stability and convergence of the peaceman rachford adi method for initial boundary value problems etude des proprietes de stabilite et de convergence de la methode des directions alternees implicite losrsqu elle est appliquee aux problemes aux valeurs initiales dans un espace bidimensionnel

tensor product generalized adi methods for separable elliptic problems we consider solving separable second order linear elliptic partial differential equations if an elliptic problem is separable then for certain discretizations the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices in the most general case the discrete problem can be written in the form a 1 otimes b 2 b 1 otimes a 2 c f we present a new tensor product generalized alternating direction implicit tpgadi iterative method for solving such discrete problems we prove convergence and establish computational efficiency the tpgadi method is applied to the hermite bicubic collocation equations we conclude that the tpgadi method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems

noncoherent detection of coherent lightwave signals corrupted by phase noise waves are treated that modulate by either on off keying ook or binary frequency shift keying fsk and are further impaired by additive gaussian noise heterodyne detection of such a waveform produces an electronic bandpass signal which to ease demodulation in the presence of phase noise is noncoherently demodulated to extract the baseband pulse stream the treatment goes beyond previous bit error rate ber analyses of optical heterodyne receivers for ook and fsk first there is full adherence to the standard brownian motion model of phase noise also the receiver structure is formulated in such a way that the probability density function of the receiver output samples can be accurately determined this permits calculations of the additive noise and phase noise tolerable when achieving bit error rates as small as 10 sup 9 finally the study is comprehensive regarding the range of parameters explored filtering at an intermediate frequency if alone as well as if filtering plus postdetection low pass filtering is considered when the receiver parameters decision threshold for ook and if filter bandwidth are optimized large amounts of phase noise can be accommodated with only minor increases in required signal to noise ratio this is especially important when the bit rate is moderate compared to the laser linewidth

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

implicit time stepping methods for the navier stokes equations a novel two factor approximate factorization for three dimensional flows is described the method uses a preconditioned conjugate gradient solver for one of the factors and this solution is based on a method developed for two dimensional flows this two dimensional method is described and its performance is summarized for steady and unsteady flows the two factor method is then given larger time steps can be used with this method than is possible with the more conventional three factor method test results for the transonic steady flow over the onera m6 wing are given

alternating direction collocation for separable elliptic partial differential equations a complete theory is given for the convergence of alternating direction collocation adc applied to a large class of linear separable elliptic partial differential equations on a rectangular domain it is shown that adc using bicubic hermite splines converges in this setting and converges to the full collocation solution moreover it is shown that a complex embedding of the problem is essential to the theory and that the associated eigensplines and generalized eigensplines can be complex with complex eigenvalues whose geometric and algebraic multiplicities need not be equal nonetheless only real computations are required

fourier spectral embedded boundary solution of the poisson s and laplace equations with dirichlet boundary conditions a fourier spectral embedded boundary method for solution of the poisson s equation with dirichlet boundary conditions and arbitrary forcing functions including zero forcing function is presented in this paper this iterative method begins by transformation of the dirichlet boundary conditions from the physical boundaries to some corresponding regular grid points which are called the numerical boundaries using a second order interpolation method then the transformed boundary conditions and the forcing function are extended to a square smoothly and periodically via multiplying them by some suitable error functions instead of direct solution of the resulting extended poisson s problem it is suggested to define and solve an equivalent transient diffusion problem on the regular domain until achievement of the steady solution which is considered as the solution of the original problem without need of any numerical time integration method time advancement of the solution is obtained directly from the exact solution of the transient problem in the fourier space consequently timestep sizes can be chosen without stability limitations which it means higher rates of convergence in comparison with the classical relaxation methods the method is presented in details for one and two dimensional problems and a new emerged phenomenon which is called the saturation state is illustrated both in the physical and spectral spaces the numerical experiments have been performed on the one and two dimensional irregular domains to show the accuracy of the method and its superiority from the rate of convergence viewpoint to the other classical relaxation methods capability of the method in dealing with complex geometries and in presence of discontinuity at the boundaries has been shown via some numerical experiments on a four leaf shape geometry

stability and accuracy of time extrapolated adi fdtd methods for solving wave equations some recent work on the adi fdtd method for solving maxwell s equations in 3 d have brought out the importance of extrapolation methods for the time stepping of wave equations such extrapolation methods have previously been used for the solution of odes the present context of wave equations brings up two main questions which have not been addressed previously 1 when will extrapolation in time of an unconditionally stable scheme for a wave equation again feature unconditional stability and 2 how will the accuracy and computational efficiency depend on how frequently in time the extrapolations are carried out we analyze these issues here

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

a spectral embedding method applied to the advection diffusion equation in order to solve partial differential equations in complex geometries with a spectral type method one describes an embedding approach which essentially makes use of fourier expansions and boundary integral equations for the advection diffusion equation the method is based on an efficient helmholtz solver the accuracy of which is tested by considering 1d and 2d helmholtz like problems finally the capabilities of the method are pointed out by considering a 2d advection diffusion problem in a hexagonal geometry

multilevel polarization shift keying optimum receiver structure and performance evaluation multilevel digital coherent optical modulation schemes based on the state of polarization of a fully polarized lightwave are proposed and analyzed based on the complete statistical characterization of the stokes parameters extracted through appropriate signal processing in the presence of shot and additive gaussian noise the optimum maximum likelihood receiver operating symbol by symbol is derived the exact performance in terms of the average symbol error probability is found optimum constellations for the case of equipower 4 8 16 and 32 signals are found on the basis of the minimization of the error probability for a given average power their performance turns out to be promising as compared to other standard modulation techniques the spectral analysis of polarization modulated signals is presented a new receiver structure which solves the problem of the excess penalties incurred in the presence of channel dichroism is proposed and analyzed

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

asymptotically stable fourth order accurate schemes for the diffusion equation on complex shapes an algorithm which solves the multidimensional diffusion equation on complex shapes to fourth order accuracy and is asymptotically stable in time is presented thisbounded errorresult is achieved by constructing on arectangular grid a differentiation matrix whose symmetric part is negative definite the differentiation matrix accounts for the dirichlet boundary condition by imposing penalty like terms numerical examples in 2 d show that the method is effective even where standard schemes stable by traditional definitions fail the ability of the paradigm to be applied to arbitrary geometric domains is an important feature of the algorithm

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

strict stability of high order compact implicit finite difference schemes temporal or strict stability of approximation to pdes is much more difficult to achieve than the classical lax stability in this paper we present a class of finite difference schemes for hyperbolic initial boundary value problems in one and two space dimensions that possess the property of strict stability the approximations are constructed so that all eigenvalues of corresponding differentiation matrix have a nonpositive real part boundary conditions are imposed by using penalty like terms fourth and sixth order compact implicit finite difference schemes are constructed and analyzed computational efficacy of the approach is corroborated by a series of numerical tests in 1 d and 2 d scalar problems

a fourier penalty method for solving the time dependent maxwell s equations in domains with curved boundaries we present a high order fourier penalty method for the maxwell s equations in the vicinity of perfect electric conductor boundary conditions the approach relies on extending the smooth non periodic domain of the equations to a periodic domain by removing the exact boundary conditions and introducing an analytic forcing term in the extended domain the forcing or penalty term is chosen to systematically enforce the boundary conditions to high order in the penalty parameter which then allows for higher order numerical methods we present an efficient numerical method for constructing the penalty term and discretize the resulting equations using a fourier spectral method we demonstrate convergence orders of up to 3 5 for the one dimensional maxwell s equations and show that the numerical method does not suffer from dispersion or pollution errors we also illustrate the approach in two dimensions and demonstrate convergence orders of 2 5 for transverse magnetic modes and 1 5 for the transverse electric modes we conclude the paper with numerous test cases in dimensions two and three including waves traveling in a bent waveguide and scattering off of a windmill like geometry

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

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

accurate high order representation of complex three dimensional surfaces via fourier continuation analysis we present a new method for construction of high order parametrizations of surfaces starting from point clouds the method we propose can be used to produce full surface parametrizations by sets of local charts each one representing a large surface patch which typically contains thousands of the points in the original point cloud for complex surfaces of scientific and engineering relevance the proposed approach accurately renders both smooth and non smooth portions of a surface it yields super algebraically convergent fourier series approximations to a given surface up to and including all points of geometric singularity such as corners edges conical points etc in view of their c smoothness except at true geometric singularities and their properties of high order approximation the surfaces produced by this method are suitable for use in conjunction with high order numerical methods for boundary value problems in domains with complex boundaries including pde solvers integral equation solvers etc our approach is based on a very simple concept use of fourier analysis to continue smooth portions of a piecewise smooth function into new functions which defined on larger domains are both smooth and periodic the continuation functions arising from a function f converge super algebraically to f in its domain of definition as discretizations are refined we demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance

short note a high order pade adi method for unsteady convection diffusion equations a high order alternating direction implicit adi method for computations of unsteady convection diffusion equations is proposed by using fourth order pade schemes for spatial derivatives the present scheme is fourth order accurate in space and second order accurate in time the solution procedure consists of a number of tridiagonal matrix operations which make the computation cost effective the method is unconditionally stable and shows higher accuracy and better phase and amplitude error characteristics than the standard second order adi method d w peaceman h h rachford jr the numerical solution of parabolic and elliptic differential equations journal of the society of industrial and applied mathematics 3 1959 28 41 and the fourth order adi scheme of karaa and zhang high order adi method for solving unsteady convection diffusion problem journal of computational physics 198 2004 1 9

a second order accurate embedded boundary method for the wave equation with dirichlet data the accuracy of cartesian embedded boundary methods for the second order wave equation in general two dimensional domains subject to dirichlet boundary conditions is analyzed based on the analysis we develop a numerical method where both the solution and its gradient are second order accurate we avoid the small cell stiffness problem without sacrificing the second order accuracy by adding a small artificial term to the dirichlet boundary condition long time stability of the method is obtained by adding a small fourth order dissipative term several numerical examples are provided to demonstrate the accuracy and stability of the method the method is also used to solve the two dimensional tm z problem for maxwell s equations posed as a second order wave equation for the electric field coupled to ordinary differential equations for the magnetic field

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

trellis coded polarization shift keying modulation for digital optical communications the application of the well known technique of trellis coded modulation to coherent optical communications using polarization shift keying polsk is described and analyzed the resulting receiver is formed by a front end which performs the heterodyne detection and the stokes parameter extraction cascaded with an electronic viterbi processor operating the maximum likelihood estimate of the transmitted sequence results in terms of the error event probability using optimum as well as a simpler suboptimum branch metric show power gains of the order of 3 4 db at the expense of a reasonable increase in complexity only concerning the processing in the electronic domain these coding gains are not lost even in the presence of high levels of phase noise to which polsk in general is highly insensitive

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

an alternating direction implicit orthogonal spline collocation scheme for nonlinear parabolic problems on rectangular polygons we consider a nonlinear parabolic initial boundary value problem on a rectangular polygon with the solution satisfying robin boundary conditions with variable coefficients an approximation to the solution at the desired time value is obtained using an alternating direction implicit extrapolated crank nicolson scheme in which orthogonal spline collocation with piecewise polynomials of an arbitrary degree is used for spatial discretization at each time level the scheme determines the intermediate solution along horizontal lines and the approximate solution along vertical lines passing through the gauss points only at the last time level is the approximate solution along vertical lines converted into the approximate solution defined on the entire rectangular polygon this property of our approach leads to its efficient implementation and its applicability to rectangular polygons

strict stability of high order compact implicit finite difference schemes the role of boundary conditions for hyperbolic pdes ii abstract this paper deals with the problem of systems of hyperbolic pdes in one and two space dimensions using the theory of part i 7