experiments with implicit upwind methods for the euler equations abstract a number of implicit integration schemes for the one dimensional euler equations with conservative upwind spatial differencing are tested on a problem of steady discontinuous flow fastest convergence quadratic for the first order backward euler scheme is obtained with the upwind switching provided by van leer s differentiable split fluxes which easily linearize in time with roe s nondifferentiable split flux differences the iterations may get trapped in a limit cycle this also happens in a second order scheme with split fluxes if the matrix coefficients arising in the implicit time linearization are not properly centered in space the use of second order terms computed from split fluxes degrades the accuracy of the solution especially if these are subjected to a limiter for the sake of monotonicity preservation second order terms computed from the characteristic variables perform best

parallel domain oriented multilevel methods instead of the usual nodal basis we use a generating system for the discretization of pdes that contains not only the basis functions of the finest level of discretization but additionally the basis functions of all coarser levels of discretization the galerkin approach now results in a semidefinite system of linear equations to be solved standard iterative gs methods for this system turn out to be equivalent to elaborated multigrid methods for the fine grid system

on bi grid local mode analysis of solution techniques for 3 d euler and navier stokes equations a procedure is presented for utilizing a bi grid stability analysis as a practical tool for predicting multigrid performance in a range of numerical methods for solving euler and navier stokes equations model problems based on the convection equation the diffusion equation and burger s equation are used to illustrate the superiority of the bi grid analysis as a predictive tool for multigrid performance in comparison to the smoothing factor derived from conventional von neumann analysis for the euler equations bi grid analysis is presented for three upwind difference based factorizations namely spatial eigenvalue and combination splits and two central difference based factorizations namely lu and adi methods in the former both the steger warming and van leer flux vector splitting methods are considered for the navier stokes equations only the beam warming adi central difference scheme is considered in each case estimates of multigrid convergence rates from the bi grid analysis are compared to smoothing factors obtained from single grid stability analysis effects of grid aspect ratio and flow skewness are examined both predictions are compared with practical multigrid convergence rates for 2 d euler and navier stokes solutions based on the beam warming central difference scheme and 3 d euler solutions with various upwind difference schemes it is demonstrated that bi grid analysis can be used as a reliable tool for the prediction of practical multigrid performance

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

on the class of chebyshev approximation problems which arise in connection with a conjugate gradient type method we investigate approximation problems of the form mathop min limits p p 0 1 mathop max p lambda limits lambda in s p withp ranging over all polynomials of degree k ands any line segment of the complex plane problems of this type appear in connection with error bounds for the minimum residual algorithm for solving linear systems with coefficient matrixa ei di n wheren nh andd besides a few general results on p we present the explicit solution of p for segmentss which are parallel to the imaginary and symmetric to the real axis this result leads to an optimal semi iterative method for the class of matricesa i n wheren nt

general k part stationary iterative solutions to linear systems the solution of i b x b is sought where b is a n times n matrix i is the identity matrix b is an n vector of known constants and x is the unknown vector when n is large a direct solution of the equation is not feasible one instead seeks an approximation to x this paper discusses stationary k part methods for finding sequences of approximations to x after presenting a generalization of the companion matrix criteria for the convergence of k part stationary operator coefficient algorithms two theorems are proved which relate the spectrum of a particular form of the companion matrix to the roots of a meromorphic function evaluated at the eigenvalues of b a corollary which results from these theorems gives the convergence criteria for the most general k part stationary constant coefficient algorithm in terms of the image of the unit disc under a rational function which is easily related to the stationary algorithm it is then shown that the asymptotic rate of convergence for this most gener

robust multigrid algorithms for the navier strokes equations abstract anisotropies occur naturally in computational fluid dynamics where the simulation of small scale physical phenomena such as boundary layers at high reynolds numbers causes the grid to be highly stretched leading to a slowdown in convergence of multigrid methods several approaches aimed at making multigrid a robust solver have been proposed and analyzed in the literature using the scalar diffusion equation however they have rarely been applied to solving more complicated models such as the incompressible navier stokes equations this paper contains the first published numerical results of the behavior of two popular robust multigrid approaches alternating plane smoothers combined with standard coarsening and plane implicit smoothers combined with semi coarsening for solving the 3 d incompressible navier stokes equations in the simulation of the driven cavity and a boundary layer over a flat plate on a stretched grid grid size grid stretching and reynolds number are the factors considered in evaluating the robustness of the multigrid methods both approaches yield large increases in convergence rates over cell implicit smoothers on stretched grids the combination of plane implicit smoothers and semi coarsening was found to be fully robust in the flat plate simulation up to reynolds numbers 10 6 and the best alternative in the driven cavity simulation for reynolds numbers above 10 3 the alternating plane approach exhibits a better behavior for lower reynolds numbers below 10 3 in the driven cavity simulation

optimal multigrid convergence by elliptic hyperbolic splitting we describe a multigrid method for solving the steady euler equations that is optimal in the sense of requiring o n operations till convergence where n is the number of unknowns the method relies on an elliptic hyperbolic decomposition achieved by local preconditioning the splitting allows the embedded advection equations to be treated with streamwise semicoarsening rather than full coarsening which would not be effective a simple 2 d numerical computation is presented as proof of concept a convergence study indicates the split method has complexity o n over a wide range of grid spacings and mach numbers while the use of full coarsening for all equations makes the complexity deteriorate to almost o n1 5

design and implementation of a multigrid code for the euler equations the steady state equations of inviscid fluid flow the euler equations are a nonlinear nonelliptic system of equations admitting solutions with dis continuities for example shocks the efficient numerical solution of these equations poses a strenuous challenge to multigrid methods a multigrid code has been developed for the numerical solution of the euler equations in this paper some of the factors that had to be taken into account in the design and development of the code are reviewed these factors include the importance of choosing an appropriate difference scheme the usefulness of local mode analysis as a design tool and the crucial question of how to treat the nonlinearity sample calculations of transonic flow about airfoils will be presented no claim is made that the particular algorithm presented is optimal

directional agglomeration multigrid techniques for high reynolds number viscous flows a preconditioned directional implicit agglomeration algorithm is developed for solving two and three dimensional viscous flows on highly anisotropic unstructured meshes of mixed element types the multigrid smoother consists of a pre conditioned point or line implicit solver which operates on lines constructed in the unstructured mesh using a weighted graph algorithm directional coarsening or agglomeration is achieved using a similar weighted graph algorithm a tight coupling of the line construction and directional agglomeration algorithms enables the use of aggressive coarsening ratios in the multigrid algorithm which in turn reduces the cost of a multigrid cycle convergence rates which are independent of the degree of grid stretching are demonstrated in both two and three dimensions further improvement of the three dimensional convergence rates through a gmres technique is also demonstrated

a new multigrid approach to convection problems abstract a new multigrid method for convection problems is presented it is designed to overcome the problem of alignment the flow being aligned with the grid the technique employs semi coarsening in several directions simultaneously and gives rise to multiple coarser grids with the same total number of points per grid level but with different sizes in each co ordinate direction the amount of work per multigrid cycle is still o n as an example the method is applied to the nonlinear upwind differenced euler equations of gas dynamics in two dimensions convergence rates are estimated by two level fourier analysis for the linearised equations numerical experiments on the nonlinear equations confirm these estimates

a hybrid arnoldi faber iterative method for nonsymmetric systems of linear equations we present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations say of the formax b wherea n n witha nonsingular andb n are given this hybrid method begins with a limited number of steps of the arnoldi method to obtain some information on the location of the spectrum ofa and then switches to a richardson iterative method based on faber polynomials for a polygonal domain the faber polynomials can be constructed recursively from the parameters in the schwarz christoffel mapping function in four specific numerical examples of non normal matrices we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid gmres or restarted versions of the gmres algorithm it is however sensitive as other hybrid methods also are to the amount of information on the spectrum ofa acquired during the first arnoldi phase of this procedure

on the construction of semi iterative methods given a nonsingular linear system a bf x bf b a splitting a m n leads to the one step iteration 1 bf x m t bf x m 1 bf c with t m 1 n and c m 1 bf b we investigate semiiterative methods with respect to 1 under the assumption that the eigenvalues of t are contained in some compact set u subset mathbb c 1 notin u using results about maximal convergence of polynomials and uniformly distributed nodes from approximation theory we describe semiiterative methods which are asymptotically optimal with respect to u under mildly restrictive assumptions on u we construct semiiterative methods which are asymptotically optimal with respect to u and which can be interpreted as one step nonstationary methods

the conformal bratwurst mapsand associated faber polynomials summary conformal maps from the exterior of the closed unit disk onto the exterior of bratwurst shape sets in the complex plane are constructed using these maps coefficients for the computation of the corresponding faber polynomials are derived a bratwurst shape set is the result of deforming an ellipse with foci on the real axis by conformally mapping the real axis onto the unit circle such sets are well suited to serve as inclusion sets for sets associated with a matrix for example the spectrum field of values or a pseudospectrum hence the sets can be applied in the construction and analysis of a broad range of iterative methods for the solution of linear systems the main advantage of the approach is that the conformal maps are derived from elementary transformations allowing an easy computation of the associated transfinite diameter asymptotic convergence factor and faber polynomials numerical examples are given

accurate multiblock navier stokes solver for complex aerodynamic configurations a computational fluid dynamics tool capable to simulate accurately viscous flows around complex aerodynamic configurations is described the method combines a multiblock technique with a low dissipation numerical method incorporated into a multigrid framework structured subdomains blocks are united in multigrid multiblock structures and the blocks are treated independently at each stage of the numerical procedure maintaining a regular information exchange between the neighboring blocks in the numerical procedure the convection part of the equations is approximated by a low order upwind biased scheme employed for multigrid relaxation in combination with a higher order essentially nonoscillatory scheme used to supply a defect to the right hand side of the discrete equations on the finest multigrid level in a way ensuring the overall high accuracy of the scheme computational examples demonstrate the ability of the resulting method to perform accurate large scale computations of complex three dimensional turbulent flows around realistic aerodynamic configurations

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

multigrid applications to three dimensional elliptic coordinate generation the multigrid algorithm was applied to solve the coupled set of elliptic quasilinear partial differential equations associated with three dimensional coordinate generation the results indicate that the multigrid scheme is more than twice as fast as conventional relaxation schemes on moderate size grids convergence factors of order 0 90 per work unit were achieved on 36 000 point grids the paper covers the form of transformation develops the set of generation equations and gives details on the multigrid approach used included are a development of the full approximation storage scheme details of the smoothing rate analysis and a section devoted to rational programming techniques applicable to the multigrid algorithm

matrix analysis of acoustic wave propagation within curved ducting systems abstract a method is presented for theoretically determining the acoustic characteristics of rigid walled ducting systems which contain one or more bends a scattering matrix approach is utilized in the method and it is particularly useful for analyzing ducting systems which contain bends of variable curvature the results of numerical calculations relevant to the analysis of a section of spirally curved rectangular duct are presented these results exhibit favourable numerical characteristics and compare well with those produced by an alternative analysis the method is also used to investigate acoustic wave propagation through parabolically curved circular duct sections

the frequency decomposition multilevel method a robust additive hierarchical basis preconditioner hackbusch s frequency decomposition multi level method is characterized by the application of three additional coarse grid corrections in parallel to the standard one each coarse grid correction was designed to damp errors from a different part of the frequency spectrum in this paper we introduce a cheap variant of this method partly based on semi coarsening which demands less recursive calls than the original version using the theory of the additive schwarz methods we will prove robustness of our method as a preconditioner applied to anisotropic equations

a gmres based plane smoother in multigrid to solve 3d anisotropic fluid flow problems for a discretization of the 3d steady incompressible navier stokes equations a solution method is presented for solving flow problems on stretched grids the discretization is a vertex centered finite volume discretization with a flux splitting approach for the convective terms second order accuracy is obtained with the well known defect correction technique b koren j comput phys 87 25 1990 the solution method used is multigrid for which a plane smoother is presented for obtaining good convergence in flow domains with severely stretched grids a matrix is set up in a plane which is solved iteratively with a preconditioned gmres method here a stop criterion for gmres is tested which reduces the number of inner iterations compared to an exact plane solver without affecting the multigrid convergence rates the performance of the solution method is shown for a poisson model problem and for 3d incompressible channel flow examples

robust convergence of multilevel algorithms for convection diffusion equations a new approach is developed to analyze convergence of multilevel algorithms for convection diffusion equations this approach uses a multilevel recursion formula which can be applied to a variety of nonsymmetric problems here the recursion formula is applied to a robust multilevel algorithm for convection diffusion equations with convection in the x or y direction the multilevel algorithm uses semicoarsening line relaxation and prewavelets the convergence rate is proved to be less than 0 18 independent of the size of the convection term and the number of unknowns the assumptions allow the convection term to have a turning point so that an interior layer can appear in the solution of the convection diffusion equation the computational cost of the multilevel cycle is about o n log n independent of the size of the convection term where n is the number of unknowns it is proved that o log n multilevel cycles starting from the initial guess 0 lead to an o n 2 algebraic error with respect to the l infty norm independent of the size of the convection term

grid independent convergence of the multigrid method for first order equations the multigrid method for numerical solution of systems of first order partial differential equations is analyzed the smoothing iterations are of polynomial type and are used on all grids under certain conditions the convergence of the iterations is independent of the grid size this is explained by the propagation of smooth error modes out through the boundary and the damping of oscillatory modes an upper bound is derived on the number of multigrid v cycles needed to solve a two dimensional scalar equation numerical experiments with a simple scalar equation and the euler equations confirm the theoretical results

an adaptive method for solving nonsymmetric linear systems involving applications of scpack consider the following large system of linear equations rm x it t x bf c eqno 0 where it is assumed that t is n by n real matrix with 1 excluded from the spectrum of t and that c is a real vector of dimension n this dissertation uses the results from function theory approximation theory and conformal mapping theory and constructs an adaptive numerical method to solve equation 0 with the aid of scpack this adaptive method periodically estimates the eigenvalue set to obtain near asymptotically optimal rates of convergence of the associated k l step semiiterative methods

convergence analysis of a multigrid method for convection diffusion equations this paper is concerned with the convergence analysis of robust multigrid methods for convection diffusion problems we consider a finite difference discretization of a 2d model convection diffusion problem with constant coefficients and dirichlet boundary conditions for the approximate solution of this discrete problem a multigrid method based on semicoarsening matrix dependent prolongation and restriction and line smoothers is applied for a multigrid w cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion convection ratio for the contraction number of a multigrid v cycle a bound is proved which is uniform for a class of convection dominated problems the analysis is based on linear algebra arguments only

stationary and almost stationary iterative k l step methods for linear and nonlinear systems of equations we first discuss the solution of a fixed point equationx x of a frechet differentiable self mapping by iterative methods of the general form x m left phi left sum limits i 0 m 1 x i gamma i m 1 right sum limits j 0 m 1 x j beta j m right frac 1 beta m m m 1 2 ldots defined by two infinite nonsingular upper triangular matricesb j m andc j m with column sums 1 we show that two such methods defined byb c and tilde b tilde c respectively are in a certain sense equivalent if and only if c 1 b tilde c 1 tilde b in particular hat b c 1 and the unit matrix replacingc define an equivalent so called semiiterative method we introduce k l step methods as those whereb andc have upper bandwidthk andl 1 respectively they require storing max k l previous iterates only for stationary methodsb andc have toeplitz structure except for their first row which is chosen such that the column sum condition holds an euler method which may require to store all iterates is equivalent to a stationary k l step method if and only if the underlying conformal mapg is a rational function of the formg w w 0 k k v 0 v l 1 w 1 by choosingg withg 1 1 such that for some onto the exterior of some continuums known to contain the eigen values of the frechet derivative of one obtains a feasible procedure for designing a locally converging stationary k l step method custom made for a set of problems in the case x tx d with a linear operatort where one wants to solve a linear system of equations we show that the residual polynomials of a stationary semiiterative method are generalized faber polynomials with respect to a particular weight function using another weight function leads to what we call almost stationary methods the classical chebyshev iteration is an example of such a method we define equivalent almost stationary k l step methods and give a corresponding convergence result

on the approximation order of extremal point methods for hyperbolic minimal energy problems let be an analytic jordan curve in the unit disk we regard the hyperbolic minimal energy problem where denotes the set of all probability measures on there exist several extremal point discretizations of among others introduced by m tsuji tsuji points or by k menke hyperbolic menke points in the present article it is proven that hyperbolic menke points approach the images of roots of unity under a conformal map from onto geometrically fast if the number of points tends to infinity this establishes a conjecture of k menke in particular explicit bounds for the approximation error are given finally an effective method for the numerical determination of providing a geometrically shrinking error bound is presented

upwind relaxation methods for the navier stokes equations using inner iterations an upwind line relaxation algorithm for the navier stokes equations which employs inner iterations is applied to a supersonic and a subsonic test problem the purpose of using inner iterations is to accelerate the convergence to steady state solutions thereby reducing the overall cpu time a convergence criterion is developed to assist in automating the inner iterative procedure the ability of the line inner iterative procedure to mimic the quadratic convergence of the direct solver method is confirmed in both test problems but some of the non quadratic inner iterative results were more efficient than the quadratic results in the supersonic test case the use of inner iterations was very efficient in reducing the residual to machine zero for this test problem the inner iteration method required only about 65 of the cpu time which was required by the most efficient line relaxation method without inner iterations in the subsonic test case poor matrix conditioning forced the use of under relaxation in order to obtain convergence of the inner iterations resulting in an overall method which was less efficient than line relaxation methods which employ a more conventional cpu savings strategy 23 refs 9 figs

multigrid relaxation for the euler equations abstract a multigrid method for finding stationary solutions of the euler equations is described and tested spatial discretization is obtained by upwind differencing implicit time discretization is applied to construct a switched evolution relaxation ser scheme the multigrid method a correction scheme accelerates the inversion of the large linear system that arises in the ser scheme in a two dimensional transonic test problem the correction scheme is used in combination with symmetric block gauss seidel relaxation restriction is carried out by the addition of residuals and corresponding blocks prolongation by the simple distribution of coarse grid corrections to the finer grid it turns out that the total amount of work required to obtain a converged solution is roughly proportional to n 0 1 n being the number of zones both for a first order and a second order accurate solution the gain in efficiency with respect to a single grid scheme thereby becomes proportional to about n 0 4 in both cases

characteristics of modal wave propagation within longitudinally curved acoustic waveguides abstract a computational procedure based upon the rayleigh ritz method is presented for generating numerical approximations to the acoustic modes in a circularly curved waveguide these approximations are used to calculate matrices characterizing the modal transmission and reflection properties of a section of the curved waveguide flanked on either side by infinitely long straight waveguides the theory is applied to a waveguide of elliptic cross section and tables of eigenvalues mode contours and matrix elements are presented for specific examples

flexible multiple semicoarsening for three dimensional singularly perturbed problems we present robust parallel multigrid based solvers for 3d scalar partial differential equations the robustness is obtained by combining multiple semicoarsening strategies matrix dependent transfer operators and a krylov subspace acceleration the basis for the 3d preconditioner is a 2d method with multiple semicoarsened grids based on the mg s method from c w oosterlee appl numer math 19 1995 pp 115 128 and t washio and c w oosterlee gmd arbeitspapier 949 gmd st augustin germany 1995 the 2d method is generalized to three dimensions with a line smoother in the third dimension the method based on semicoarsening has been parallelized with the grid partitioning technique j linden b steckel and k stuben parallel comput 7 1988 pp 461 475 o a mcbryan et al impact comput sci engrg 3 1991 pp 1 75 and is evaluated as a solver and as a preconditioner on a mimd machine the robustness of the 3d method is shown for finite volume and finite difference discretizations of 3d anisotropic diffusion equations and convection dominated convection diffusion problems

efficient solution of the euler and navier stokes equations with a vectorized multiple grid algorithm a multiple grid algorithm for use in efficiently obtaining steady solutions to the euler and navier stokes equations is presented the convergence of the explicit maccormack algorithm on a fine grid is accelerated by propagating transients from the domain using a sequence of successively coarser grids both the fine and coarsegrid schemes are readily vectorizable the combination of multiple grid ding and vectorization results in substantially reduced computational times for the numerical solution of a wide range of flow problems results are presented for subsonic transonic and supersonic inviscid flows and for subsonic attached and separated laminar viscous flows work reduction factors over a scalar single grid algorithm range as high as 76 8

on hybrid iterative methods for nonsymmetric systems of linear equations hybrid methods for the solution of systems of linear equations consist of a first phase where some information about the associated coefficient matrix is acquired and a second phase in which a polynomial iteration designed with respect to this information is used most of the hybrid algorithms proposed recently for the solution of nonsymmetric systems rely on the direct use of eigenvalue estimates constructed by the arnoldi process in phase i we will show the limitations of this approach and propose an alternative also based on the arnoldi process which approximates the field of values of the coefficient matrix and of its inverse in the krylov subspace we also report on numerical experiments comparing the resulting new method with other hybrid algorithms

an implicit form for the osher upwind scheme conservative upwind schemes for the euler equations such as the osher scheme accurately resolve flow discontinuities and correctly model the physics of the problem however these schemes require many more arithmetic operations per integration step than simple central difference schemes and hence result in large computing times an implicit version of the first order and second order accurate osher schemes in two spatial dimensions and generalized coordinates is developed in this study because implicit schemes permit the use of large integration steps in many cases they require fewer integration steps to reach steady state especially in calculations on grids with widely varying mesh cell sizes the implicit scheme developed in this study accelerated convergence speeds by almost an order of magnitude in the problems considered test cases include quasi one dimensional nozzle flow and supersonic flow past a cylinder

multigrid method and fourth order compact scheme for 2d poisson equation with unequal mesh size discretization abstract a fourth order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two dimensional poisson equation in a rectangular domain multigrid methods using a partial semicoarsening strategy and line gauss seidel relaxation are designed to solve the resulting sparse linear systems numerical experiments are conducted to test the accuracy of the fourth order compact difference scheme and to compare it with the standard second order difference scheme convergence behavior of the partial semicoarsening and line gauss seidel relaxation multigrid methods is examined experimentally

a high resolution euler solver based on multigrid semi coarsening and defective correction abstract in an earlier paper an o n method for the computation of stationary solutions to the euler equations of inviscid compressible gas dynamics has been described the method is a variant of the multigrid technique and employs semi coarsening in all co ordinate directions simultaneously it provides good convergence rates for first order upwind discretisations even in the case of alignment the flow being aligned with the grid here we discuss the application of this scheme to higher order discretisations two grid analysis for the linear constant coefficient case shows that it is difficult to obtain uniformly good convergence rates for a higher order scheme because of waves perpendicular to stream lines the defect correction technique suffers from the same problem however convergence to a point where the residual of the total error the sum of the iteration error and the discretisation error is of the order of the truncation error can be obtained in about seven defect correction cycles according to estimates for the linear constant coefficient equations this result is explored for the nonlinear case by some illustrative numerical experiments

a dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems we present a novel multigrid mg procedure for the efficient solution of the large non symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi layered elastic medium the governing equations involve a highly non linear coupled system of integro partial differential equations along with the fracture front free boundary problem the conditioning of the algebraic equations typically degrades as o n3 a number of characteristics of this problem present significant new challenges for designing an effective mg strategy large changes in the coefficients of the pde are dealt with by taking the appropriate harmonic averages of the discrete coefficients coarse level green s functions for multiple elastic layers are constructed using a single dual mesh and superposition coarse grids that are sub sets of the finest grid are used to treat mixed variable problems associated with pinch points localized approximations to the jacobian at each mg level are used to devise efficient gauss seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements the performance of the mg preconditioner is demonstrated in a number of numerical experiments copyright 2005 john wiley sons ltd

new quadrature formulas from conformal maps gauss and clenshaw curtis quadrature like legendre and chebyshev spectral methods make use of grids strongly clustered at boundaries from the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal nevertheless such methods may waste a factor of pi 2 with respect to each space dimension we propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight sided domain the new methods are compared with related ideas of bakhvalov kosloff and tal ezer rokhlin and alpert and others an advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands for example one of the formulas presented is proved to converge 50 faster than gauss quadrature for functions analytic in an varepsilon neighborhood of 1 1

the faber polynomials for annular sectors a conformai mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the faber polynomials for that region these polynomials are of interest in providing near optimal polynomial approximations in a variety of contexts including the construction of semi iterative methods for linear equations the relevant conformai map for an annular sector z r z 1 0 argz it with 0 9 it is derived here and a recurrence relation is established for the coefficients of its laurent expansion about the point at infinity the recursive evaluation of scaled faber polynomials is formulated in such a way that an algebraic manipulation package may be used to generate explicit expressions for their coefficients in terms of two parameters which are determined by the interior angle of the annular sector and the ratio of its radii properties of the coefficients of the scaled faber polynomials are established and those for polynomials of degree 15 are tabulated in a supplement at the end of this issue a simple closed form is obtained for the coefficients of the faber series for 1 z known results for an interval a circular arc and a circular sector are reproduced as special cases

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

transmission and reflection of higher order acoustic modes in a mitred duct bend a two dimensional finite element technique was used to investigate the characteristics of a 90 mitred bend for higher order modes experiments were used to corroborate the numerical results and good agreement was obtained between the two approaches for values of the wave number parameter extending to a little above the cut on value of the second cross mode the results are presented as modal contributions of the reflected and transmitted sounds when a propagating mode is incident on the bend the magnitude of the energy transfer is also examined for the case where an evanescent mode is incident on the bend

an analysis of equivalent operator preconditioning for equation free newton krylov methods we consider the computation of a fixed point of a time stepper using newton krylov methods and propose and analyze equivalent operator preconditioning for the resulting linear systems for a linear scalar advection reaction diffusion equation we investigate in detail how the convergence rate depends upon the choice of preconditioner parameters and upon the time discretization the results are especially valuable when computing fixed points of a coarse time stepper in the equation free multiscale framework in which one simulates an unavailable coarse scale model by wrapping a set of computational routines around appropriately initialized fine scale simulations both analytical results and numerical experiments are presented showing that one can speed up the convergence of iterative methods significantly for a wide range of parameter values in the preconditioner

on semiiterative methods generated by faber polynomials consider a nonsingular system of linear algebraic equationsa x b or in fixed point formx tx c where the eigenvalues oft are contained in some compact subset of the complex plane using faber polynomials iteration methods of the form y m mu m 0 left c ty m 1 right mu m 1 y m 1 ldots mu m k y m k m 0 m j 1 for allm 1 are constructed which include the chebyshev iterative method as well as the stationaryk step methods when is a rectangle we also develop four step methods which yield fast convergence in cases when the speed of convergence of the optimized chebyshev method is poor

multigrid based shifted laplacian preconditioning for the time harmonic elastic wave equation we investigate the numerical performance of an iterative solver for a frequency domain finite difference discretization of the isotropic elastic wave equation the solver is based on the shifted laplacian preconditioner originally designed for the acoustic wave equation this preconditioner represents a discretization of a heavily damped wave equation and can be efficiently inverted by a multigrid iteration however the application of multigrid to the elastic case is not straightforward because standard methods such as point jacobi fail to smooth the s wave wavenumber components of the error when high p to s velocity ratios are present we consider line smoothers as an alternative and apply local mode analysis to evaluate the performance of the various components of the multigrid preconditioner numerical examples in 2 d demonstrate the efficacy of our method

fourier analysis of multigrid methods for general systems of pdes most iteration methods for solving boundary value problems can be viewed as approximations of a time dependent differential equation in this paper we show that the multigrid method has the effect of increasing the timestep for the smooth part of the solution leading back to an increase of the convergence rate for the nonsmooth part the convergence is an effect of damping fourier analysis is used to find the relation between the convergence rate for multigrid methods and singlegrid methods the analysis is performed for general partial differential equations and an arbitrary number of grids the difference in the behavior of the iterations between firstand second order equations is discussed the theoretical results are confirmed in simple numerical experiments

on multigrid solution of high reynolds incompressible entering flows an approach is presented for effectively separating the solution process of the elliptic component of high reynolds incompressible steady entering flow for which classical multigrid techniques are well suited from that of the non elliptic part for which other methods are more effective it is shown by analysis and numerical calculations that such an approach is very effective in terms of asymptotic convergence as well as reduction of errors well below discretization level in a 1 fmg algorithm

a hybrid multilevel method for high order discretization of the euler equations on unstructured meshes higher order discretization has not been widely successful in industrial applications to compressible flow simulation among several reasons for this one may identify the lack of tailor suited best practice relaxation techniques that compare favorably to highly tuned lower order methods such as finite volume schemes in this paper we investigate solution algorithms in conjunction with high order spectral difference discretization for the euler equations using such techniques as multigrid and matrix free implicit relaxation methods in particular we present a novel hybrid multilevel relaxation method that combines optionally matrix free implicit relaxation techniques with explicit multistage smoothing using geometric multigrid furthermore we discuss efficient implementation of these concepts using such tools as automatic differentiation

convergence analysis of projection methods for the numerical solution of large lyapunov equations the numerical solution of large scale continuous time lyapunov matrix equations is of great importance in many application areas assuming that the coefficient matrix is positive definite but not necessarily symmetric in this paper we analyze the convergence of projection type methods for approximating the solution matrix under suitable hypotheses on the coefficient matrix we provide new asymptotic estimates for the error matrix when a galerkin method is used in a krylov subspace numerical experiments confirm the good behavior of our upper bounds when linear convergence of the solver is observed

a multilevel algorithm for the solution of second order elliptic differential equations on sparse grids a multilevel algorithm is presented that solves general second order elliptic partial differential equations on adaptive sparse grids the multilevel algorithm consists of several v cycles in x and y direction a suitable discretization provide that the discrete equation system can be solved in an efficient way numerical experiments show a convergence rate of order o 1 for the multilevel algorithm