an implicit fourth order compact finite difference scheme for one dimensional burgers equation abstract a fourth order compact finite difference method is proposed in this paper to solve one dimensional burgers equation the newly proposed method is based on the hopf cole transformation which transforms the original nonlinear burgers equation into a linear heat equation and transforms the dirichlet boundary condition into the robin boundary condition the linear heat equation is then solved by an implicit fourth order compact finite difference scheme a compact fourth order formula is also developed to approximate the robin boundary conditions while the initial condition for the heat equation is approximated using simpson s rule to maintain overall fourth order accuracy numerical experiments have been conducted to demonstrate the efficiency and high order accuracy of this method the numerical results also show that the method is unconditionally stable as there is no constraint on time step size

finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms abstract in this paper a finite difference scheme is presented for the generalized time fractional telegraph equation gtfte defined using generalized fractional derivative gfd terms introduced recently the generalization of fractional derivatives is done by introducing scale and weight functions and for their particular choices gfd reduces to caputo and riemann liouville derivatives we present the solution behaviour of the gtfte by changing the weight and scale functions in gfd the convergence and the stability of the finite difference scheme fds are also presented and for the numerical simulation of the fds we consider examples which validate our numerical method

adaptive mesh and algorithm refinement using direct simulation monte carlo adaptive mesh and algorithm refinement amar embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement amr hierarchy the coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and coarse levels of amr direct simulation monte carlo dsmc is used as the particle algorithm embedded within a godunov type compressible navier stokes solver several examples are presented and compared with purely continuum calculations

a new explicit method for the diffusion convention equation abstract we consider here the finite difference approximation to the diffusion convection equation from which new explicit formulation are obtained which are asymmetric these explicit schemes can then be used to develop a new class of methods called group explicit as introduced in 2 theoretical aspects of the stability consistency convergence and truncation errors of this new class of methods is briefly discussed and numerical evidence presented to confirm our recomendations

hybrid continuum molecular simulations of transient gas flows with rarefaction an adaptively coupled continuum molecular approach for compressible viscous flows in transient calculations is presented the continuum domain is described by the unsteady compressible navier stokes equations and the molecular domain is solved by direct simulation monte carlo a strategy is described to extend the overlapped schwarz method with dirichlet dirichlet boundary condition coupling procedure to transient simulations the method has been successfully validated against full direct simulation monte carlo results for transient simulations of a one dimensional shock tube and a two dimensional pressure driven slit flow a sensitivity analysis showed that the used overlapped schwarz coupling method with dirichlet dirichlet boundary conditions is only weakly sensitive to various parameters e g the exact position of the continuum molecular interface the size of the overlap region and the scatter in the molecular solution which is a clear advantage over the more commonly used flux based coupling technique another advantage of the chosen coupling method is that the continuum and molecular time steps can be decoupled and that in general a coupling time step can be used that is much larger than the molecular time step the study performed also highlighted a limit of the method it was found to be necessary to keep the courant number based on the coupling time step the cell size in the continuum region and the molecular most probable velocity below 1 to avoid instabilities

generation of the chapman enskog distribution a pair of simple efficient and robust algorithms for generating random velocities sampled from the chapman enskog distribution is presented

development and verification of a coupled dsmc ns scheme using unstructured mesh an efficient and accurate parallel coupled dsmc ns method using three dimensional unstructured grid topology is proposed and verified for the simulation of high speed gas flows involving continuum and rarefied regimes a domain overlapping strategy taking advantage of unstructured data format with dirichlet dirichlet type boundary conditions based on two breakdown parameters is used iteratively to determine the choice of solvers in the spatial domain the selected breakdown parameters for this study include 1 a local maximum knudsen number defined as the ratio of the local mean free path and local characteristic length based on property gradient and 2 a thermal non equilibrium indicator defined as the ratio of the difference between translational and rotational temperatures to the translational temperature a supersonic flow m 4 over a quasi 2 d 25 wedge is employed as the first step in verifying the present coupled method the results of simulation using the coupled method are in excellent agreement with those of the pure dsmc method which is taken as the benchmark solution effects of the size of overlapping regions and the choice of breakdown parameters on the convergence history are discussed results show that the proposed iteratively coupled method predicts the results more accurately as compared to the one shot coupled method which has been often used in practice further a realistic 3 d nitrogen flow which two near continuum parallel orifice jets underexpand into a near vacuum environment is simulated using the present coupled method to demonstrate its capability finally developments in extending the present method are also outlined in this paper

an h adaptive mesh method for boltzmann bgk hydrodynamics coupling we introduce a coupled method for hydrodynamic and kinetic equations on 2 dimensional h adaptive meshes we adopt the euler equations with a fast kinetic solver in the region near thermodynamical equilibrium while use the boltzmann bgk equation in kinetic regions where fluids are far from equilibrium a buffer zone is created around the kinetic regions on which a gradually varying numerical flux is adopted based on the property of a continuously discretized cut off function which describes how the flux varies the coupling will be conservative in order for the conservative 2 dimensional specularly reflective boundary condition to be implemented conveniently the discrete maxwellian is approximated by a high order continuous formula with improved accuracy on a disc instead of on a square domain the h adaptive method can work smoothly with a time split numerical scheme through h adaptation the cell number is greatly reduced this method is particularly suitable for problems with hydrodynamics breakdown on only a small part of the whole domain so that the total efficiency of the algorithm can be greatly improved three numerical examples are presented to validate the proposed method and demonstrate its efficiency

a numerical solution of burgers equation in this paper the burgers equation which is one dimensional quasi linear parabolic partial differential equation was solved by a variational method constructed on the method of discretization in time the numerical results obtained by these ways for various values of viscosity have been compared with the exact solution it was seen that they were in good agreement with each other

unimodular eigenvalues uniformly distributed sequences and linear dynamics we study increasing sequences of positive integers nk k 1 with the following property every bounded linear operator t acting on a separable banach or hilbert space with supk 1 tnk has a countable set of unimodular eigenvalues whether this property holds or not depends on the distribution modulo one of sequences nk k 1 r or on the growth of nk 1 nk counterexamples to some conjectures in linear dynamics are given for instance a hilbert space operator which is frequently hypercyclic chaotic but not topologically mixing is constructed the situation of c0 semigroups is also discussed

invariant measures for frequently hypercyclic operators abstract we investigate frequently hypercyclic and chaotic linear operators from a measure theoretic point of view among other things we show that any frequently hypercyclic operator t acting on a reflexive banach space admits an invariant probability measure with full support which may be required to vanish on the set of all periodic vectors for t that there exist frequently hypercyclic operators on the sequence space c 0 admitting no ergodic measure with full support and that if an operator admits an ergodic measure with full support then it has a comeager set of distributionally irregular vectors we also give some necessary and sufficient conditions which are satisfied by all the known chaotic operators for an operator t to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures finally we give a baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support

invariant gaussian measures for operators on banach spaces and linear dynamics we give conditions for an operator t on a complex separable banach space x with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non degenerate invariant gaussian measure with respect to which it is weak mixing the existence of such a measure depends on the geometry of the banach space and on the possibility of parametrizing the mathbb t eigenvector fields of t in a regular way we also investigate the connection with frequent hypercyclicity

flow simulation using generalized static and dynamic grids a new approach is presented for a e ow simulation system using generalized grids in a generalized grid the physical domain of interest is decomposed into cells with an arbitrary number of edges or faces the grid can be of structured unstructured or hanging node type or an arbitrary combination of the types a cell face based data structure is used to storethegrid information a e ow simulation system is developed forgeneralized gridsthat can handle static and dynamic grids the full navier stokes equations in the integral form are taken as the relations that govern the e uid e ow a cell centered e nite volume scheme is developed for solving these governing equations the numerical e ux across the cell faces is evaluated by an upwind scheme based on roe s approximate riemann solver a higher order scheme is formulated by utilizing taylor s series expansion and green s theorem limiter functions are used to preserve monotonocity the skin friction coefe cient is used to evaluate the accuracy of the limiterfunctions thegeneralized minimalresidualmethod isutilized tosolvethesparselinearsystem ofequations resulting from thelinearization of the e ux vectors thespalart allmaras one equation turbulence model has been implemented for the generalized grids and is used to evaluate the turbulent viscosity for dynamically moving bodies the equations of classical mechanics are used to predict the trajectory based on the external aerodynamic and body forces a variety of computational examples are presented to demonstratethe wide rangeof applications and the results are compared with experimental data whenever available

a hybrid method for hydrodynamic kinetic flow part i a particle grid method for reducing stochastic noise in kinetic regimes in this work we present a hybrid particle grid monte carlo method for the boltzmann equation which is characterized by a significant reduction of the stochastic noise in the kinetic regime the hybrid method is based on a first order splitting in time to separate the transport from the relaxation step the transport step is solved by a deterministic scheme while a hybrid dsmc based method is used to solve the collision step such a hybrid scheme is based on splitting the solution in a collisional and a non collisional part at the beginning of the collision step and the dsmc method is used to solve the relaxation step for the collisional part of the solution only this is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution performing collisions as in a standard dsmc method and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution the latter is added to a piecewise constant reconstruction of the non collisional part of the solution which in fact remains unchanged during the relaxation step numerical results show that the stochastic noise is significantly reduced at large knudsen numbers with respect to the standard dsmc method indeed in this algorithm the particle scheme is applied only on the collisional part of the solution so only this fraction of the solution is affected by stochastic fluctuations but since the collisional part of the solution reduces as the knudsen number increases stochastic noise reduces as well at large knudsen numbers

dynamics of non convolution operators and holomorphy types abstract in this article we study the hypercyclic behavior of non convolution operators defined on spaces of analytic functions of different holomorphy types over banach spaces the operators in the family we analyze are a composition of differentiation and composition operators and are extensions of operators in h c studied by aron and markose in 2004 the dynamics of this class of operators in the context of one and several complex variables was further investigated by many authors it turns out that the situation is somewhat different and that some purely infinite dimensional difficulties appear for example in contrast to the several complex variable case it may happen that the symbol of the composition operator has no fixed points and still the operator is not hypercyclic we also prove a runge type theorem for holomorphy types on banach spaces

a new difference scheme with high accuracy and absolute stability for solving convection diffusion equations in this paper we use a semi discrete and a pade approximation method to propose a new difference scheme for solving convection diffusion problems the truncation error of the difference scheme is o h email protected 5 it is shown through analysis that the scheme is unconditionally stable numerical experiments are conducted to test its high accuracy and to compare it with crank nicolson method

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

hypercyclicity and unimodular point spectrum abstract we show that an operator on a separable complex banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic we apply this result to construct hypercyclic operators with prescribed k unimodular point spectrum we show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators and prove that the family of composition operators c on h 2 d where is a disk automorphism having 1 as attractive fixed point has a residual set of common hypercyclic vectors

chaoticity and invariant measures for a cell population model abstract we present a structured model of a cell reproduction system given by a partial differential equations with a nonlocal division term this equation generates semiflows acting on some subspaces of locally integrable functions we show that these semiflows possess invariant mixing measures positive on open sets from this it follows that the system is chaotic i e it has dense trajectories and each trajectory is unstable we also show the chaoticity of this system in the sense of devaney

new lifting relations for estimating lbm distribution functions from corresponding macroscopic quantities based on equilibrium and non equilibrium moments due to superior accuracy and stability of multiple relaxation time mrt collision operator over its single relaxation time srt counterpart new lifting relations are proposed here to construct single particle distribution functions for mrt lbm from macroscopic variables using these lifting relations a new hybrid fvm lb method is presented called finite type lb hybrid method which is consistent with mrt lbm in this new hybrid method single particle distribution functions in mrt lbm sub domain boundaries are computed using equilibrium and non equilibrium moments these moments are computed in navier stokes fvm sub domain boundaries using macroscopic variables and their derivatives the new method is validated by solving three benchmark problems i e two and three dimensional lid driven cavity flows and two dimensional unsteady flow around a squared section cylinder these problems are analyzed with pure fvm pure lbm and finite type lb hybrid method ftlbhm and the related results are compared with each other and with benchmark data these comparisons clearly demonstrate the accuracy of the present novel methodology for simulating steady unsteady flow fields in two and three dimensions new lifting relations are developed to construct a hybrid ns fvm lbm mrt method hybrid method is implemented to a 3 dimensional problem for the first time based on authors knowledge using new hybrid fvm lbm mrt method to simulate flow around complex geometries such as porous media

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

a fourth order method of the convection diffusion equations with neumann boundary conditions abstract in this paper we have developed a fourth order compact finite difference scheme for solving the convection diffusion equation with neumann boundary conditions firstly we apply the compact finite difference scheme of fourth order to discrete spatial derivatives at the interior points then we present a new compact finite difference scheme for the boundary points which is also fourth order accurate finally we use a pade approximation method for the resulting linear system of ordinary differential equations the presented scheme has fifth order accuracy in the time direction and fourth order accuracy in the space direction it is shown through analysis that the scheme is unconditionally stable numerical results show that the compact finite difference scheme gives an efficient method for solving the convection diffusion equations with neumann boundary conditions

strong mixing measures for linear operators and frequent hypercyclicity abstract we construct strongly mixing invariant measures with full support for operators on f spaces which satisfy the frequent hypercyclicity criterion for unilateral backward shifts on sequence spaces a slight modification shows that one can even obtain exact invariant measures

parallel dsmc method using dynamic domain decomposition a general parallel direct simulation monte carlo method using unstructured mesh is introduced which incorporates a multi level graph partitioning technique to dynamically decompose the computational domain the current dsmc method is implemented on an unstructured mesh using particle raytracing technique which takes the advantages of the cell connectivity information in addition various strategies applying the stop at rise sar ieee trans comput 1988 39 1073 1087 scheme is studied to determine how frequent the domain should be re decomposed a high speed bottom driven cavity flow including small medium and large problems based on the number of particles and cells are simulated corresponding analysis of parallel performance is reported on ibm sp2 parallel machine up to 64 processors analysis shows that degree of imbalance among processors with dynamic load balancing is about 1 6 1 2 of that without dynamic load balancing detailed time analysis shows that degree of imbalance levels off very rapidly at a relatively low value with increasing number of processors when applying dynamic load balancing which makes the large problem size fairly scalable for processors more than 64 in general optimal frequency of activating sar scheme decreases with problem size at the end the method is applied to compute two two dimensional hypersonic flows a three dimensional hypersonic flow and a three dimensional near continuum twin jet gas flow to demonstrate its superior computational capability and compare with experimental data and previous simulation data wherever available

a combined continuum dsmc technique for multiscale analysis of microfluidic filters a multiscale method that combines continuum fluid models with the direct simulation monte carlo dsmc method is presented the continuum regions are treated by stokes equations and a scattered point based finite cloud method is employed to solve the stokes equations the continuum and dsmc regions are combined by an overlapped schwarz alternating method with dirichlet dirichlet type boundary conditions a scattered point interpolation scheme is developed to interpolate the solution between subdomains the convergence characteristics of the multiscale approach are investigated in detail specifically the dependence of convergence on the overlap size the dsmc noise and the number of time steps employed in the dsmc algorithm are studied while the convergence depends weakly on the dsmc noise and the overlap size the number of dsmc time steps simulated in each coupling iteration should be selected so that the total time steps simulated until convergence of the coupled process is close to the time constant of the dsmc subsystem steady state analysis of microfluidic filters is studied in detail using the multiscale approach the multiscale approach is also applied for the simulation of a membrane with an array of microfluidic filters and a dual stage microfluidic device with an array of microfluidic filters for particle trapping and sorting

restrictive taylor s approximation for solving convection diffusion equation in this paper we shall develop a new explicit method for solve the convection diffusion equation which will exhibit several advantageous features highly accurate fast and with good results whatever the exact solution is too large the stability region is discussed the error upper bound is proved the obtained results for a test problem compared with the exact solution and its douglas approximation it proves the mentioned advantages

chaotic semigroups from second order partial differential equations abstract we give general conditions on given parameters to ensure devaney and distributional chaos for the solution c 0 semigroup corresponding to a class of second order partial differential equations we also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups in the case of chaos we prove that the c 0 semigroup admits a strongly mixing measure with full support we also give concrete examples of partial differential equations such as the telegraph equation whose solutions satisfy these properties

weighted finite difference techniques for the one dimensional advection diffusion equation various numerical techniques will be developed and compared for solving the one dimensional advection diffusion equation with constant coefficient these techniques are based on the two level finite difference approximations the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyett this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes the new methods are more accurate and are more efficient than the conventional techniques these schemes are free of numerical diffusion the results of a numerical experiment are presented and the accuracy and central processor cpu time needed are discussed and compared

extended one step time integration schemes for convection diffusion equations abstract we first describe a one parameter family of unconditionally stable third order time integration schemes for the convection diffusion equation u t cu x vu xx based on the extended trapezoidal formulas of usmani and agarwal 1 interestingly there exists a method that is fourth order as well as unconditionally stable we then describe a one parameter family of unconditionally stable fourth order time integration schemes based on the extended simpson rules of chawla et al 2 again there exists a method that is fifth order as well as unconditionally stable the stability and accuracy of the obtained methods are tested and compared with the widely used method of crank nicolson by considering three problems of practical interest

on the finite difference approximation to the convection diffusion equation abstract in this paper the system of ordinary differential equations arisen from discretizing the convection diffusion equation with respect to the space variable to compute its approximate solution along a time level is considered this system involves in computing e k a y for some vector y where k is the time step size and a is a large tridiagonal toeplitz matrix the common ways to compute an approximate solution of the convection diffusion equation are based on replacing e k a y by an its approximation we give an explicit expression for the exact value of e k a y and then the numerical results of our method with that of some well known methods are compared

a compact finite volume method and its extrapolation for elliptic equations with third boundary conditions a fourth order compact finite volume method is constructed for one and two dimensional elliptic equations with third boundary conditions in this paper taking two point boundary value problem of third kind as an example we derive some useful high accuracy post processing formulas to recover the numerical derivatives at the nodes or midpoints of the elements we also improve the accuracy of the compact finite volume scheme from order 4 to 6 based on richardson extrapolation by rigorously proving the scheme has error asymptotic expansion numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme as well as its post processing formulas and extrapolation

a finite volume method for one dimensional reaction diffusion problems abstract a truly conservative finite volume method for one dimensional reaction diffusion equations is presented the method is based on piecewise second degree polynomials approximations that result in linear fluxes a symmetric mass matrix and a semi discrete system of nonlinear ordinary differential equations in time it is shown that the semi discrete method is linearly stable provided that the problem is mathematically well posed it is also shown that implicit discretizations of the time variable are stable whereas the explicit one is conditionally stable and has a smaller domain of linear stability than the standard explicit finite difference method it is also illustrated that upon lumping the mass matrix the method coincides with a finite difference discretization of the original problem a detailed formulation of the dirichlet neumann and robin boundary conditions is presented such a formulation makes use of control volumes which are here referred to as incomplete ones although the finite volume technique presented here uses a node centered formulation it can be easily extended to cell centered ones the technique is also developed for multidimensional problems

high order unconditionally stable difference schemes for the riesz space fractional telegraph equation in this paper a class of unconditionally stable difference schemes based on the pade approximation is presented for the riesz space fractional telegraph equation firstly we introduce a new variable to transform the original differential equation to an equivalent differential equation system then we apply a second order fractional central difference scheme to discretize the riesz space fractional operator finally we use 1 1 2 2 and 3 3 pade approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations matrix analysis is used to show the unconditional stability of the proposed algorithms two examples with known exact solutions are chosen to assess the proposed difference schemes numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space fractional hyperbolic equation