constructing spectral schemes of the immersed interface method via a global description of discontinuous functions a global description of discontinuous functions is introduced in this paper by expressing a discontinuous function as the sum of a smooth function and a correction term determined by jump conditions we turn the unknown function from a discontinuous one into a sufficiently smooth one when solving a differential equation spectral schemes are developed based on this concept with the intention of eliminating or reducing the gibbs oscillation finite difference schemes are also constructed as an alternative of the current immersed interface methods both spectral and finite difference schemes are tested on one and two dimensional cases

simulating the dynamics of inextensible vesicles by the penalty immersed boundary method in this paper we develop an immersed boundary ib method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid in order to take into account the inextensibility constraint of the vesicle the penalty immersed boundary pib method is used to virtually decouple the fluid and vesicle dynamics as numerical tests of our current pib method the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored and compared with the previous literature the method is also validated by a series of convergence study which confirms its consistent first order accuracy on the velocity field the vesicle configuration the vesicle area and the perimeter errors in addition the method is also applied to study a binary component vesicle problem

an efficient coding system for long source sequences the elias source coding scheme is modified to permit a source sequence of practically unlimited length to be coded as a single codeword using arithmetic of only limited precision the result is shown to be a nonblock arithmetic code of the first in first out fifo type source symbols are decoded in the same order as they were encoded codeword lengths which are near optimum for the specified statistical properties of the source can be achieved explicit encoding and decoding algorithms are provided which effectively implement the coding scheme applications to data compression and cryptography are suggested

fluid structure interactions research in the courant institute s applied mathematics laboratory the applied mathematics laboratory is a research laboratory within the mathematics department of the courant institute it was established to carry out physical experiments modeling and associated numerical studies in a variety of problems of interest to courant faculty postdocs and graduate and undergraduate students most of the research to date has involved fluid mechanics and we focus in this paper on the work that relates to the interaction of fluids with rigid movable or flexible bodies 2012 wiley periodicals inc

dynamics of a closed rod with twist and bend in fluid we investigate the instability and subsequent dynamics of a closed rod with twist and bend in a viscous incompressible fluid a new version of the immersed boundary ib method is used in which the immersed boundary applies torque as well as force to the surrounding fluid and in which the equations of motion of the immersed boundary involve the local angular velocity as well as the local linear velocity of the fluid an important feature of the ib method in this context is that self crossing of the rod is automatically avoided because the rod moves in a continuous interpolated velocity field a rod with a uniformly distributed twist that has been slightly perturbed away from its circular equilibrium configuration is used as an initial condition with the fluid initially at rest if the twist in the rod is sufficiently small the rod simply returns to its circular equilibrium configuration but for larger twists that equilibrium configuration becomes unstable and the rod undergoes large excursions before relaxing to a stable coiled configuration

an improvement of error exponents at low rates for the generalized version of concatenated codes corresp the lower bound m the error exponent for the generalized version of concatenated codes is shown to be improved at low rates for binary input memoryless channels

implicit second order immersed boundary methods with boundary mass abstract the immersed boundary method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures immersed boundary computations typically evaluate the elastic forces explicitly in the configuration of the immersed elastic structure in many applications this results in a severe restriction on the time step we present a semi implicit and a fully implicit second order accurate immersed boundary method the methods provide a natural way to handle mass on the immersed elastic structures we demonstrate their performance for a prototypical fluid structure interaction problem the methods are shown to possess superior stability properties that significantly alleviate the typically severe time step restriction of explicit computations

a wave dynamics criterion for optimization of mammalian cardiovascular system the cardiovascular system in mammals follows various optimization criteria covering the heart the vascular network and the coupling of the two through a simple dimensional analysis we arrived at a non dimensional number wave condition number that can predict the optimum wave state in which the left ventricular lv pulsatile power lv workload is minimized in a mammalian cardiovascular system this number is also universal among all mammals independent of animal size maintaining a value of around 0 1 by utilizing a unique in vitro model of human aorta we tested our hypothesis against a wide range of aortic compliance pulse wave velocity we concluded that the optimum value of the wave condition number remains to be around 0 1 for a wide range of aorta compliance that we could simulate in our in vitro system

a class of distortionless codes designed by bayes decision theory the problem of distortionless encoding when the parameters of the probabilistic model of a source are unknown is considered from a statistical decision theory point of view a class of predictive and nonpredictive codes is proposed that are optimal within this framework specifically it is shown that the codeword length of the proposed predictive code coincides with that of the proposed nonpredictive code for any source sequence a bound for the redundancy for universal coding is given in terms of the supremum of the bayes risk if this supremum exists then there exists a minimax code whose mean code length approaches it in the proposed class of codes and the minimax code is given by the bayes solution relative to the prior distribution of the source parameters that maximizes the bayes risk

an efficient semi implicit immersed boundary method for the navier stokes equations the immersed boundary method is one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to require small time steps to maintain stability when solved with an explicit method many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint but none of them give satisfactory performance in this paper we propose an efficient semi implicit scheme to remove this stiffness from the immersed boundary method for the navier stokes equations the construction of our semi implicit scheme consists of two steps first we obtain a semi implicit discretization which is proved to be unconditionally stable this unconditionally stable semi implicit scheme is still quite expensive to implement in practice next we apply the small scale decomposition to the unconditionally stable semi implicit scheme to construct our efficient semi implicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit scheme can be solved explicitly in the spectral space thus the computational cost of our semi implicit schemes is comparable to that of an explicit scheme our extensive numerical experiments show that our semi implicit scheme has much better stability property than an explicit scheme this offers a substantial computational saving in using the immersed boundary method

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

validation of a simple method for representing spheres and slender bodies in an immersed boundary method for stokes flow on an unbounded domain we test the efficacy of using a single lagrangian point to represent a sphere and a one dimensional array of such points to represent a slender body in a new immersed boundary method for stokes flow a numerical parameter the spacing of the eulerian grid is used to determine the effective radius of the immersed sphere or slender body such representations are much less expensive computationally than those with two or three dimensional meshes of lagrangian points to perform this test we develop a numerical method to solve the discretized stokes equations on an unbounded eulerian grid which contains an arbitrary configuration of lagrangian points that apply force to the fluid and that move with the fluid we compare results computed with this new immersed boundary method to known results for spheres and rigid cylinders in stokes flow in r 3 we find that for certain choices of parameters the interactions with the fluid of a single lagrangian point accurately replicate those of a sphere of some particular radius independent of the location of the point with respect to the eulerian grid the interactions of a linear array of lagrangian points for certain choices of parameters accurately replicate those of a cylinder of some particular radius independent of the position and orientation of the array with respect to the eulerian grid the effective radius of the sphere and the effective radius of the cylinder turn out to be related in a simple and natural way our results suggest recipes for choosing parameters that should be useful to practitioners one surprising result is that one must not use too many lagrangian points in an array another is that the approximate delta functions traditionally used in the immersed boundary method perform much better than higher order delta functions with the same support

on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems the immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a visco elastic structure in m c lai simulations of the flow past an array of circular cylinders as a test of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1998 m c lai c s peskin an immersed boundary method with formal second order accuracy and reduced numerical viscosity j comput phys 160 2000 705 719 lai and peskin introduced a formally second order accurate immersed boundary method but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions consequently in practice only first order convergence rates have been observed in the present work we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid structure interaction problem involving an immersed viscoelastic shell of finite thickness studied over a broad range of reynolds numbers we consider two sets of material properties for the viscoelastic structure including a case where the material properties of the coupled system are discontinuous at the fluid structure interface for both sets of material properties the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations

removing the stiffness of elastic force from the immersed boundary method for the 2d stokes equations the immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization in this paper we propose several efficient semi implicit schemes to remove this stiffness from the immersed boundary method for the two dimensional stokes flow first we obtain a novel unconditionally stable semi implicit discretization for the immersed boundary problem using this unconditionally stable discretization as a building block we derive several efficient semi implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization our stability analysis and extensive numerical experiments show that our semi implicit schemes offer much better stability property than the explicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit schemes can be solved explicitly in the spectral space thus the computational cost of our semi implicit schemes is comparable to that of an explicit scheme but with a much better stability property

two dimensional simulations of valveless pumping using the immersed boundary method flow driven by pumping without valves is examined motivated by biomedical applications cardiopulmonary resuscitation cpr and the human fetus before the development of the heart valves the direction of flow inside a loop of tubing which consists of almost rigid and flexible parts is investigated when the boundary of one end of the flexible segment is forced periodically in time despite the absence of valves net flow around the loop may appear in these simulations the magnitude and even the direction of this flow depend on the driving frequency of the periodic forcing

pumping effects in models of periodically forced flow configurations a periodically forced system of differential equations is defined to be a pump if there exists an asymptotically periodic solution with non equilibrium mean it is proved that such systems exist the definition is based on physical and numerical observations of pumping in models of asymmetric flow configurations for models with rigid pipes and tanks physical explanations for the pumping effects are derived one of the pumps is an internally forced linear system for externally forced nonlinear rigid pipe models necessary and sufficient conditions for pumping are given it is then demonstrated in a general setting that no externally forced linear pump exists

universal modeling and coding the problems arising in the modeling and coding of strings for compression purposes are discussed the notion of an information source that simplifies and sharpens the traditional one is axiomatized and adaptive and nonadaptive models are defined with a measure of complexity assigned to the models a fundamental theorem is proved which states that models that use any kind of alphabet extension are inferior to the best models using no alphabet extensions at all a general class of so called first in first out fifo arithmetic codes is described which require no alphabet extension devices and which therefore can be used in conjunction with the best models because the coding parameters are the probabilities that define the model their design is easy and the application of the code is straightforward even with adaptively changing source models

2 d parachute simulation by the immersed boundary method parachute aerodynamics involves an interaction between the flexible elastic porous parachute canopy and the high speed airflow relative to the parachute through which the parachute falls computer simulation of parachute dynamics typically simplifies the problem in various ways e g by considering the parachute as a rigid bluff body here we avoid such simplification by using the immersed boundary ib method to study the full fluid structure interaction the ib method is generalized to handle porous immersed boundaries and the generalized method is used to study the influence of porosity on parachute stability

simulations of the whirling instability by the immersed boundary method when an elastic filament spins in a viscous incompressible fluid it may undergo a whirling instability as studied asymptotically by wolgemuth powers and goldstein phys rev lett 84 2000 pp 16 23 we use the immersed boundary ib method to study the interaction between the elastic filament and the surrounding viscous fluid as governed by the incompressible navier stokes equations this allows the study of the whirling motion when the shape of the filament is very different from the unperturbed straight state

simulations of valveless pumping in an open elastic tube mathematical models and numerical simulations of flows driven by pumping without valves valveless pumping are presented this work has been originally motivated by a biomedical objective to explain the complicated valveless blood flow mechanism in the circulation for instance the fetal circulation before the formation of valves and blood circulation during cardiopulmonary resuscitation a mathematical model of valveless pumping either in a closed loop system or in an open system consists of a couple of tubes with different elasticities or radii in this work we develop new models which are not necessarily connected by two different materials only one open elastic tube and an open elastic tube contained in a closed almost rigid tank although only one soft material is used or the soft and rigid materials are not connected we have observed the existence of a net flow driven by the periodic compress and release action and the important features of valveless pumping that have been observed in earlier models or experiments we have also shown that there exist mostly one directional net flows inside the elastic tube if a long and thin tube is considered and the pumping is applied to the short portion near the edge of the tube this might explain why most earlier reports on the physical experiments could not easily observe the existence of both directional net flows depending on the parameters such as frequency another important result is that the direction and the magnitude of a net flow can be explained by the sign and the amount of power which is work done on the fluid by the fluid pressure and the elastic wall over one period respectively a new feature in this work is that only one elastic material is sufficient to show the existence of a net flow in a valveless pump system because of a simple structure in our new model it is much easier to construct a valveless pump system applicable to real world applications such as a microelectromechanical system device

linear time adaptive arithmetic coding the issue of how arithmetic coding should be implemented is addressed a data structure is described and shown to support adaptive arithmetic coding on an arbitrary sized alphabet in time linear in the size of the inputs and outputs experimental results that show the method to be useful even on relatively small alphabets are given

a comparison of enumerative and adaptive codes the problem of noiselessly encoding a message when prior statistics are known is considered the close relationship of arithmetic and enumerative coding for this problem is shown by computing explicit arithmetic coding probabilities for various enumerative coding examples this enables a comparison to be made of the coding efficiency of markov models and enumerative codes as well as a new coding scheme intermediate between the two these codes are then extended to messages whose statistics are not known em a priori two adaptive codes are described for this problem whose coding efficiency is upper bounded by the extended enumerative codes on some practical examples the adaptive codes perform significantly better than the nonadaptive ones

a discrete forcing immersed boundary method for the fluid structure interaction of an elastic slender body we present an immersed boundary ib method for the simulation of flow around an elastic slender body the present method is based on the discrete forcing ib method for a stationary rigid body proposed by kim kim and choi 2001 25 the discrete forcing approach is used to relieve the limitation on the computational time step size the incompressible navier stokes equations are implicitly coupled with the dynamic equation for an elastic slender body motion the first is solved in the eulerian coordinate and the latter is described in the lagrangian coordinate the elastic slender body is modeled as a thin and flexible solid and is segmented by finite number of thin blocks each block is moved by external and internal forces such as the hydrodynamic elastic and buoyancy forces where the hydrodynamic force is obtained directly from the discrete forcing used in the ib method all the spatial derivative terms are discretized with the second order central difference scheme the present method is applied to three different fluid structure interaction problems flows around a flexible filament a flapping flag in a free stream and a flexible flapping wing in normal hovering respectively computations are performed at maximum cfl numbers of 0 75 1 the results obtained agree very well with those from previous studies

nonblock coding of binary sources by probabilistic enumeration a simple and efficient nonblock coding scheme for binary sources is suggested it uses a probability model to map any source string to a unique integer and thereby defines an enumeration of all possible strings contrary to normal enumerative coding the new method is noncombinatorial and operates sequentially incrementing the code value symbol by symbol by simple arithmetic it is akin to arithmetic coding but does not use intervals for memoryless sources the redundancy per symbol is shown to be asymptotically less than p3 bits where p is the smaller of symbol probabilities especially for integer valued 1 p the code is asymptotically optimal these results are confirmed by both analysis and experiments with arbitrary precision arithmetic in a practical implementation the source is partitioned into substrings enabling restricted precision arithmetic thanks to subtle implicit coding the additional redundancy is marginal the source model can be also context based and even adaptivity can be incorporated a peculiarity of the method is that decoding is done backwards lifo hence for higher order models the string suffix not prefix is used as the context domain the speed of the practical version is close to that of binary arithmetic coders

spectral element simulations of interactive particles in a fluid abstract this paper presents three dimensional simulations of interactive particles in a fluid the originality of this paper is the integrated eulerian lagrangian algorithms implemented with high order methods in both space and time for numerical solutions to navier stokes equations with additional source terms implementation of fourth order stiffly stable schemes was tested for time integration before they were used in computations in order to provide the h p type mesh refinement for extra flexibility to achieve higher spatial resolution a modal spectral element method was used to solve governing equations in three dimensions another originality is the efficiency in handling moving particles without adaptive or moving mesh simulation results were validated with experimental data and verified with exact solutions in addition numerical results were also compared with solutions at higher resolutions and good agreement was accomplished results indicate that algorithms and implementations are accurate and appropriate for investigating three dimensional interactive particles in a fluid involving many moving objects

an immersed boundary method based on the lattice boltzmann approach in three dimensions with application the immersed boundary ib method originated by peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid the navier stokes n s equations in the ib method are usually solved using numerical methods such as fft and projection methods here in our work the n s equations are solved by an alternative approach the lattice boltzmann method lbm compared to many conventional n s solvers the lbm can be easier to implement and more convenient for modeling additional physics in a problem this alternative approach adds extra versatility to the immersed boundary method in this paper we discuss the use of a 3d lattice boltzmann model d3q19 within the ib method we use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3d channel and determine a drag scaling law for the sheet our main conclusions are 1 the hybrid method is convergent with first order accuracy which is consistent with the immersed boundary method in general 2 the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow

a galerkin energy preserving method for two dimensional nonlinear schrodinger equation abstract in this paper a galerkin energy preserving scheme is proposed for solving nonlinear schrodinger equation in two dimensions the nonlinear schrodinger equation is first rewritten as an infinite dimensional hamiltonian system following the method of lines the spatial derivatives of the nonlinear schrodinger equation are approximated with the aid of the galerkin methods the resulting ordinary differential equations can be cast into a canonical hamiltonian system a fully discretized scheme is then devised by considering an average vector field method in time moreover based on the fast fourier transform and the matrix diagonalization method a fast solver is developed to solving the resulting algebraic equations finally the proposed scheme is employed to capture the blow up phenomena of the nonlinear schrodinger equation