equations by a certain nonlinear galerkin method and by the traditional galerkin method the boundary conditions are periodic and the forcing is in a single mode two different choices of the forcing mode are tested we introduce and implement a homotopy which allows continuation between approximations of different size

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

filter accuracy for the lorenz 96 model fixed versus adaptive observation operators in the context of filtering chaotic dynamical systems it is well known that partial observations if sufficiently informative can be used to control the inherent uncertainty due to chaos the purpose of this paper is to investigate both theoretically and numerically conditions on the observations of chaotic systems under which they can be accurately filtered in particular we highlight the advantage of adaptive observation operators over fixed ones the lorenz 96 model is used to exemplify our findings we consider discrete time and continuous time observations in our theoretical developments we prove that for fixed observation operator the 3dvar filter can recover the system state within a neighbourhood determined by the size of the observational noise it is required that a sufficiently large proportion of the state vector is observed and an explicit form for such sufficient fixed observation operator is given numerical experiments where the data is incorporated by use of the 3dvar and extended kalman filters suggest that less informative fixed operators than given by our theory can still lead to accurate signal reconstruction adaptive observation operators are then studied numerically we show that for carefully chosen adaptive observation operators the proportion of the state vector that needs to be observed is drastically smaller than with a fixed observation operator indeed we show that the number of state coordinates that need to be observed may even be significantly smaller than the total number of positive lyapunov exponents of the underlying system

on the construction of restricted denominator exponential w methods in this paper we consider the practical construction of exponential w methods for the solution of large stiff nonlinear initial value problems based on the restricted denominator rational approach for the computation of the functions of matrices required this approach is employed together with the krylov subspace method based on the arnoldi algorithm two integrators are constructed and tested on some classical stiff equations arising from the semidiscretization of parabolic problems

the fidelity of exponential and imex integrators for wave turbulence introduction of a new near minimax integrating factor scheme abstract exponential and imex integrators are considered for wave turbulence problems which are stiff systems that often require long time simulations applying a multi scale asymptotic analysis to the difference equations for a simple wave turbulence problem reveals that exponential runge kutta methods incorrectly model interactions between resonant waves unless they are specifically designed otherwise and implicit explicit imex integrators introduce dispersive and dissipative errors that severely degrade the accuracy of slow energy exchange between resonant waves the same analysis shows integrating factor if methods to be the most suitable numerical experiments performed on the resonant triad model and the majda mclaughlin tabak mmt model verify the discrete asymptotic analysis in the mmt problem exponential time differencing methods require a time step more than two orders of magnitude smaller than if methods to achieve comparable accuracy and imex methods require more than six orders of magnitude smaller time steps a novel near minimax rational approximation of the matrix exponential is proposed to be used for integrating factor methods and is demonstrated to be as accurate as the if method but without the need to compute a matrix exponential

spectral decomposition of fractional operators and a reflected stable semigroup abstract in this paper we provide the spectral decomposition in hilbert space of the c 0 semigroup p and its adjoint p having as generator respectively the caputo and the right sided riemann liouville fractional derivatives of index 1 2 these linear operators which are non local and non self adjoint appear in many recent studies in applied mathematics and also arise as the infinitesimal generators of some substantial processes such as the reflected spectrally negative stable process our approach relies on intertwining relations that we establish between these semigroups and the semigroup of a bessel type process whose generator is a self adjoint second order differential operator in particular from this commutation relation we characterize the positive real axis as the continuous point spectrum of p and provide a power series representation of the corresponding eigenfunctions we also identify the positive real axis as the residual spectrum of the adjoint operator p and elucidate its role in the spectral decomposition of these operators by resorting to the concept of continuous frames we proceed by investigating the domain of the spectral operators and derive two representations for the heat kernels of these semigroups as a by product we also obtain regularity properties for these latter and also for the solution of the associated cauchy problem

low dimensional attractors arise from forcing at small scales abstract standard estimates of the dimension of the attractor of the 2d navier stokes equations are given in terms of a dimensionless grashof number that measures the total amount of energy injected into the flow however this result takes no account of whether the forcing is concentrated at large or small scales by a simple modification of the usual argument this paper provides a bound that is linear in a modified grashof number that reflects the spatial structure of the forcing in particular as a fixed amount of energy is injected at progressively smaller scales then the dimension of the attractor decreases

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

generalized integrating factor methods for stiff pdes the integrating factor if method for numerical integration of stiff nonlinear pdes has the disadvantage of producing large error coefficients when the linear term has large norm we propose a generalization of the if method and in particular construct multistep type methods with several orders of magnitude improved accuracy we also consider exponential time differencing etd methods and point out connections with a particular application of the commutator free lie group methods we present a new fourth order etdrk method with improved accuracy the methods considered are compared in several numerical examples

a class of explicit multistep exponential integrators for semilinear problems a class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed it is shown that the k step method achieves order k for appropriate starting values which can be computed by auxiliary routines or by one strategy proposed in the paper together with some implementation issues numerical illustrations are also provided

b series and order conditions for exponential integrators we introduce a general format of numerical ode solvers which include many of the recently proposed exponential integrators we derive a general order theory for these schemes in terms of b series and bicolored rooted trees to ease the construction of specific schemes we generalize an idea of zennaro em math comp 46 1986 pp 119 133 and define natural continuous extensions in the context of exponential integrators this leads to a relatively easy derivation of some of the most popular recently proposed schemes the general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces we will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes finally we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

on restart and error estimation for krylov approximation of w f a v krylov algorithms are widely used for the solution of large scale linear systems a krylov approach is implemented also for the approximation of the vector w which results from operating with a function of a matrix on a vector v for example when solving a set of linear odes we have w f a v where f z rm exp tz the main drawback of the krylov algorithm lies in the need to store all the vectors spanning the approximation space when solving linear systems it is possible to overcome this drawback by restarting in this paper we show how to apply restarts in the general case of approximating w f a v this new algorithm allows a much more efficient approximation of the exponential operator than the standard krylov algorithm and it is especially useful in the case of functions which cannot be factored into a product of functions yet another interesting subject is the error estimate the scheme given here provides error estimates for free a trait which enables us to stop the algorithm whenever the desired accuracy is achieved

rooted tree analysis of runge kutta methods with exact treatment of linear terms we investigate a class of time discretization schemes called etd runge kutta methods where the linear terms of an ordinary differential equation are treated exactly while the other terms are numerically integrated by a one step method these schemes proposed by previous authors can be regarded as modified runge kutta methods whose coefficients are matrices instead of scalars from this viewpoint we reexamine the notion of consistency convergence and order to provide a mathematical foundation for new methods applying the rooted tree analysis expansion theorems of both the strict and numerical solutions are proved and two types of order conditions are defined several classes of formulas with up to four stages that satisfy the conditions are constructed and it is shown that the power series of matrices employed as their coefficients can be determined using the order conditions

discrete data assimilation in the lorenz and 2d navier stokes equations consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing in this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time this paper studies discrete data assimilation for the lorenz equations and the incompressible two dimensional navier stokes equations in both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

boundary conditions for two sided fractional diffusion abstract this paper develops appropriate boundary conditions for the two sided fractional diffusion equation where the usual second derivative in space is replaced by a weighted average of positive left and negative right fractional derivatives mass preserving reflecting boundary conditions for two sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary stable consistent explicit and implicit euler methods are detailed and steady state solutions are derived steady state solutions for two sided fractional diffusion equations using both riemann liouville and caputo flux are computed for riemann liouville flux and reflecting boundary conditions the steady state solution is singular at one or both of the end points for caputo flux and reflecting boundary conditions the steady state solution is a constant function numerical experiments illustrate the convergence of these numerical methods finally the influence of the reflecting boundary on the steady state behavior subject to both the riemann liouville and caputo fluxes is discussed

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

reprint of boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

observers for compressible navier stokes equation we consider a multidimensional model of a compressible fluid in a bounded domain we want to estimate the density and velocity of the fluid based on the observations for only velocity we build an

beyond conventional runge kutta methods in numerical integration of odes and daes by use of structures and local models there are two parts in this paper in the first part we consider an overdetermined system of differential algebraic equations daes we are particularly concerned with hamiltonian and lagrangian systems with holonomic constraints the main motivation is in finding methods based on gauss coefficients preserving not only the constraints symmetry symplecticness and variational nature of trajectories of holonomically constrained hamiltonian and lagrangian systems but also having optimal order of convergence the new class of s s gauss lobatto specialized partitioned additive runge kutta spark methods uses greatly the structure of the daes and possesses all desired properties in the second part we propose a unified approach for the solution of ordinary differential equations odes mixing analytical solutions and numerical approximations the basic idea is to consider local models which can be solved efficiently for example analytically and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction in order to preserve also symmetry we define the new class of symmetrized runge kutta methods with local model srklm

an operator splitting algorithm for coupled one dimensional advection diffusion reaction equations abstract an operator splitting algorithm for a system of one dimensional advection diffusion reaction equations describing the transport of non conservative pollutants is presented in this paper the algorithm is a strang type splitting procedure incorporating contributions from the inhomogeneous terms by the duhamel s principle the associated homogeneous equations are split into advection diffusion and reaction equations and solved by a backward method of characteristic a finite element method and an explicit runge kutta method respectively the boundary conditions applicable to the split equations are derived numerical analyses of the algorithm consisting of the stability the accuracy and the convergence of the solution procedure are presented the composite algorithm is second order accurate in time and space and conditionally stable the numerical characteristics of the algorithm are demonstrated by several examples

data assimilation algorithm for 3d benard convection in porous media employing only temperature measurements abstract in this paper we propose a continuous data assimilation downscaling algorithm for the benard convection in porous media using only discrete spatial mesh measurements of the temperature in this algorithm we incorporate the observables as a feedback nudging term in the evolution equation of the temperature we show that under an appropriate choice of the nudging parameter and the size of the mesh and under the assumption that the observed data is error free the solution of the proposed algorithm converges at an exponential rate asymptotically in time to the unique exact unknown reference solution of the original system associated with the observed finite dimensional projection of temperature data moreover we note that in the case where the observational measurements are not error free one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements

on a prey predator reaction diffusion system with holling type iii functional response in this paper we study a prey predator model defined by an initial boundary value problem whose dynamics is described by a holling type iii functional response we establish global existence and uniqueness of the strong solution we prove that if the initial data are positive and satisfy a certain regularity condition the solution of the problem is positive and bounded on the domain q 0 t x w and then we deduce the continuous dependence on the initial data a numerical approximation of the system is carried out with a spectral method coupled with the fourth order runge kutta time solver the biological relevance of the comparative numerical results is also presented

determining modes for continuous data assimilation in 2d turbulence we study the number of determining modes necessary for continuous data assimilation in the two dimensional incompressible navier stokes equations our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes we treat this problem analytically by proving a convergence result depending on the h 1 norm of f and computationally by considering a family of forcing functions with identical grashof numbers that are supported on different annuli in fourier space the rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

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

order conditions for commutator free lie group methods we derive order conditions for commutator free lie group integrators for certain problems these schemes can be good alternatives to the runge kutta munthe kaas schemes especially when applied to stiff problems or to homogeneous manifolds with large isotropy groups the order conditions correspond to certain subsets of the set of ordered rooted trees we discuss ways to select these subsets and their combinatorial properties we also suggest how the reuse of flow calculations can be included in order to reduce the computational cost in the case that at most two flow calculations are admitted in each stage the order conditions simplify substantially we derive families of fourth order schemes which effectively use only five flow calculations per step

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

exponential integrator methods for systems of non linear space fractional models with super diffusion processes in pattern formation abstract nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in science and engineering however solving these type of models is computationally demanding in this paper we propose an exponential integrator method for space fractional models as an attractive and easy to code alternative for other existing second order exponential integrator methods this scheme is based on using a real distinct poles discretization for the underlying matrix exponentials one of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency the scheme is established to be second order convergent and proven to be robust for nonlinear space fractional reaction diffusion problems involving non smooth initial data our approach is exhibited by solving a system of two dimensional problems which exhibits pattern formation and has applications in cell division empirically super diffusion processes are displayed by investigating the effect of the fractional power of the underlying laplacian operator on the pattern formation found in these models furthermore the superiority of our method over competing second order etd schemes bdf2 scheme and imex schemes is demonstrated

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

an analytic semigroup generated by a fractional differential operator abstract we study a space fractional diffusion problem where the non local diffusion flux involves the caputo derivative of the diffusing quantity we prove the unique existence of regular solutions to this problem by means of the semigroup theory we show that the operator defined as divergence of product of a positive function with the caputo derivative is a generator of analytic semigroup

a class of numerical algorithms for large time integration the nonlinear galerkin methods the nonlinear galerkin methods are numerical schemes well adapted to the long term integration of evolution partial differential equations they consist of looking for approximate solutions lying in nonlinear manifolds that are close in some sense to the attractor the aim of this paper is to extend these schemes to a large class of manifolds convergence results are derived for the schemes introduced

determination of the solutions of the navier stokes equations by a set of nodal values we consider the navier stokes equations of a viscous incompresible fluid and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions the results presented here are exact results and not approximate ones we show in several cases that the solutions are entirely determined by their values on a discrete set provided this set contains enough points and these points are sufficiently densely distributed in a sense described in the article two typical results are the following ones two stationary solutions coincide if they coincide on a set sufficiently dense but finite similarly if the large time behavior of the solutions to the navier stokes equations is known on an appropriate discrete set then the large time behavior of the solution itself is totally determined

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

on the number of determining nodes for the 2d navier stokes equations abstract it is known that the solutions of the 2d navier stokes equations in bounded domains are determined by a finite discrete set of nodal values that is if the large time behavior of the solutions to the navier stokes equations is known on an appropriate finite discrete set then the large time behavior of the solution itself is totally determined here an upper bound is rigorously established for the number of nodes needed to determine the solutions of the navier stokes equations in two dimensions with periodic boundary conditions

data assimilation for the navier stokes α equations abstract the well posedness of the data assimilation problem for the navier stokes equations on a bounded three dimensional domain is investigated the data assimilation procedures under consideration are the adjoint method of variational data assimilation 4d var and the method of continuous data assimilation concerning the adjoint method the existence of optimal initial conditions with respect to an observation dependent cost functional is proven the optimizers are characterized by a first order necessary condition involving the adjoint linearized navier stokes equations and conditions for the uniqueness of the initial conditions are given well posedness of the continuous data assimilation problem is proven and convergence rates in terms of observational resolution are provided

accuracy and stability of filters for dissipative pdes data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system filters refer to a class of data assimilation algorithms designed to update the estimation of the state in an on line fashion as data is acquired sequentially for linear problems subject to gaussian noise filtering can be performed exactly using the kalman filter for nonlinear systems filtering can be approximated in a systematic way by particle filters however in high dimensions these particle filtering methods can break down hence for the large nonlinear systems arising in applications such as oceanography and weather forecasting various ad hoc filters are used mostly based on making gaussian approximations the purpose of this work is to study the accuracy and stability properties of these ad hoc filters we work in the context of the 2d incompressible navier stokes equation although the ideas readily generalize to a range of dissipative partial differential equations pdes by working in this infinite dimensional setting we provide an analysis which is useful for the understanding of high dimensional filtering and is robust to mesh refinement we describe theoretical results showing that in the small observational noise limit the filters can be tuned to perform accurately in tracking the signal itself filter accuracy provided the system is observed in a sufficiently large low dimensional space roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics the tuning corresponds to what is known as variance inflation in the applied literature numerical results are given which illustrate the theory the positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters can perform poorly in reproducing statistical variation about the true signal

fractional partial differential equations with boundary conditions abstract we identify the stochastic processes associated with one sided fractional partial differential equations on a bounded domain with various boundary conditions this is essential for modelling using spatial fractional derivatives we show well posedness of the associated cauchy problems in c 0 and l 1 in order to do so we develop a new method of embedding finite state markov processes into feller processes on bounded domains and then show convergence of the respective feller processes this also gives a numerical approximation of the solution the proof of well posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the lax richtmyer equivalence theorem to prove convergence without checking well posedness

an exponential integrator for advection dominated reactive transport in heterogeneous porous media we present an exponential time integrator in conjunction with a finite volume discretisation in space for simulating transport by advection and diffusion including chemical reactions in highly heterogeneous porous media representative of geological reservoirs these numerical integrators are based on the variation of constants solution and solving the linear system exactly this is at the expense of computing the exponential of the stiff matrix comprising the finite volume discretisation using real leja points or a krylov subspace technique compared to standard finite difference based time integrators we observe for a variety of example applications that numerical solutions with exponential methods are generally more accurate and require less computational cost they hence comprise an efficient and accurate method for simulating non linear advection dominated transport in geological formations

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

higher order grunwald approximations of fractional derivatives and fractional powers of operators we give stability and consistency results for higher order grunwaldtype formulae used in the approximation of solutions to fractional in space partial differential equations we use a new carlson type inequality for periodic fourier multipliers to gain regularity and stability results we then generalise the theory to the point where the first derivative operator is replaced by the generator of a bounded group on an arbitrary banach space

reflected spectrally negative stable processes and their governing equations this paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one reflected at its infimum first we derive the forward equation using the theory of sun dual semigroups the resulting forward equation is a boundary value problem on the positive half line that involves a negative riemann liouville fractional derivative in space and a fractional reflecting boundary condition at the origin then we apply numerical methods to explicitly compute the transition density of this space inhomogeneous markov process for any starting point to any desired degree of accuracy finally we discuss an application to fractional cauchy problems which involve a positive caputo fractional derivative in time

a new class of split exponential propagation iterative methods of runge kutta type sepirk for semilinear systems of odes abstract in this paper the framework of the exponential propagation iterative methods of runge kutta type epirk is extended to construct split epirk sepirk integrators for semilinear systems of odes the structure of the sepirk methods possesses the flexibility and generality that allows construction of very efficient schemes we demonstrate how bicolored trees based b series can be used to derive the order conditions for the new integrators an algorithm is developed to solve the order conditions up to order three and several new schemes with desirable properties are proposed the numerical results illustrate the advantages offered by the new class of integrators the experiments also address the comparative performance of split vs non split epirk methods and the question of improving efficiency by optimizing coefficients of the sepirk schemes it is shown that specific schemes can be custom built to improve computational efficiency for a particular problem

on a criterion for locating stable stationary solutions to the navier stokes equations on donne une estimation superieure uniforme explicite a la borne de constantin foias ternam et on donne une forme applicable explicite aux conditions suffisantes on considere seulement le cas a 2 dimensions

on group fourier analysis and symmetry preserving discretizations of pdes in this paper we review some group theoretic techniques applied to discretizations of pdes inspired by the recent years active research in lie group and exponential time integrators for differential equations we will in the first part of the paper present algorithms for computing matrix exponentials based on fourier transforms on finite groups as an example we consider spherically symmetric pdes where the discretization preserves the 120 symmetries of the icosahedral group this motivates the study of spectral element discretizations based on triangular subdivisions in the second part of the paper we introduce novel applications of multivariate non separable chebyshev polynomials in the construction of spectral element bases on triangular and simplicial sub domains these generalized chebyshev polynomials are intimately connected to the theory of root systems and weyl groups used in the classification of semi simple lie algebras and these polynomials share most of the remarkable properties of the classical chebyshev polynomials such as near optimal lebesgue constants for the interpolation error the existence of fft based algorithms for computing interpolants and pseudo spectral differentiation and existence of gaussian integration rules the two parts of the paper can be read independently

exponential rosenbrock type methods we introduce a new class of exponential integrators for the numerical integration of large scale systems of stiff differential equations these so called rosenbrock type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the jacobian in contrast to standard integrators the methods are fully explicit and do not require the numerical solution of linear systems we analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in banach spaces in particular we derive an abstract stability and convergence result for variable step sizes this analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods we present a third order method with two stages and a fourth order method with three stages respectively the application of the required matrix functions to vectors are computed by krylov subspace approximations we briefly discuss these implementation issues and we give numerical examples that demonstrate the efficiency of the new integrators

variable step implementation of etd methods for semilinear problems abstract spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations exponential integrators solve exactly the linear part which is assumed to be responsible of the stiffness of the problem while the nonlinear terms are approximated numerically in this paper a variable step implementation of the exponential time differencing methods of multistep type is considered a local error estimate of exponential type whose computation is for free is introduced and some implementation issues for diagonalizable matrices are addressed numerical results showing the behaviour of the proposed variable step implementation are provided too

continuous data assimilation for the 2d benard convection through velocity measurements alone 2015 elsevier b v an algorithm for continuous data assimilation for the two dimensional benard convection problem is introduced and analyzed it is inspired by the data assimilation algorithm developed for the navier stokes equations which allows for the implementation of variety of observables low fourier modes nodal values finite volume averages and finite elements the novelty here is that the observed data is obtained for the velocity field alone i e no temperature measurements are needed for this algorithm we provide conditions on the spatial resolution of the observed data under the assumption that the observed data is free of noise which are sufficient to show that the solution of the algorithm approaches at an exponential rate the unique exact unknown solution of the benard convection problem associated with the observed finite dimensional projection of velocity

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

exponential rosenbrock methods of order five construction analysis and numerical comparisons the aim of this paper is to construct high order exponential rosenbrock methods and to analyze their convergence properties for the time discretization of large scale systems of stiff differential equations we present a new and simple approach for deriving the stiff order conditions these conditions allow us to construct new pairs of embedded methods of high order as an example we present a fifth order method with five stages for particular problems the order conditions can be simplified it is then even possible to construct a method of order 5 with three stages only the error analysis is performed in an abstract framework of strongly continuous semigroups that allows us to treat semilinear evolution equations in banach spaces convergence results are proved for variable step size implementations to demonstrate the efficiency of the new integrators we give some numerical experiments in matlab in particular numerical comparisons for semilinear parabolic pdes in one and two space dimensions are presented

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

a new class of time discretization schemes for the solution of nonlinear pdes we consider issues of stability of time discretization schemes withexacttreatment of thelinear part elp schemes for solving nonlinear pdes a distinctive feature of elp schemes is the exact evaluation of the contribution of the linear term that is if the nonlinear term of the equation is zero then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient it turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices for a finite but arbitrary accuracy this observation makes our approach practical in a number of applications in particular we consider applications of elp schemes to advection diffusion equations we study the stability of these schemes and show that both explicit and implicit elp schemes have distinctly different stability properties if compared with known implicit explicit schemes for example we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection diffusion equations

higher order time stepping methods for time dependent reaction diffusion equations arising in biology abstract this paper demonstrates the use of higher order methods to solve some time dependent stiff pdes in the past the most popular numerical methods for solving system of reaction diffusion equations was based on the combination of low order finite difference method with low order time stepping method we extend in this report the compatibility of fourth order finite difference scheme in space coupled with fourth order time stepping methods such as imexlm4 imexpc4 imexrk4 and etdrk4 b in time for direct integration of reaction diffusion equations in one space dimension some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio temporal dynamical structures

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

global in time stability and accuracy of imex fem data assimilation schemes for navier stokes equations abstract we study numerical schemes for incompressible navier stokes equations using imex temporal discretizations finite element spatial discretizations and equipped with continuous data assimilation a technique recently developed by azouani et al 2014 we analyze stability and accuracy of the proposed methods and are able to prove well posedness long time stability and long time accuracy estimates under restrictions of the time step size and data assimilation parameter we give results for several numerical tests that illustrate the theory and show that for good results the choice of discretization parameter and element choices can be critical

the locally extrapolated exponential time differencing lod scheme for multidimensional reaction diffusion systems this paper introduces the local extrapolation of first order locally one dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction diffusion systems this novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system the storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems the stability monotonicity and convergence of the locally extrapolated exponential time differencing scheme have been examined stability analysis shows that the scheme is strongly stable l stable and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution the performance of the novel scheme has been investigated by testing it on a two dimensional schnakenberg model two and three dimensional brusselator models and a three dimensional enzyme kinetics of michaelis menten type reaction diffusion model numerical experiments demonstrate the efficiency accuracy and reliability of the scheme locally extrapolated lod exponential time differencing scheme is developed multidimensional nonlinear reaction diffusion systems are considered monotonicity and stability are analyzed the order of convergence is examined