on the spurious solutions in the high order finite difference methods for eigenvalue problems in this paper the origins of spurious solutions occurring in the high order finite difference methods are studied based on a uniform mesh spurious modes are found in the high order one sided finite difference discretizations of many eigenvalue problems spurious modes are classified as spectral pollution and non spectral pollution the latter can be partially avoided by mesh refinement while the former persists when the mesh is refined through numerical studies of some prototype eigenvalue problems such as those of the helmholtz and beam equations we show that perfect central differentiation schemes do not produce any spurious modes nevertheless high order central difference schemes encounter difficulties in implementing complex boundary conditions we further show that the central difference schemes will produce spurious modes as well if asymmetric approximation is involved in boundary treatments in general central difference schemes are less likely to produce spurious modes than one sided difference schemes 2007 elsevier b v all rights reserved

synchronization of stochastic bistable systems by biperiodic signals we study the nonlinear response of a noisy bistable system to a biperiodic signal through experiments with an electronic circuit schmitt trigger the signal we use is a biharmonic one i e a superposition of low and high frequency harmonic components it is shown that the mean switching frequency msf of the system can be locked at both low and high frequencies moreover the phenomenon of msf locking at the lower frequency can be induced and enhanced by the higher frequency excitation thus high frequency bias can control synchronization at the low frequency

on the validity of a proof that the discrete singular convolution dsc lagrange distributed approximation function ldaf method is inferior to high order finite differences a few families of counterexamples are provided to a proof that the discrete singular convolution dsc lagrange distributed approximation function ldaf method is inferior to high order finite differences journal of computational physics 214 538 549 2006

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

spatial finite difference approximations for wave type equations the simplest finite difference approximations for spatial derivatives are centered explicit and applied to regular equispaced grids well established generalizations include the use of implicit compact approximations and staggered grids we find here that the combination of these two concepts together with high formal order of accuracy is very effective for approximating the first derivatives in space that occur in many wave type pdes

convection diffusion equation with absorption and non decaying initial data in this paper we give the precise description of the large time behavior of the solution u of the cauchy problem tu u a u u in rn 0 u x 0 x 0in rn by using the ordinary differential equation and a linear parabolic equation here n 1 a rn 1 1 0 and is a bounded continuous function such that lp rn for some 1 p furthermore we study the large time behavior of the hot spots for the solution u

gain in stochastic resonance precise numerics versus linear response theory beyond the two mode approximation in the context of the phenomenon of stochastic resonance sr we study the correlation function the signal to noise ratio snr and the ratio of output over input snr i e the gain which is associated to the nonlinear response of a bistable system driven by time periodic forces and white gaussian noise these quantifiers for sr are evaluated using the techniques of linear response theory lrt beyond the usually employed two mode approximation scheme we analytically demonstrate within such an extended lrt description that the gain can indeed not exceed unity we implement an efficient algorithm based on work by greenside and helfand detailed in the appendix to integrate the driven langevin equation over a wide range of parameter values the predictions of lrt are carefully tested against the results obtained from numerical solutions of the corresponding langevin equation over a wide range of parameter values we further present an accurate procedure to evaluate the distinct contributions of the coherent and incoherent parts of the correlation function to the snr and the gain as a main result we show for subthreshold driving that both the correlation function and the snr can deviate substantially from the predictions of lrt and yet the gain can be either larger or smaller than unity in particular we find that the gain can exceed unity in the strongly nonlinear regime which is characterized by weak noise and very slow multifrequency subthreshold input signals with a small duty cycle this latter result is in agreement with recent analog simulation results by gingl et al icnf 2001 edited by g bosman world scientific singapore 2002 pp 545 548 fluct noise lett 1 l181 2001

equivalent accuracy at a fraction of the cost overcoming temporal dispersion abstractnumerical dispersion in finite difference fd modeling produces coherent artifacts severely constraining the resolution of advanced imaging and inversion techniques conventionally numerical dispersion is reduced by increasing the order of accuracy of the fd operators and we resign ourselves to paying the high computational cost that is incurred assuming no spatial dispersion we have found that fd time dispersion is independent of the medium velocity and the spatial grid for propagation and only depends on the time stepping scheme and the propagation time based on this observation we have devised postpropagation filters to collapse the time dispersion effect of fd modeling our dispersion correction filters are designed by comparing the input waveform with dispersive waveforms obtained by 1d forward modeling these filters are then applied on multidimensional shot records to eliminate the time dispersion by two schemes 1 stationary filtering plus interpolation and 2 nonstationary fil

euler maclaurin and gregory interpolants let a sufficiently smooth function f f on 1 1 1 1 be sampled at n 1 n 1 equispaced points and let k ge 0 k 0 be given an euler maclaurin interpolant to the data is defined consisting of a sum of a degree k k algebraic polynomial and a degree n n trigonometric polynomial which deviates from f f by o n k o n k and whose integral is equal to the order k k euler maclaurin approximation of the integral of f f this interpolant makes use of the same derivatives f j pm 1 f j 1 as the euler maclaurin formula a variant gregory interpolant is also defined based on finite difference approximations to the derivatives whose integral for k k odd is equal to the order k k gregory approximation to the integral

eventual c regularity and concavity for flows in one dimensional porous media we study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation u t eft u m ight xx inq r imes eft 0 nfty ight ueft x 0 ight u 0 eft x ight forx n r

well posedness for the navier slip thin film equation in the case of partial wetting we are interested in the thin lm equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid lm driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourthorder parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with p 13 1 4 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l 2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

large systems of interacting particles and the porous medium equation abstract we study the empirical processes of systems of interacting particles whose time evolution is given by coupled ordinary differential equations in particular as the population size tends to infinity we derive different versions of the porous medium equation as limit dynamics

interfaces with a corner point in one dimensional porous medium flow we prove that for a large class of initial distributions the solutions of the initial value problem for the one dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time this happens for example if the initial pressure is o x 2 as x 0 we also give sufficient conditions for the smoothness of the interface which improve previous results

how parabolic free boundaries approximate hyperbolic fronts a rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the cauchy problem for nonlinear first order and second order scalar conservation laws is presented among other properties it is shown that under appropriate assumptions parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

the number of peaks of nonnegative solutions to some nonlinear degenerate parabolic equations abstract we consider the cauchy problems for some nonlinear degenerate parabolic equations in r n for nonzero bounded nonnegative initial data having compact support and show that the set of peaks of the nonnegative solution consists of one point after a finite time

a highly efficient time space domain optimized method with lax wendroff type time discretization for the scalar wave equation abstract in this paper we propose an efficient time space domain optimized optts finite difference scheme to model 2d and 3d scalar wave propagation it adopts piecewise constant interpolation coefficients for several consecutive courant number ranges which avoids the extra time costs caused by loading the coefficients consecutively according to different wave velocities in heterogeneous media the new scheme is capable of suppressing numerical dispersion errors using a large time interval because we introduce a lax wendroff type time discretization which utilizes the wave field information of o m 1 m 2 neighboring grid points in each step with merely o m 1 m 2 computation where m 1 and m 2 are the stencil radii to generate the 2nd order and 4th order spatial derivatives the improved computational efficiency and accuracy of optts are validated by substantial experiments using both theoretical analysis and numerical modeling they show that in heterogeneous models the maximal courant numbers reach 1 0 and 0 85 in 2d and 3d cases respectively without severe temporal error and the scheme is computationally nearly three times faster than its high order counterpart providing a powerful tool for large scale modeling and high resolution imaging

nonlinear stochastic resonance with subthreshold rectangular pulses we analyze the phenomenon of nonlinear stochastic resonance sr in noisy bistable systems driven by pulsed time periodic forces the driving force contains within each period two pulses of equal constant amplitude and duration but opposite signs each pulse starts every half period and its duration is varied for subthreshold amplitudes we study the dependence of the output signal to noise ratio and the sr gain on the noise strength and the relative duration of the pulses we find that the sr gains can reach values larger than unity with maximum values showing a nonmonotonic dependence on the duration of the pulses

parabolic approach to nonlinear elliptic eigenvalue problems we consider the asymptotic profiles of the nonlinear parabolic flows ut um to show the geometric properties of the following elliptic nonlinear eigenvalue problems posed in a strictly convex domain for different values of p in the classical case p 1 we establish that log f is a strictly concave function when 0 p 1 the function is strictly concave while for there is at least a solution such that is strictly convex moreover eventual convexity properties for the parabolic flows are proved

a spatially second order alternating direction implicit adi method for solving three dimensional parabolic interface problems abstract a new matched alternating direction implicit adi method is proposed in this paper for solving three dimensional 3d parabolic interface problems with discontinuous jumps and complex interfaces this scheme inherits the merits of its ancestor for two dimensional problems while possesses several novel features such as a non orthogonal local coordinate system for decoupling the jump conditions two side estimation of tangential derivatives at an interface point and a new douglas rachford adi formulation that minimizes the number of perturbation terms to attack more challenging 3d problems in time discretization this new adi method is found to be of first order and stable in various experiments in space discretization the matched adi method achieves the second order accuracy based on simple cartesian grids for various irregularly shaped surfaces and spatial temporal dependent jumps computationally the matched adi method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3d parabolic equations in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom n i e o n furthermore unlike iterative methods the adi method is an exact or non iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed n therefore the proposed matched adi method provides a very promising tool for solving 3d parabolic interface problems

on the fictitious domain and interpolation formulations of the matched interface and boundary mib method this work overcomes the difficulty of dealing with large curvatures in a high order matched interface and boundary mib method proposed for solving elliptic interface problems the mib method smoothly extends the solution across the interface so that standard high order central finite difference schemes can be used without the loss of accuracy one feature of the mib is that it disassociates the discretization of the elliptic equation from the enforcement of interface jump conditions the other is to make iterative use of only the lowest order jump conditions to determine the fictitious values on extended domains it is of arbitrarily high order in convergence in principle however its applicability was hindered by the lack of sufficiently many grid points to determine all the fictitious values required for high order schemes at the location where the curvature of the interface is relatively large we remove this obstacle by introducing a new concept the disassociation between the discretization and the domain extension we show that the improved mib method is robust for handling general irregular interfaces by extensive numerical experiments on the poisson equation and the helmholtz equation to better understand the mib method and other potential high order interface schemes we propose an alternative interpolation formulation of the mib method and show that the new formulation is essentially equivalent to the improved one

optimal lower bounds on asymptotic support propagation rates for the thin film equation abstract we derive lower bounds on asymptotic support propagation rates for strong solutions of the cauchy problem for the thin film equation the bounds coincide up to a constant factor with the previously known upper bounds and thus are sharp our results hold in case of at most three spatial dimensions and n 1 2 92 the result is established using weighted backward entropy inequalities with singular weight functions to yield a differential inequality combined with some entropy production estimates the optimal rate of propagation is obtained to the best of our knowledge these are the first lower bounds on asymptotic support propagation rates for higher order nonnegativity preserving parabolic equations

well posedness for the navier slip thin film equation in the case of complete wetting we are interested in the thin film equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid film driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourth order parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with 13 14 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

implicit finite difference simulations of seismic wave propagation we propose a new finite difference modeling method implicit both in space and in time for the scalar wave equation we use a three level implicit splitting time integration method for the temporal derivative and implicit finite difference operators of arbitrary order for the spatial derivatives both the implicit splitting time integration method and the implicit spatial finite difference operators require solving systems of linear equations we show that it is possible to merge these two sets of linear systems one from implicit temporal discretizations and the other from implicit spatial discretizations to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm we give the complete derivations of the implicit splitting time integration method and the implicit spatial finite difference operators and present the resulting discretized formulas for the scalar wave equation we conduct a thorough numerical analysis on grid dispersions of this new implicit

role of fluctuations in the response of coupled bistable units to weak time periodic driving forces we analyze the stochastic response of a finite set of globally coupled noisy bistable units driven by rather weak time periodic forces we focus on the stochastic resonance and phase frequency synchronization of the collective variable defined as the arithmetic mean of the variable characterizing each element of the array for single unit systems stochastic resonance can be understood with the powerful tools of linear response theory proper noise induced phase frequency synchronization for a single unit system in this linear response regime does not exist for coupled arrays our numerical simulations indicate an enhancement of the stochastic resonance effects leading to gains larger than unity as well as genuine phase frequency synchronization the nonmonotonicity of the response with the strength of the coupling strength is investigated comparison with simplifying schemes proposed in the literature to describe the random response of the collective variable is carried out

regularity of solutions and interfaces of the porous medium equation via local estimates on demontre la regularite c pour des interfaces mobiles de solutions locales de l equation de milieu poreux

large time geometrical properties of solutions of the barenblatt equation of elasto plastic filtration abstract we study the geometric behavior for large times of the solutions of the following equation u t u t u 0 1 posed in the whole space r n for n 1 when the initial data are nonnegative continuous and compactly supported we prove that after a finite time log u becomes a concave function in the space variable and converges to all orders of differentiability to a certain parabolic shape so called barenblatt type profile which was described in kamin et al 1991 20 extensions to more general fully nonlinear equations are considered

highly accurate compact mixed methods for two point boundary value problems finite difference method and finite element method are widely used partly for their simplicity though these methods can obtain first order or second order of accuracy in this paper we give two highly accurate fourth order and sixth order of accuracy respectively while still quite simple schemes for two point boundary value problems we call them compact mixed methods since they can obtain numerical solutions for unknown function and its first derivative simultaneously to yield best numerical solutions we combine our compact mixed schemes with a direct solver and an iterative solver then we employ fourier method to analyze differencing errors shows that our compact mixed formulae are closer to the true wavenumber numerical experiments show compact mixed schemes are very efficient and highly accurate techniques

behaviour of the velocity of one dimensional flows in porous media the one dimensional flow of gas of density u through a porous medium obeys the equation u um x where m 1 x e r and t 0 we prove that the local velocity of the gas given by v mu 2u1 not only is bounded for t t 0 but approaches an n wave profile as t oo n waves are the typical asymptotic profiles for some first order conservation laws a class of nonlinear hiperbolic equations the case m 0 0 2 u x 0 u0 x forx e r where m is a positive constant and u0 satisfies 0 3 u0 e l r u0 0 a e u0 0 apart from its mathematical interest as a simple model of nonlinear evolution equation of possibly degenerate or singular parabolic type equation 0 1 and its n dimensional analogue appear in a number of applications cf 18 for a reference in the typical application to the flow of gas through a porous medium we have m 1 and u stands for the density of the gas writing 0 1 as a conservation law for the mass

concavity of solutions of the porous medium equation abstract the flow of a gas through a porous medium is governed by a degenerate quasilinear parabolic equation it is known that the nonnegative solutions to this equation possess a lower bound for the second derivative of the pressure in the spatial variables this bound plays an important role in the mathematical treatment and is related to the entropy of the flow since the solutions exhibit interfaces across which v sub x jumps positively no upper bound is possible globally for v subxx nevertheless it is proven that the concavity of v t in the region where v is positive is preserved in time this is in itself an interesting geometric property of the solution it also allows one to obtain precise information about the asymptotic behaviour of the flow

global 2d digital image correlation for motion estimation in a finite element framework a variational formulation and a regularized pyramidal multi grid implementation in this study the inverse problem of reconstructing the in plane 2d displacements of a monitored surface through a sequence of two dimensional digital images severely ill posed in hadamard s sense is deeply investigated a novel variational formulation is presented for the continuum 2d digital image correlation problem and critical issues such as semi coercivity and solution multiplicity are discussed by functional analysis tools in the framework of a galerkin finite element discretization of the displacement field a robust implementation for 2d digital image correlation is outlined aiming to attenuate the spurious oscillations which corrupt the deformation scenario especially when very fine meshes are adopted recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid multi grid cycles are performed ascending and descending along the pyramid with only one newton iteration per level irrespective of the tolerance satisfaction as if the problem were linear at each level the conventional least square matching functional is herein enriched by a tychonoff regularization provision preserving the solution against an unstable response the algorithm is assessed on the basis of both synthetic and truly experimental image pairs copyright 2013 john wiley sons ltd

approximation of eigenvalues of sturm liouville problems defined on a semi infinite domain abstract in this paper we describe how to approximate numerically the eigenvalues of a sturm liouville problem defined on a semi infinite interval the key idea is to transform the problem in such a way as to compress the semi infinite interval in a finite interval by applying a suitable change of the independent variable then we approximate each derivative in the sturm liouville equation thus obtained with finite difference schemes consequently we convert the sturm liouville problem into an algebraic eigenvalue problem the numerical results of the experiments show that the proposed approach is promising

derivative formulae and errors for non uniformly spaced points lagrange interpolation estimates a function at a reference position from known values of the function at distinct non uniformly spaced points x 1 x n here the corresponding n point finite difference formulae are derived to estimate derivatives up to order n 1 at a recurrence relation is derived that permits the errors to be determined as a taylor series to any accuracy the error coefficient multiplying the n j th derivative is a polynomial of order j 1 in the elementary symmetric functions for the displacements x 1 x n appendices state finite difference formulae to estimate the derivatives and the first four error terms for n 1 5

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

second order method for solving 3d elasticity equations with complex interfaces elastic materials are ubiquitous in nature and indispensable components in man made devices and equipments when a device or equipment involves composite or multiple elastic materials elasticity interface problems come into play the solution of three dimensional 3d elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equations this work introduces the matched interface and boundary mib method for solving 3d elasticity interface problems the proposed mib elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface the interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines such an enforcement determines the fictitious values a number of new techniques have been developed to construct efficient mib elasticity interface schemes for dealing with cross derivative in coupled governing equations the proposed method is extensively validated over both weak and strong discontinuities of the solution both piecewise constant and position dependent material parameters both smooth and nonsmooth interface geometries and both small and large contrasts in the poisson s ratio and shear modulus across the interface numerical experiments indicate that the present mib method is of second order convergence in both l and l 2 error norms for handling arbitrarily complex interfaces including biomolecular surfaces to our best knowledge this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins

long time behaviour of solutions of a class of nonlinear parabolic equations abstract in this paper the long time behaviour of solutions of a class of nonlinear parabolic equations is studied it is shown that the solutions of initial boundary value problem to the equations converge to a travelling wave solution of the equation or a self similar solution of a hamilton jacobi equation under certain conditions on initial and boundary values of the solutions

maximizing the cfl number of stable time space domain explicit finite difference modeling abstract the time space domain explicit finite difference fd method numerically solves wave equation by applying fd to approximate its spatial and temporal derivatives this method always faces the stability problem i theoretically analyze stability condition of this method for the acoustic wave equation and find that the maximum courant friedrichs lewy cfl number of stable modeling depends on the maximum value of the spatial fd dispersion relation the conventional method commonly determines spatial fd coefficients by approximating the spatial fd dispersion to the exact relation within the given wavenumber area however outside this area the dispersion relation and hence the maximum cfl number is uncontrollable to make it controllable i propose to simultaneously approximate the spatial fd dispersion relation to a given function outside this area further i develop an algorithm to maximize the maximum cfl number stability analysis and modeling examples demonstrate that the proposed method has better stability than the conventional method especially when the maximum effective wavenumber frequency is small for the 2d case the conventional method performs stable simulation using a cfl number of no more than 0 707 but the new method can perform stable simulation using a large cfl number for example 1 6 4 8 and 9 0 and non long spatial fd stencils

time space domain implicit finite difference methods for modeling acoustic wave equations it was proved that the implicit spatial finite difference fd method can obtain a higher accuracy than the explicit fd by using even a smaller operator length however when only second order fd in time is used the combined fd is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used the time space domain fd can suppress the temporal dispersion however because the spatial derivatives are solved explicitly the method suffers from spatial dispersion and a large spatial operator length has to be adopted we propose two effective time space domain implicit fd methods for modeling 2d and 3d acoustic wave equations the high order fd is first incorporated into the discretization for the second order temporal derivative and is combined with the implicit spatial fd the plane wave analysis method is used to derive the time space domain dispersion relations and two novel methods are proposed to determine the spatial and temporal fd coefficients in the joint time

local existence and regularity for a class of degenerate parabolic equations on etudie des equations paraboliques scalaires a une variable d espace de la forme u t f u u x uu xx ou f f u p s est une fonction lisse de u p s r 3 avec f s strictement positive

a pseudospectral fictitious point method for high order initial boundary value problems when pseudospectral approximations are used for space derivatives one often encounters spurious eigenvalues these can lead to severe time stepping difficulties for pdes this is especially the case for equations with high order derivatives in space requiring multiple conditions at one or both boundaries we note here that a very simple to implement fictitious point approach circumvents most of these difficulties the new approach is tested on the kuramoto sivashinsky equation and on a dispersive linear pde featuring a time space corner singularity

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

drift and diffusion in periodically driven renewal processes we consider the drift and diffusion properties of periodically driven renewal processes these processes are defined by a periodically time dependent waiting time distribution which governs the interval between subsequent events we show that the growth of the cumulants of the number of events is asymptotically periodic and develop a theory which relates these periodic growth coefficients to the waiting time distribution defining the periodic renewal process the first two coefficients which are the mean frequency and effective diffusion coefficient of the number of events are considered in greater detail they may be used to quantify stochastic synchronization

two state markovian theory of input output frequency and phase synchronization abstract a markovian dichotomic system driven by a deterministic time periodic force is analyzed in terms of the statistical properties of the switching events between the states the consideration of the counting process of the switching events leads us to define a discrete phase we obtain expressions for the instantaneous output frequency and phase diffusion associated to the dichotomic process as well as for their cycle averages these expressions are completely determined by the rates of escape from both states they are a convenient starting point for the study of the stochastic frequency and phase synchronization in a wide range of situations both classical and quantum in which two metastable states are involved

blow up analysis for a quasilinear parabolic system with multi coupled nonlinearities this paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux as the results of the interaction among the multi coupled nonlinearities in the system some appropriate conditions for global existence and global nonexistence of solutions are determined respectively

observability of lag synchronization of coupled chaotic oscillators lag synchronization is a recently discovered theoretical phenomenon where the dynamical variables of two coupled nonidentical chaotic oscillators are synchronized with a time delay relative to each other we investigate experimentally and numerically to what extent lag synchronization can be observed in physical systems where noise is inevitable our measurements and numerical computation suggest that lag synchronization is typically destroyed when the noise level is comparable to the amount of average system mismatch at small noise levels lag synchronization occurs in an intermittent fashion

array enhancement of stochastic synchronization and signal to noise ratio gain in the nonlinear regime of signal transmission the nonlinear transformation of an external noisy signal by an array of noninteracting elements with internal noise is considered the array simulates a neuronal system that processes spike trains it is shown that increasing the number of array elements entails significant extending of the stochastic synchronization region and improvement of the signal to noise ratio snr the effects are demonstrated for arrays of triggers overdamped bistable oscillators and fitzhugh nagumo systems the interrelation between snr improvement and the efficiency of information processing is discussed

method of lines solutions of the extended boussinesq equations a numerical solution procedure based on the method of lines for solving the nwogu one dimensional extended boussinesq equations is presented the numerical scheme is accurate up to fifth order in time and fourth order accurate in space thus reducing all truncation errors to a level smaller than the dispersive terms retained by most extended boussinesq models exact solitary wave solutions and invariants of motions recently derived by the authors are used to specify initial data for the incident solitary waves in the numerical model of nwogu and for the verification of the associated computed solutions the invariants of motions and several error measures are monitored in order to assess the conservative properties and the accuracy of the numerical scheme the proposed method of lines solution procedure is general and can be easily modified to solve a wide range of boussinesq like equations in coastal engineering

method of lines study of nonlinear dispersive waves in this study we consider partial differential equation problems describing nonlinear wave phenomena e g a fully nonlinear third order korteweg de vries kdv equation the fourth order boussinesq equation the fifth order kaup kupershmidt equation and an extended kdv5 equation first we develop a method of lines solution strategy using an adaptive mesh refinement algorithm based on the equidistribution principle and spatial regularization techniques on the resulting highly nonuniform spatial grids the computation of high order derivative terms appears particularly delicate and we focus attention on the selection of appropriate approximation techniques finally we solve several illustrative problems and compare our computational approach to conventional solution techniques

oscillatory systems driven by noise frequency and phase synchronization the phenomenon of effective phase synchronization in stochastic oscillatory systems can be quantified by an average frequency and a phase diffusion coefficient a different approach to compute the noise averaged frequencyis put forward the method is based on a threshold crossing rate pioneered by rice after the introduction of the rice frequency for noisy systems we compare this quantifier with those obtained in the context of other phase concepts such as the natural and the hilbert phase respectively it is demonstrated that the average rice frequency r typically supersedes the hilbert frequency h i e r h we investigate next the rice frequency for the harmonic and the damped bistable kramers oscillator both without and with external periodic driving exact and approximative analytic results are corroborated by numerical simulation results our results complement and extend previous findings for the case of noise driven inertial systems

asymptotic preserving and well balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes abstract the original well balanced wb framework of greenberg and leroux 1996 24 and gosse 2002 18 relying on non conservative nc products see lefoch and tzavaras 1999 41 is set up in order to efficiently treat the so called cattaneo model of chemotaxis in 1d see hillen and stevens 2000 29 it proceeds by concentrating the source terms onto dirac masses this allows to handle them by nc jump relations based on steady state equations which can be integrated explicitly a riemann solver is deduced and the corresponding wb godunov scheme completed with the standard hoff smoller theory see hillen and stevens 2000 29 for the diffusion reaction equation ruling the evolution of the chemoattractant concentration is studied in detail later following former results of gosse and toscani 2002 2003 21 22 a simple rewriting of the nc jump relations allows to generate another version of the same godunov scheme which is well adapted to the parabolic scaling involving a small parameter e the standard bv framework is used to study the uniform stability of this asymptotic preserving ap scheme with respect to e allows to pass to the limit and derive a simple centered discretization of the keller segel model finally results by filbet 2006 14 permit to pass to the complementary limit when the space step h is sent to zero numerical results are included to illustrate the feasibility and the efficiency of the method

the interfaces of one dimensional flows in porous media abstract a porous media equation pme has been used as a model for a number of physical phenomena heat diffusion at high temperatures boundary layer theory spread of a thin layer of viscous material and mainly the flow of gas in a porous medium the most distinctive characteristic of the solutions to pme as compared with the linear heat equation is the finite speed of propagation in this paper the properties of the interfaces are studied in terms of the initial data sometimes the interface is stationary for a certain time and then begins to move we characterize the existence of a positive waiting time and give bounds for it

compact finite difference methods for high order integro differential equations high order integro differential equations ides especially nonlinear are usually difficult to solve even numerically in this chapter we give a highly accurate compact finite difference method to efficiently solve integro differential equations including high order and nonlinear problems by numerical experiments we show that compact finite difference method for integro differential equations is fast and can obtain a high accuracy

spurious resonance in semidiscrete methods for the korteweg de vries equation a multiple scales analysis of semidiscrete methods for the korteweg de vries equation is conducted methods that approximate the spatial derivatives by finite differences with arbitrary order accuracy and the limiting method the fourier pseudospectral method are considered the analysis reveals that a resonance effect can occur in the semidiscrete solution but not in the solution of the continuous equation it is shown for the fourier pseudospectral discretization that resonance can only be caused by aliased modes the spurious semidiscrete solutions are investigated in numerical experiments and we suggest methods for avoiding spurious resonance

waiting time solutions of a nonlinear diffusion equation similarity solutions of the equation partial c partial t partial partial x c n partial c partial x n 0 are studied for both t 0 and t 0 and there is a stationary free boundary at x 0 and for t 0 have support in x at beta where beta 1 a 0 i e there is a moving free boundary x at beta by prescribing appropriate initial data c c 0 x x 0 at t t 0 these solutions demonstrate the existence of a waiting time t 0 before the free boundary moves a comparison theorem then enables a number of results to be obtained and conjectures to be made about the existence and length of waiting times for general initial data a three parameter family of solutions which exhibit blow up at t 0 is also constructed applications of these results to models for the spreading of a viscous blob on a hor

staircase free finite difference time domain formulation for general materials in complex geometries a stable cartesian grid staircase free finite difference time domain formulation for arbitrary material distributions in general geometries is introduced it is shown that the method exhibits higher accuracy than the classical yee 1966 scheme for complex geometries since the computational representation of physical structures is not of a staircased nature furthermore electromagnetic boundary conditions are correctly enforced the method significantly reduces simulation times as fewer points per wavelength are needed to accurately resolve the wave and the geometry both perfect electric conductors and dielectric structures have been investigated numerical results are presented and discussed

linear and nonlinear experimental regimes of stochastic resonance we investigate the stochastic resonance phenomenon in a physical system based on a tunnel diode the experimental control parameters are set to allow the control of the frequency and amplitude of the deterministic modulating signal over an interval of values spanning several orders of magnitude we observe both a regime described by the linear response theory and the nonlinear deviation from it in the nonlinear regime we detect saturation of the power spectral density of the output signal detected at the frequency of the modulating signal and a dip in the noise level of the same spectral density when these effects are observed we detect a phase and frequency synchronization between the stochastic output and the deterministic input

stochastic resonance and noise delayed extinction in a model of two competing species abstract we study the role of the noise in the dynamics of two competing species we consider generalized lotka volterra equations in the presence of a multiplicative noise which models the interaction between the species and the environment the interaction parameter between the species is a random process which obeys a stochastic differential equation with a generalized bistable potential in the presence of a periodic driving term which accounts for the environment temperature variation we find noise induced periodic oscillations of the species concentrations and stochastic resonance phenomenon we find also a nonmonotonic behavior of the mean extinction time of one of the two competing species as a function of the additive noise intensity

first applications of a formula for the error of finite sinc interpolation in former articles we have given a formula for the error committed when interpolating a several times differentiable function by the sinc interpolant on a fixed finite interval in the present work we demonstrate the relevance of the formula through several applications correction of the interpolant through the insertion of derivatives to increase its order of convergence improvement of the barycentric formula rational sinc interpolants with and without replacement of the usually unknown derivatives with finite differences convergence acceleration through extrapolation and improvement of one sided interpolants

theory of frequency and phase synchronization in a rocked bistable stochastic system we investigate the role of noise in the phenomenon of stochastic synchronization of switching events in a rocked overdamped bistable potential driven by white gaussian noise the archetype description of stochastic resonance we present an approach to the stochastic counting process of noise induced switching events starting from the markovian dynamics of the nonstationary continuous particle dynamics one finds upon contraction onto two states a non markovian renewal dynamics a proper definition of an output discrete phase is given and the time rate of change of its noise average determines the corresponding output frequency the phenomenon of noise assisted phase synchronization is investigated in terms of an effective instantaneous phase diffusion the theory is applied to rectangular shaped rocking signals versus increasing input noise strengths in this case for an appropriate choice of the parameter values the system exhibits a noise induced frequency locking accompanied by a very pronounced suppression of the phase diffusion of the output signal precise numerical simulations corroborate very favorably our analytical results the novel theoretical findings are also compared with prior ones

time space domain dispersion relation based finite difference method with arbitrary even order accuracy for the 2d acoustic wave equation high order finite difference fd methods have been widely used for numerical solution of acoustic wave equations it has been reported that the modeling accuracy is of 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation under the same discretization the present version of time space domain dispersion relation based fd method can improve the accuracy from 2nd order to 2m th order along eight directions for the 2d acoustic wave equation to increase the accuracy further we propose a new fd stencil for 2d acoustic wave equation modeling this new time space domain dispersion relation based fd stencil can reach the same arbitrary even order accuracy along all directions and is more accurate and more stable than the conventional one for the same m dispersion analysis and modeling examples demonstrate its advantages

adaptive method of lines solutions for the extended fifth order korteweg de vries equation in this paper we investigate the dynamics and interaction properties of recently discovered embedded solitons in an extended fifth order kdv model inspired by water waves in the presence of surface tension the dynamical behaviour of the solitons can be efficiently followed by using a moving or an adaptive finite difference mesh in combination with a suitable time integrator we will demonstrate this numerically for different types of wave solutions such as solitary waves multihumped waves and interacting waves

c regularity for the porous medium equation on etablit un resultat de regularite optimale pour l equation d un milieu poreux u t u m xx 0 x r t 0 u 0 u 0 pour m 1 et la donnee initiale u 0 positive continue a support compact connexe

finite difference modeling with adaptive variable length spatial operators most finite difference simulation algorithms use fixed length spatial operators to compute spatial derivatives the choice of length is dictated by computing cost stability and dispersion criteria that are satisfied globally we propose finite difference schemes with adaptive variable length spatial operators to decrease computing costs significantly without reducing accuracy these schemes adopt long operators in regions of low velocity and short operators in regions of high velocity two methods automatically determine variable operator lengths dispersion analysis along with 1d and 2d modeling demonstrates the validity and efficiency of our schemes in addition a hybrid absorbing boundary condition helps reduce unwanted reflections from model boundaries our scheme is more efficient than those based on variable grid methods for modeling migration and inversion of models with complex velocity structures because the latter require local grid refinement which usually increases memory requirements

theory of non markovian stochastic resonance we consider a two state model of non markovian stochastic resonance sr within the framework of the theory of renewal processes residence time intervals are assumed to be mutually independent and characterized by some arbitrary nonexponential residence time distributions which are modulated in time by an externally applied signal making use of a stochastic path integral approach we obtain general integral equations governing the evolution of conditional probabilities in the presence of an input signal these equations generalize earlier integral renewal equations by cox and others to the case of driving induced nonstationarity on the basis of these equations a response theory of two state renewal processes is formulated beyond the linear response approximation moreover a general expression for the linear response function is derived the connection of the developed approach with the phenomenological theory of linear response for manifest non markovian sr put forward i goychuk and p hanggi phys rev lett 91 070601 2003 is clarified and its range of validity is scrutinized the theory is then applied to sr in symmetric non markovian systems and to the class of single ion channels possessing a fractal kinetics

phase velocity and phase diffusion in periodically driven discrete state systems we develop a theory to calculate the effective phase diffusion coefficient and the mean phase velocity in periodically driven stochastic models with two discrete states this theory is applied to a dichotomically driven markovian two state system explicit expressions for the mean phase velocity the effective phase diffusion coefficient and the pe clet number are analytically calculated the latter indicates as a measure of phase coherence forced synchronization of the stochastic system with respect to the periodic driving and exhibits a bona fide resonance in a second step the theory is applied to a non markovian two state system modeling excitable systems the results prove again stochastic synchronization to the periodic driving and are in good agreement with simulations of a stochastic fitzhugh nagumo system

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

an augmented matched interface and boundary mib method for solving elliptic interface problem abstract in this paper a second order accurate augmented matched interface and boundary mib is introduced for solving two dimensional 2d elliptic interface problems with piecewise constant coefficients the augmented mib seamlessly combines several key ingredients of the standard mib augmented immersed interface method iim and explicit jump iim to produce a new fast interface algorithm based on the mib zeroth and first order jump conditions are enforced across an arbitrarily curved interface which yields fictitious values on cartesian nodes near the interface by using such fictitious values a simple procedure is proposed to reconstruct cartesian derivative jumps as auxiliary variables and couple them with the jump corrected taylor series expansions which allow us to restore the order of the central difference across the interface to two moreover by using the schur complement to disassociate the algebraic computation of auxiliary variables and function values the discrete laplacian can be efficiently inverted by using the fast fourier transform fft it is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size as a consequence the total computational cost of the augmented mib is about o n 2 log n for a cartesian grid with dimension n n in 2d therefore the augmented mib outperforms the classical mib in all cases by significantly reducing the cpu time while keeping the same second order of accuracy in dealing with complicated interfaces

an interpolation matched interface and boundary method for elliptic interface problems an interpolation matched interface and boundary imib method with second order accuracy is developed for elliptic interface problems on cartesian grids based on original mib method proposed by zhou et al y zhou g wei on the fictious domain and interpolation formulations of the matched interface and boundary method j comput phys 219 2006 228 246 explicit and symmetric finite difference formulas at irregular grid points are derived by virtue of the level set function the difference scheme using imib method is shown to satisfy the discrete maximum principle for a certain class of problems rigorous error analyses are given for the imib method applied to one dimensional 1d problems with piecewise constant coefficients and two dimensional 2d problems with singular sources comparison functions are constructed to obtain a sharp error bound for 1d approximate solutions furthermore we compare the ghost fluid method gfm immersed interface method iim mib and imib methods for 1d problems finally numerical examples are provided to show the efficiency and robustness of the proposed method

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

noise sensitivity of phase synchronization time in stochastic resonance theory and experiment recent numerical and heuristic arguments have revealed that the average phase synchronization time between the input and the output associated with stochastic resonance is highly sensitive to noise variation in particular there is evidence that this average time exhibits a cusplike behavior as the noise strength varies through the optimal value here we present an explicit formula for the average phase synchronization time in terms of the phase diffusion coefficient and the average frequency difference between the input and the output signals we also provide experimental evidence for the cusplike behavior by using a bistable microelectronic circuit system

determination of finite difference weights using scaled binomial windows the finite difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes conventional finite difference weights can be calculated either from taylor series expansions or by lagrange interpolation polynomials the finite difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain for this reason we also can derive finite difference operators by truncating the convolution series of the pseudospectral method various truncation windows can be employed for this purpose and they result in finite difference operators with different dispersion properties we found that there exists two families of scaled binomial windows that can be used to derive conventional finite difference operators analytically with a minor change these scaled binomial windows can also be used to derive optimized finite difference operators with enhanced dispersion properties

fast high dimensional node generation with variable density abstract we present an algorithm for producing discrete distributions with a prescribed nearest neighbor distance function our approach is a combination of quasi monte carlo q mc methods and weighted riesz energy minimization the initial distribution is a stratified q mc sequence with some modifications a suitable energy functional on the configuration space is then minimized to ensure local regularity the resulting node sets are good candidates for building meshless solvers and interpolants as well as for other purposes where a point cloud with a controlled separation covering ratio is required applications of a three dimensional implementation of the algorithm in particular to atmospheric modeling are also given