an algebraic multigrid method for a class of elliptic differential systems the development numerical investigation and analysis of an algebraic multigrid amg method for the numerical solution of a representative class of elliptic differential systems are presented our amg scheme is based on the amg strategy of brandt ruge and stuben however the present approach may be implemented within other amg schemes by means of numerical experiments with model problems arising from some application areas it is demonstrated that the computational performance of the proposed amg scheme is comparable to that of amg applied to a single scalar equation an amg convergence theory is presented that extends convergence results for amg methods for scalar problems to the case of amg applied to elliptic systems some details of our amg implementation are given

automated approach to very high order aeroacoustic computations computational aeroacoustics requires efficient high resolution simulation tools for smooth problems this is best accomplished with very high order in space and time methods on small stencils however the complexity of highly accurate numerical methods can inhibit their practical application especially in irregular geometries this complexity is reduced by using a special form of hermite divided difference spatial interpolation on cartesian grids and a cauchy kowalewski recursion procedure for time advancement in addition a stencil constraint tree reduces the complexity of interpolating grid points that am located near wall boundaries these procedures are used to develop automatically and to implement very high order methods 15 for solving the linearized euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point the accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

improved block interface conditions for aeroacoustic problems with grid discontinuities aeroacoustic simulations are extremely sensitive to numerical errors to achieve a high order of accuracy optimized finite difference schemes such as the well known dispersion relation preservation schemes and multiblock structured meshes are often used challenges appear in communicating the flow information near grid discontinuities between neighboring blocks and sophisticated interface treatments are required the characteristic interface condition has been successful in applications in previous publications and yet limitations exist that are identified in the present study this paper proposes a new interface treatment based on the analogy between finite difference and finite volume discretizations one sided finite difference schemes are used to calculate the spatial derivatives in the normal direction and the residuals are reconstructed from a finite volume perspective applications to several benchmark cases show an encouraging improvement as compared to the previous characteristic interface

a comparison of algebraic multigrid and geometric immersed interface multigrid methods for interface problems in l adams and z li siam j sci comput 24 2002 pp 463 479 a multigrid method was designed specifically for interface problems that have been discretized using the methods described in l adams and z li siam j sci comput 24 2002 pp 463 479 and in z li and k ito siam j sci comput 23 2001 pp 339 361 for elliptic interface problems using the maximum principle preserving schemes in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 a new method was introduced that utilizes a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse grid matrices are m matrices this paper compares the immersed interface multigrid methods introduced in l adams and z li siam j sci comput 24 2002 pp 463 479 and l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 with algebraic multigrid which uses no geometric information to set up the multigrid components for coarse grid correction we show that algebraic multigrid is a robust solver for our test problems it outperforms the method in l adams and z li siam j sci comput 24 2002 pp 463 479 and performs nearly as well as the method in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 which is shown to be the most efficient for all problem parameters and sizes

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

high order low dispersive and low dissipative explicit schemes for multiple scale and boundary problems explicit high order numerical schemes are proposed for the accurate computation of multiple scale problems and for the implementation of boundary conditions specific high order node centered finite differences and selective filters removing grid to grid oscillations are first designed for the discretization of the buffer region between a x grid domain and 2 x grid domain the coefficients of these matching schemes are chosen so that the maximum order of accuracy is reached non centered finite differences and selective filters are then developed with the aim of accurately computing boundary conditions they are constructed by minimizing the dispersion and the dissipation errors in the wave number space for waves down to four points per wavelength the dispersion and dissipation properties of the matching and the boundary schemes are described in detail and their accuracy limits are determined to show that these schemes calculate accurately waves with at least five points per wavelength test problems including linear convection wall reflection and acoustic scattering around a cylinder are finally solved to illustrate the accuracy of the schemes

applications of high order optimized upwind schemes for computational aeroacoustics high order optimized upwind dispersion relation preserving finite difference schemes are used for solving multidimensional acoustic wave propagation problems the accuracy and the robustness of the schemes are tested for acoustic problems with various geometry and boundary conditions excellent agreements between the numerical and the analytical solutions indicate that the high order optimized upwind schemes minimize not only the dispersion error but also the dissipation error while retaining the numerical stability the simulation results have also demonstrated that the optimized upwind schemes damp out the spurious short waves automatically without the need of filter or explicit dissipation terms the overall effectiveness of the high order optimized upwind schemes is verified the good numerical result of the problem of acoustic scattering by a sphere suggests that the optimized upwind schemes may be used to solve acoustic problems involving a relatively complex geometry

uniformly accurate finite difference schemes for p refinement the accuracy deficiency of high order conventional finite difference schemes in the high frequency range is attributed to the approximation property of a power polynomial interpolation over equispaced nodes to improve the numerical representation of high frequency solutions a new family of high order finite difference schemes is constructed by trigonometric polynomial interpolation as expected these new difference schemes can resolve a wider frequency range than conventional finite difference schemes with the same stencil in particular they remove the possibility of runge s phenomenon for uniform mesh the recently proposed optimized difference schemes are indeed types of mixed difference schemes which make a compromise between power and trigonometric polynomial interpolation

towards adaptive smoothed aggregation alpha sa for nonsymmetric problems applying smoothed aggregation sa multigrid to solve a nonsymmetric linear system a mathbf x mathbf b is often impeded by the lack of a minimization principle that can be used as a basis for the coarse grid correction process this paper proposes a petrov galerkin pg approach based on applying sa to either of two symmetric positive definite spd matrices sqrt a t a or sqrt aa t these matrices however are typically full and difficult to compute so it is not computationally efficient to use them directly to form a coarse grid correction the proposed approach approximates these coarse grid corrections by using sa to accurately approximate the right and left singular vectors of a that correspond to the lowest singular value these left and right singular vectors are used to construct the restriction and interpolation operators respectively a preliminary two level convergence theory is presented suggesting that more relaxation should be applied than for an spd problem additionally a nonsymmetric version of adaptive sa alpha sa is given that automatically constructs sa multigrid hierarchies using a stationary relaxation process on all levels numerical results are reported for convection diffusion problems in two dimensions with varying amounts of convection for constant variable and recirculating convection fields the results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar pdes with the exception of recirculating flow this paper serves as a first step for nonsymmetric alpha sa the long term goal of this effort is to develop nonsymmetric alpha sa for systems of pdes where the sa framework has proven to be well suited for adaptivity in spd problems

robust norm equivalencies for diffusion problems additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem these solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition to obtain a robust iterative scheme it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem in this paper we present such a robust norm equivalence for the model problem w u with a scalar diffusion coefficient in c r 2 our estimates involve only very weak information about w and the results are applicable for a large class of diffusion coefficients namely we require to be in the muckenhoupt class a 1 a function class well studied in harmonic analysis the presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems

on the effective accuracy of spectral like optimized finite difference schemes for computational aeroacoustics the present article aims at highlighting the strengths and weaknesses of the so called spectral like optimized explicit central finite difference schemes when the latter are used for numerically approximating spatial derivatives in aeroacoustics evolution problems with that view we first remind how differential operators can be approximated using explicit central finite difference schemes the possible spectral like optimization of the latter is then discussed the advantages and drawbacks of such an optimization being theoretically studied before they are numerically quantified for doing so two popular spectral like optimized schemes are assessed via a direct comparison against their standard counterparts such a comparative exercise being conducted for several academic test cases at the end general conclusions are drawn which allows us discussing the way spectral like optimized schemes shall be preferred or not to standard ones when it comes to simulate real life aeroacoustics problems

first order system least squares for incompressible resistive magnetohydrodynamics magnetohydrodynamics mhd is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles hence the equations that describe the plasma form a nonlinear system that couples navier stokes equations with maxwell s equations this paper shows that the first order system least squares fosls finite element method is a viable discretization for these large mhd systems to solve this system a nested iteration newton fosls amg approach is taken most of the work is done on the coarse grid including most of the linearizations we show that at most one newton step and a few v cycles are all that are needed on the finest grid here we describe how the fosls method can be applied to incompressible resistive mhd and how it can be used to solve these mhd problems efficiently a 3d steady state and a reduced 2d time dependent test problem are studied the latter equations can simulate a large aspect ratio tokamak the goal is to resolve as much physics from the test problems with the least amount of computational work we show that this is achieved in a few dozen work units or fine grid residual evaluations

adaptive algebraic multigrid efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort overcoming the limitations of geometric multigrid methods to simple geometries and differential equations algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix while this allows for efficient black box solution of the linear systems associated with discretizations of many elliptic differential equations it also results in a lack of robustness due to unsatisfied assumptions made on the near null spaces of these matrices this paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process emphasis is on the principles that guide the adaptivity and their application to algebraic multigrid solution of certain symmetric positive definite linear systems

radiated noise from airfoils in realistic mean flows the long term objective of the research described is to use computational aeroacoustics methodology and parallel computers to increase the understanding of broadband blade noise in a systematic progression toward simulations of completely realistic configurations and conditions some simplified problems that address the important features of the flow are investigated a two dimensional navier stokes code implemented using the message passing library and fortran 90 on the ibm sp2 is used to perform the calculations results are presented for the interaction of a vortical gust and naca airfoils including nonlinear effects the influence of gust frequency and airfoil thickness is described a multigrid method is used to obtain converged steady state solutions before the gust is introduced in a source region inside the domain

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

a new algebraic multigrid approach for stokes problems standard discretizations of stokes problems lead to linear systems of equations in saddle point form making difficult the application of algebraic multigrid methods in this paper a new approach is proposed it consists in first transforming the system by pre and post multiplication with simple algebraic sparse block triangular matrices this is a form of pre conditioning in the literal sense designed to make sure that the transformed matrix is well adapted to multigrid in particular after transformation all the diagonal blocks are symmetric and positive definite and correspond to or resemble a discrete laplace operator then to each of these diagonal blocks is associated a prolongation that works well for it using any relevant algebraic or geometric multigrid method next a multigrid scheme for the global system is naturally set up by combining these partial prolongations with a galerkin coarse grid matrix for this approach combined with damped jacobi smoothing a uniform two grid convergence bound is derived for the global system under the assumption that the two grid schemes for the different diagonal blocks are themselves uniformly convergent this result is illustrated by a few examples showing further that time dependent problems and variable viscosity can be handled in a natural way without requiring parameter adjustment a numerical comparison also shows that the new approach can be more effective than state of the art block preconditioning techniques

accurate gradient approximation for complex interface problems in 3d by an improved coupling interface method most elliptic interface solvers become complicated for complex interface problems at those exceptional points where there are not enough neighboring interior points for high order interpolation such complication increases especially in three dimensions usually the solvers are thus reduced to low order accuracy in this paper we classify these exceptional points and propose two recipes to maintain order of accuracy there aiming at improving the previous coupling interface method 26 yet the idea is also applicable to other interface solvers the main idea is to have at least first order approximations for second order derivatives at those exceptional points recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point whenever this is possible recipe 2 is to flip domain signatures and introduce a ghost state so that a second order method can be applied this ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface the original state is recovered by a post processing using nearby states and jump conditions the choice of recipes is determined by a classification scheme of the exceptional points the method renders the solution and its gradient uniformly second order accurate in the entire computed domain numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions and a real molecule 1d63 which is double helix shape and composed of hundreds of atoms

a new compact spectral scheme for turbulence simulations we propose a new kind of compact difference scheme for the computation of the first and second derivatives in the simulation of high reynolds number turbulent flows the scheme combines and truncates the pseudospectral representation of derivative for convergence acceleration comparison of the wave resolution property with available optimized compact schemes minimizing the prescribed wave resolution error reveals our scheme s superiority for the same size of stencils without introducing optimization parameters an accompanying boundary scheme is also proposed with the stability analysis the proposed scheme is tested for the evaluation of derivatives of a function that decays very slowly in the wavenumber space and for the simulation of three dimensional isotropic turbulence

an algebraic study of a local multigrid method for variational problems a two level iterative method for solving systems of equations arising from local refinement of finite element meshes is proposed the smoothing steps consists of gauss seidel relaxations applied only to the local mesh a direct solver is required on the global coarse mesh we analyze this scheme by modifying the standard algebraic convergence theory the estimates of the convergence factor do not require any regularity assumption the estimates can be evaluated element by element the evaluated estimates are compared with the actual convergence factor and with another bound in several numerical examples the theory also applies to a method with a direct solver on the local fine mesh in this case we have obtained very sharp and in fact optimal estimates of the convergence factors

stability analysis of a controlled mechanical system with parametric uncertainties in lugre friction model abstractparameters of friction model identified for a specific control system development are not constants they vary over time and have a significant effect on the control system stability altho

convergence of algebraic multigrid methods for symmetric positive definite matrices with weak diagonal dominance for symmetric positive definite matrices with weak diagonal dominance the algebraic sense of smooth errors is discussed and convergence of algebraic multigrid amg methods is proved it is found that the results of amg for symmetric positive definite m matrices with weak diagonal dominance also hold without the m matrix assumption the key result is the new formula x email protected ia i email protected j email protected email protected j a i j a j i x 2 email protected email protected j email protected email protected j a i j x email protected email protected ia i j a i j x j 2

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

high accuracy algorithms for computational aeroacoustics this paper presents an analysis of high bandwidth operators developed for use with an essentially nonoscillatory eno method the spatial operators of a sixth order eno code are modified to resolve waves with as few as 7 points per wavelength ppw by decreasing the formal order of the algorithm numerical and analytical solutions are compared for the model problems of plane wave propagation and sound generation by an oscillating sphere these problems involve linear propagation wave steepening and shock formation an analysis of the ppw required for sufficient accuracy shows that low order algorithms need an excessive number of grid points to produce acceptable solutions in contrast high order codes provide good predictions on relatively coarse grids the high bandwidth operators produce only modest improvements over the original sixth order operators for nonlinear problems in which wave steepening is significant however they clearly outperform the original operators for long distance linear propagation because the high bandwidth operators have the same stencil as the original sixth order operators these gains are achieved with no additional computational work

numerical solution of the time domain maxwell equations using high accuracy finite difference methods high accuracy finite difference schemes are used to solve the two dimensional time domain maxwell equations for electromagnetic wave propagation and scattering the high accuracy schemes consist of a seven point spatial operator coupled with a six stage runge kutta time marching method two methods are studied one of which produces the maximum order of accuracy and one of which is optimized for propagation distances smaller than roughly 300 wavelengths boundary conditions are presented which preserve the accuracy of these schemes when modeling interfaces between different materials numerical experiments are performed which demonstrate the utility of the high accuracy schemes in modeling waves incident on dielectric and perfect conducting scatterers using cartesian and curvilinear grids the high accuracy schemes are shown to be substantially more efficient in both computing time and memory than a second order and a fourth order method the optimized scheme can lead to a reduction in error relative to the maximum order scheme with no additional expense especially when the number of wavelengths of travel is large

a time domain method for the prediction of sound attenuation in lined ducts abstract this paper describes the development and application of a time domain acoustic liner model which is suitable for the simulation of sound propagation and attenuation in lined ducts the fluid flow within the duct domain is represented by the non linear unsteady euler equations while the liner model consists of a resistive frequency independent part and a reactive part which is obtained by solving the one dimensional euler equations within the liner cavity specialized boundary conditions are used for matching the 1d cavity flow and the 2d duct flow the liner model has been formulated in order to predict the sound attenuation with or without mean flow as well as for linear or non linear sound propagation the model has been validated in the case of linear pure tone and n wave signals by checking against analytical formulations obtained via eigensolutions of the linearized inviscid flow equations very good agreement was obtained for both zero and subsonic mean flows

stability analysis via the concept of lyapunov exponents a case study in optimal controlled biped standing balancing control is important for biped standing in spite of large efforts it is very difficult to design balancing control strategies satisfying three requirements simultaneously maintaining postural stability improving energy efficiency and satisfying the constraints between the biped feet and the ground in this article a proportional derivative pd controller is proposed for a standing biped which is simplified as a two link inverted pendulum with one additional rigid foot link the genetic algorithm ga is used to search for the control gain meeting all three requirements the stability analysis of such a deterministic biped control system is carried out using the concept of lyapunov exponents les based on which the system stability where the disturbance comes from the initial states and the structural stability where the disturbance comes from the pd gains are examined quantitively in terms of stability region this article contributes to the biped balancing control more significan

algebraic multigrid theory the symmetric case a rigorous two level theory is developed for general symmetric matrices and nonsymmetric ones using kaczmarz relaxation without assuming any regularity not even any grid structure of the unknowns the theory applies to algebraic multigrid amg processes as well as to the usual geometric multigrid it yields very realistic estimates and precise answers to basic algorithmic questions such as in what algebraic sense does gauss seidel or jacobi kaczmarz line gauss seidel etc relaxation smooth the error when is it appropriate to use block relaxation what algebraic relations must be satisfied by the coarse to fine interpolations what is the algorithmic role of the geometric origin of the problem the theory helps to rigorize local mode analyses and locally analyze cases where the latter is inapplicable

a proof that the discrete singular convolution dsc lagrange distributed approximating function ldaf method is inferior to high order finite differences finite differences approximate the mth derivative of a function u x by a series j n nd j m u x j where the x j are the grid points the closely related discrete singular convolution dsc and lagrange distributed approximating function ldaf methods treated here as a single algorithm approximate derivatives in the same way as finite differences but with different numerical weights that depend upon a free parameter a by means of fourier analysis and error theorems we show that the dsc is worse than the standard finite differences in differentiating exp ikx for all k when a a f d where a f d 1 n 1 with n as the stencil width is the value of the dsc parameter that makes its weights most closely resemble those of finite differences for a

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow