rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

solution of the implicitly discretised reacting flow equations by operator splitting abstract a non iterative method for handling the coupling of the implicitly discretised time dependent fluid flow equations is described the method is based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations the main feature of the technique is the splitting of the solution process into a series of steps whereby operations on pressure are decoupled from those on velocity at each step with the split sets of equations being amenable to solution by standard techniques at each time step the procedure yields solutions which approximate the exact solution of the difference equations the accuracy of this splitting procedure is assessed for a linearised form of the discretised equations and the analysis indicates that the solution yielded by it differs from the exact solution of the difference equations by terms proportional to the powers of the time step size by virtue of this it is possible to dispense with iteration thus resulting in an efficient implicit scheme while retaining simplicity of implementation relative to contemporary block simultaneous methods this is verified in a companion paper which presents results of computations carried out using the method

finite surface discretization for incompressible navier stokes equations and coupled conservation laws abstract we present a new finite surface discretization fsd aiming at the incompressible navier stokes equations nse and other coupled conservation laws this discretization defines the velocities as surfaced averaged values living on the faces of the pressure volumes in which the mass is set to be conserved consequently the calculation of the mass balance on these control volumes is exact which allows more accurate information to be kept in the velocity field and produces a very accurate prediction of the pressure in the next time step the proposed discretization reduces the stencil size of the poisson equation in the projection method compared to the finite volume and finite difference discretizations due to highly accurate mass conservation the compact sixth order approximation of fsd can be used with an explicit fourth order pressure treatment this property greatly reduces the cost and complexity of the implementation we present the discrete evolution equation of the surface averaged velocities together with the enforcement of mass conservation and the solution procedure for the pressure the approximation of the nse under this new discretization uses a combination of finite difference and finite volume methods the proposed method is validated using standard laminar test cases we identify the conditions under which a fourth order pressure treatment can support the sixth order and eighth order approximations of the convection term using fourier analysis the performance of the method is evaluated on turbulent channel flows up to friction reynolds number of 950 the quantitative relationships between the accuracy of the solution and grid size are identified we present two performance indices for comparison with other methods at the error level of 1 per mille the proposed method is 28 times faster than the classic second order scheme

pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations a calculation procedure for viscous flows at all mach numbers has been described the scheme has been developed for a nonorthogonal coordinate system in order to handle both incompressible and compressible flows pressure has been selected as a primary dependent variable in preference to density the pressure field is evaluated using the compressible form of the simpler algorithm the accuracy of the proposed scheme has been assessed by comparing the results with experimental data or other numerical results available in the literature

efficient implementation of thinc scheme in this paper we improve the multi dimensional thinc tangent of hyperbola for interface capturing scheme f xiao y honma t kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer meth fluid 48 2005 1023 the thinc scheme is a vof volume of fluid type method in the original thinc scheme one dimensional thinc scheme was straightforwardly used for multi dimensional cases in this paper we propose the wlic weighed line interface calculation method to extend the thinc scheme to multi dimension in the wlic method the interface is reconstructed by taking an average of line interfaces along x y and z coordinates with the weights calculated from surface normal the wlic method can extend the thinc scheme to multi dimension while maintaining simplicity of implementation and achieve a higher accuracy than the original thinc scheme the wlic method can readily extend the thinc scheme to three dimension

the 2 norm of random matrices abstract numerical experiments show that it is possible to derive simple estimates for the expected 2 norm of random matrices a with elements from a normal distribution with zero mean and standard deviation and from a poisson distribution with mean value these estimates are max m n 2 max m n and e 2 mn respectively where m and n are the dimensions of a

compressible multiphase semi implicit method with moment of fluid interface representation a unified method for simulating multiphase flows using an exactly mass momentum and energy conserving cell integrated semi lagrangian advection algorithm is presented the deforming material boundaries are represented using the moment of fluid method the new algorithm uses a semi implicit pressure update scheme that asymptotically preserves the standard incompressible pressure projection method in the limit of infinite sound speed the asymptotically preserving attribute makes the new method applicable to compressible and incompressible flows including stiff materials enabling large time steps characteristic of incompressible flow algorithms rather than the small time steps required by explicit methods shocks are captured and material discontinuities are tracked without the aid of any approximate or exact riemann solvers simulations of underwater explosions and fluid jetting in one two and three dimensions are presented which illustrate the effectiveness of the new algorithm at efficiently computing multiphase flows containing shock waves and material discontinuities with large impedance mismatch

solution of low mach number aeroacoustic flows using a variational multi scale finite element formulation of the compressible navier stokes equations written in primitive variables abstract in this work we solve the compressible navier stokes equations written in primitive variables in order to simulate low mach number aeroacoustic flows we develop a variational multi scale formulation to stabilize the finite element discretization by including the orthogonal dynamic and non linear subscales together with an implicit scheme for advancing in time three additional features define the proposed numerical scheme the splitting of the pressure and temperature variables into a relative and a reference part the definition of the matrix of stabilization parameters in terms of a modified velocity that accounts for the local compressibility and the approximation of the dynamic stabilization matrix for the time dependent subscales we also include a weak imposition of implicit non reflecting boundary conditions in order to overcome the challenges that arise in the aeroacoustic simulations at low compressibility regimes the order of accuracy of the method is verified for two and three dimensional linear and quadratic elements using steady manufactured solutions several benchmark flow problems are studied including transient examples and aeroacoustic applications

the computation of compressible and incompressible recirculating flows by a non iterative implicit scheme abstract the piso algorithm which is presented in a companion paper is a non iterative method for solving the implicity discretised time dependent fluid flow equations the algorithm is here applied in conjunction with a finite volume technique employing a backward temporal difference scheme to the computation of compressible and incompressible flow cases the results of calculations are compared with similar ones obtained with an existing iterative method it is shown that for time evolving flows the splitting error of piso is negligibly small at the level of time step required to eliminate the temporal truncation error while the avoidance of iteration results in a substantial reduction in computing effort over that required by iterative methods it is also demonstrated that piso is stable for fairly large time steps which renders it useful for steady state calculations as well

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

a numerical method for the simulation of low mach number liquid gas flows this work is devoted to the numerical simulation of liquid gas flows the liquid phase is considered as incompressible while the gas phase is treated as compressible in the low mach number approximation a single fluid two pressure model is developed and the front tracking method is used to track the interface navier stokes equations coupled with that of temperature are solved in the whole computational domain velocity pressure and temperature fields are computed yielding a complete description of the dynamics for both phases we show that our method is much more efficient than the so called all mach methods involving a single pressure since large time steps can be used while retaining time accuracy the model is first validated on a reference test problem solved using an accurate ale technique to track the interface numerical examples in two space dimensions are next presented they consist of air bubbles immersed in a closed cavity filled up with liquid water the forced oscillations of the system consisting of the air bubbles and the liquid water are investigated they are driven by a heat supply or a thermodynamic pressure difference between the bubbles

reprint of atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers for computational efficiency conservative multi tracer semi lagrangian type transport schemes are appropriate global models based on high order galerkin approach employ highly non uniform spectral element grids and semi lagrangian transport is a challenge on those grids a conservative semi lagrangian scheme spelt spectral element lagrangian transport employing a multi moment compact reconstruction procedure is developed for non uniform quadrilateral grids the scheme is based on a characteristic semi lagrangian method that avoids complex and expensive upstream area computations the spelt scheme has been implemented in the high order method modeling environment homme which is based on a cubed sphere grid with spectral element spatial discretization additionally we show the strong scalability and multi tracer efficiency using several benchmark tests the spelt solution can be made monotonic positivity preserving by combining the flux corrected transport algorithm which is demonstrated on a uniform resolution grid in particular spelt can be efficiently used for non uniform grids and provides accurate and stable results for high resolution meshes

unified formulation for compressible and incompressible flows by using multi integrated moments ii multi dimensional version for compressible and incompressible flows following our previous paper on a novel finite volume formulation for computing flows of any mach number we present the multi dimensional extension of the method in this paper treating two kinds of averages which are generically called moments and defined as the volume integrated average via and the surface integrated average sia in this paper as the prognostic variables makes the present method different from the conventional finite volume method where only via is the basic variable the resulting discretization approach which is a type of the cip multi moment finite volume method and called vsiam3 volume surface integrated average based multi moment method is combined with a pressure based projection formulation for the time integration to enable the simulations for both compressible and incompressible flows the numerical algorithm for multi dimensions will be reported in this paper numerical experiments with flows of a wide range mach numbers will be also presented

unified adaptive variational multiscale method for two phase compressible incompressible flows in this paper we present a new stabilized finite element method to solve the two phase compressible incompressible fluid flow problems using the level set method an anisotropic mesh adaptation with a regularized interface is adopted to deal with the high discontinuity in the material properties the coupling between the pressure and the flow velocity is ensured by introducing mass conservation terms in the momentum equation therefore the same set of primitive unknowns and equations is described for both phases the unified system is solved using a new derived variational multiscale stabilized finite element method it is tested on three time dependent liquid gas interface problems the numerical results show good stability and accuracy properties permitting also to deal with important compressibility effect high density ratio and extremely stretched anisotropic elements

eb d a new clustering approach for signed social networks based on both edge betweenness centrality and density of subgraphs abstract clustering algorithms for unsigned social networks which have only positive edges have been studied intensively however when a network has like dislike love hate respect disrespect or trust distrust relationships unsigned social networks with only positive edges are inadequate thus we model such kind of networks as signed networks which can have both negative and positive edges detecting the cluster structures of signed networks is much harder than for unsigned networks because it not only requires that positive edges within clusters are as many as possible but also requires that negative edges between clusters are as many as possible currently we have few clustering algorithms for signed networks and most of them requires the number of final clusters as an input while it is actually hard to predict beforehand in this paper we will propose a novel clustering algorithm called eb d for signed networks where both the betweenness of edges and the density of subgraphs are used to detect cluster structures a hierarchically nested system will be constructed to illustrate the inclusion relationships of clusters to show the validity and efficiency of eb d we test it on several classical social networks and also hundreds of synthetic data sets and all obtain better results compared with other methods the biggest advantage of eb d compared with other methods is that the number of clusters do not need to be known prior

a unified method for computing incompressible and compressible flows in boundary fitted coordinates a unified method for computing incompressible and compressible flows with mach uniform accuracy and efficiency is described the method is equally applicable to stationary and nonstationary flows a pressure based discretisation on a staggered grid in general boundary fitted coordinates is used for the euler equations extension to navier stokes is straightforward dimensionless variables that remain finite for all mach numbers are used mach number independent accuracy and efficiency is shown by numerical experiments

a simple stabilized finite element method for solving two phase compressible incompressible interface flows abstract when computing interface flows between compressible gas and incompressible liquid fluids one faces at least to the following difficulties 1 transition from a gas density linked to the local temperature and pressure by an equation of state to a liquid density mainly constant in space 2 proper approximation of the divergence constraint in incompressible regions and 3 wave transmission at the interface the aim of the present paper is to design a global i e the same for each phase numerical method to address easily this coupling to this end the same set of primitive unknowns and equations is used everywhere in the flow but with a dynamic parameterization that changes from compressible to incompressible regions on one hand the compressible navier stokes equations are considered under weakly compressibility assumption so that a non conservative formulation can be used on the other hand the incompressible non isothermal model is retained in addition the level set transport equation is used to capture the interface position needed to identify the local characteristics of the fluid and to recover the adequate local modelling for space approximation of navier stokes equations a galerkin least squares finite element method is used two essential elements for defining this numerical scheme are the stabilization and the computation of element integral of the approximated weak form since very different concerns motivate the need for stabilization in compressible and incompressible flows the first difficulty is to design a stabilization operator suitable for both types of flows especially in mixed elements in addition some integral of discontinuous functions must be correctly computed to ensure interfacial wave transmission to overcome these two difficulties specific averages are computed especially near the interface finally the level set transport equation is computed by a quadrature free discontinuous galerkin method numerical strategies are performed and validated for 1d and 2d applications

refined error analyses of cholesky factorization this paper concentrates on the practical cholesky process for solving positive definite symmetric systems of linear equations in standard floating point arithmetic two self contained round off analyses are presented which lead to novel a posteriors error bounds of practical use and to refined a priori results of more theoretical significance unmotivated assumptions are avoided with the aim of being constructive throughout

global shallow water models based on multi moment constrained finite volume method and three quasi uniform spherical grids this is a review article to present several accurate and computationally efficient global models for shallow water equations recently developed under a general numerical framework the multi moment constrained finite volume mcv method the multi moment constrained finite volume method defines the unknowns prognostic variables as the point values at the solution points located over each mesh element the time evolution equations to update these unknowns are derived through the constraint conditions on different moments e g the point value pv and the volume integrated average via rigorous numerical conservation is guaranteed by the constraint on the via moment through a finite volume formulation of flux form the resulted numerical schemes are very simple efficient and easy to implement for both structured and unstructured grids we have implemented the mcv method to three major spherical grids yin yang overset grid cubed sphere grid and geodesic icosahedral grid which have overall quasi uniform grid spacings and are highly popular in the community of global modeling for atmospheric and oceanic dynamics in this paper we present the global shallow water models based on these three spherical grids and the third order mcv scheme we evaluate and compare the models by widely used benchmark tests which show the third order convergence rate for all models and the numerical results are competitive to other exiting models using mcv method as a numerical formulation is well balanced between solution quality and computational simplicity the proposed models provide accurate and practical bases for developing dynamic core of general circulation models on different spherical grids

an ale pairwise relaxing meshless method for compressible flows abstract in this paper we present a pairwise relaxing meshless prm method for solving the euler equations of compressible flows within the arbitrary lagrangian eulerian ale framework derived from the moving particle semi implicit mps method and the finite volume particle fvp method the prm approximates the derivatives from the value defined at the midpoint of each interacting particle pairs through a kernel based formulation pairwise relaxing constants are introduced to the kernels to provide degree of freedom to enforce the taylor series consistency condition while mass momentum and energy are conserved exactly an upwind high order reconstruction scheme via a corrective procedure and variable cut off radius is also developed for this prm method the hllc approximate riemann solver is adopted to solve riemann problem one and two dimensional numerical tests are presented to demonstrate the performance of the prm method

a multi moment finite volume formulation for shallow water equations on unstructured mesh a novel and accurate finite volume method has been presented to solve the shallow water equations on unstructured grid in plane geometry in addition to the volume integrated average via moment for each mesh cell the point values pv moment defined on cell boundary are also treated as the model variables the volume integrated average is updated via a finite volume formulation and thus is numerically conserved while the point value is computed by a point wise riemann solver the cell wise local interpolation reconstruction is built based on both the via and the pv moments which results in a scheme of almost third order accuracy efforts have also been made to formulate the source term of the bottom topography in a way to balance the numerical flux function to satisfy the so called c property the proposed numerical model is validated by numerical tests in comparison with other methods reported in the literature

detection of near singularity in cholesky and ldl t factorizations abstract in sparse matrix applications it is often important to implement the cholesky and ldl t factorization methods without pivoting in order to avoid excess fillin we consider methods for detection of a nearly singular matrix by means of these factorizations without pivoting and demonstrate that a technique based on estimation of the smallest eigenvalue via inverse iteration will always reveal a nearly singular matrix

rank and null space calculations using matrix decomposition without column interchanges abstract the most widely used stable methods for numerical determination of the rank of a matrix a are the singular value decomposition and the qr algorithm with column interchanges here two algorithms are presented which determine rank and nullity in a numerically stable manner without using column interchanges one algorithm makes use of the condition estimator of cline moler stewart and wilkinson and relative to alternative stable algorithms is particularly efficient for sparse matrices the second algorithm is important in the case that one wishes to test for rank and nullity while sequentially adding columns to a matrix

a conservative multi tracer transport scheme for spectral element spherical grids atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers for computational efficiency conservative multi tracer semi lagrangian type transport schemes are appropriate global models based on high order galerkin approach employ highly non uniform spectral element grids and semi lagrangian transport is a challenge on those grids a conservative semi lagrangian scheme spelt spectral element lagrangian transport employing a multi moment compact reconstruction procedure is developed for non uniform quadrilateral grids the scheme is based on a characteristic semi lagrangian method that avoids complex and expensive upstream area computations the spelt scheme has been implemented in the high order method modeling environment homme which is based on a cubed sphere grid with spectral element spatial discretization additionally we show the strong scalability and multi tracer efficiency using several benchmark tests the spelt solution can be made monotonic positivity preserving by combining the flux corrected transport algorithm which is demonstrated on a uniform resolution grid in particular spelt can be efficiently used for non uniform grids and provides accurate and stable results for high resolution meshes

preconditioning applied to variable and constant density flows a time derivative preconditioning of the navier stokes equations suitable for both variable and constant density fluids is developed the ideas of low mach number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density based time marching schemes for solving flows of incompressible and variable density fluids at all speeds the preconditioning is coupled with a dual time stepping scheme implemented within an explicit multistage algorithm for solving time accurate flows the resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured solution adaptive mesh topologies this method is shown to provide accurate steady state solutions for transonic and low speed flow of variable density fluids the time accurate solution of unsteady incompressible flow is also demonstrated