analysis of an explicit and matrix free fractional step method for incompressible flows in this paper we present the analysis of an explicit and matrix free fractional step method for incompressible flows the presented method can be employed in either conservation or non conservation form the stabilization convergence and conservation aspects of the presented method are discussed a procedure for eliminating the first order error in time introduced by the split is proposed some benchmark steady and unsteady state examples are presented to demonstrate the proposed new aspects of the matrix free scheme

a characteristic finite element algorithm for time dependent 3 d advection dominated transport using unstructured grids abstract an algorithm based on operator splitting has been successfully implemented for solving unsteady advection dominated transport problems in 3 d specifically the general operator integration factor splitting method of maday et al is applied to the unsteady advection diffusion equation with source sink terms the algorithm incorporates a 3 d characteristic galerkin scheme to treat advection and a standard galerkin treatment of the diffusion and source sink terms up to third order operator splitting was implemented and validated against several analytical solutions the algorithm showed the expected error behaviour and good performance in modeling advection dominated transport problems the practical utility and effectiveness of the proposed numerical scheme was further demonstrated by solving the graetz nusselt problem i e high peclet number mass heat transport in a fully developed pipe flow

exact projections and the lagrange galerkin method a realistic alternative to quadrature abstract the lagrange galerkin method has proved a very successful method in computational fluid dynamics cfd in practice but it has suffered from question marks about its stability when the integrals are approximated by quadratures in this paper we will prove instability for wider ranges of quadrature formulae than previously considered we will also introduce on approximate integration technique on triangular grids that cheaply recovers the unconditional stability of the lagrange galerkin method with no loss in accuracy the same technique can also be used for the exact projection of data from one arbitrary triangular grid to another different arbitrary triangular grid this could be used in grid adaptation algorithms or in triangular multigrid procedures

the characteristic based split cbs scheme a unified approach to fluid dynamics this paper presents a comprehensive overview of the characteristic based methods and characteristic based split cbs scheme the practical difficulties of employing the original characteristic schemes are discussed the important features of the cbs scheme are brought out by studying several problems of compressible and incompressible flows all special consideration necessary for solving these problems are thoroughly discussed the cbs scheme is presented in such a way that any interested researcher should be able to develop a code using the information provided several invicid and viscous flow examples are also provided to demonstrate the unified cbs approach for sample two dimensional codes input files and instructions the readers are referred to www nithiarasu co uk copyright 2006 john wiley sons ltd

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

hybrid eulerian lagrangian semi implicit time integrators hybrid eulerian lagrangian semi implicit time integrators helsi are presented which use the standard semi implicit formulation as their starting point the advantage of such an approach is that existing models which employ eulerian semi implicit time integrators can easily be augmented to use the helsi approach that is provided that either a series of advection problems can be solved efficiently as in the operator integration factor splitting oifs method or that the grid and underlying numerics allow the construction of efficient search and interpolation stencils typically required of semi lagrangian methods once the advection problem is solved the approach then follows the standard semi implicit approach an important caveat of this approach is that the semi implicit time integrator must be constructed around the implicit backward difference formulas

an analysis of time discretization in the finite element solution of hyperbolic problems abstract the problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined a modified equation analysis reveals that the classical second order accurate time stepping algorithms i e the lax wendroff leap frog and crank nicolson methods properly combine with piecewise linear finite elements in advection problems only for small values of the time step on the contrary as the courant number increases the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit cfl property are not achieved these fundamental numerical properties can however be recovered while still remaining in the standard galerkin finite element setting by increasing the order of accuracy of the time discretization this is accomplished by exploiting the taylor series expansion in the time increment up to the third order before performing the galerkin spatial discretization using piecewise linear interpolations as a result it appears that the proper finite element equivalents of second order finite difference schemes are implicit methods of incremental type having third and fourth order global accuracy on uniform meshes taylor galerkin methods numerical results for several linear examples are presented to illustrate the properties of the taylor galerkin schemes in one and two dimensional calculations

a lagrangian method for the shallow water equations based on a voronoi mesh one dimensional results abstract a lagrangian method for the shallow water equations in one dimension is presented the method is based on the following idea place n points on a line to represent the fluid each point represents all the fluid that is closer to this point than to any other this partitions the line into intervals the principle of least action yields a set of difference equations which can be easily solved at each time step to give new values for the height and velocity of the fluid particles in each interval the points are then moved to their new locations and the intervals reconstructed in the event that two points get too close to each other consider the points to have collided the collision is treated as an inelastic collision and the two points are merged into one according to certain conservation laws it is then possible to handle flows with shocks by modelling a shock as a collision between two particles that is a shock is thought of as one fluid particle overtaking another one and colliding with it the results of this method on the dam breaking problem are presented the solution is compared to the exact solution and to a solution by the random choice method

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

bernstein widths of hardy type operators in a non homogeneous case abstract let i a b r let 1 p q let u and v be positive functions with u l p i v l q i and let t l p i l q i be the hardy type operator given by t f x v x a x f t u t d t x i we show that the bernstein numbers b n of t satisfy lim n n b n c p q i u v r d t 1 r 1 r 1 p 1 q where c p q is an explicit constant depending only on p and q

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

a nearly conservative high order lagrange galerkin method for the resolution of scalar convection dominated equations in non divergence free velocity fields abstract in this work we present a novel lagrange galerkin method for the resolution of the scalar pure convection and convection dominated diffusion equations in non divergence free velocity fields the scheme has been formulated so that i time integration is carried out by means of an arbitrary order backward differentiation formula ii finite element space discretizations of any order can be employed and iii it satisfies a discrete mass balance equation at each simulation instant and preserves mass in those cases in which no fluid enters nor leaves the control domain as long as the integrals involved are computed exactly these properties are achieved by posing a conservation law for a weighted mass from which the standard mass can be seen as a particular case that can be easily discretized in time and in space with any order of accuracy numerical experiments which include a three dimensional test show that the method is as accurate in time as the employed time marching formula as accurate in space as the finite element discretization and generally more accurate especially in terms of mass conservation errors and efficient than the classic non conservative lagrange galerkin method to the best of our knowledge this is the first conservative lagrange galerkin method for convection dominated equations that considers both non divergence free velocity fields and time and space discretizations of any order

behaviour of the approximation numbers of a sobolev embedding in the one dimensional case we consider the sobolev embeddings e1 w01 p a b lp a b and e2 l1 p a b 1 lp a b 1 with a b and 1 p we show that the approximation numbers an ei of ei have the property that where cp is a constant dependent only on p moreover we show the precise value of an e1 and we study the unbounded sobolev embedding e3 l1 p a b lp a b and determine precisely how closely it may be approximated by n dimensional linear maps mathematical subject codes 47g10 47b10

operators of hardy type we provide a survey of the contributors of des evans dealing with operators t of the form acting between lehesgue spaces and also report on new results concerning the bernstein widths and kolmogorov numbers of t

a high order characteristics method for the incompressible navier stokes equations abstract we analyze a high order characteristics method for the navier stokes equations we focus on the cases of the first second and third order in time schemes with finite element spatial discretization a numerical comparison between the first and second order schemes is done for steady or transient states flows

unconditional stability and error estimates of modified characteristics fems for the navier stokes equations the paper is concerned with the unconditional stability and convergence of characteristics type methods for the time dependent navier stokes equations we present optimal error estimates in l 2 l2 and h 1 h1 norms for a typical modified characteristics finite element method unconditionally while all previous works require certain time step restrictions the analysis is based on an iterated characteristic time discrete system with which the error function is split into a temporal error and a spatial error with a rigorous analysis to the characteristic time discrete system we prove that the difference between the numerical solution and the solution of the time discrete system is tau independent where tau denotes the time stepsize thus numerical solution in w 1 infty w1 is bounded and optimal error estimates can be obtained in a traditional way numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time dependent navier stokes equations the approach used in this paper can be easily extended to many other characteristics based methods

the class number one problem for some non abelian normal cm fields of degree 48 we prove that there is precisely one normal cm field of degree 48 with class number one which has a normal cm subfield of degree 16 the narrow hilbert class field of q 5 101 with 3 2 5 1 0

time accurate solution of advection diffusion problems by finite elements abstract the taylor galerkin method recently proposed for the spatial and temporal discretization of hyperbolic equations is employed to derive accurate and efficient numerical schemes for the solution of time dependent advection diffusion problems two distinct numerical strategies are discussed the first is suitable for evolutionary problems while the second is appropriate for situations in which a steady state is eventually reached for purely evolutionary problems the taylor galerkin method is applied to the complete advection diffusion equation and through modifications of the standard mass matrix is shown to generate an incremental form of the crank nicolson time stepping method such an incremental form also lends itself to the derivation of computationally simple but accurate explicit algorithms to deal with transient situations which evolve towards a highly convective steady state the global taylor galerkin method is found to be ineffective since it reduces to the standard galerkin formulation as the temporal term vanishes for such cases we suggest the use of a splitting up method in which advection and diffusion are treated separately by appropriate taylor galerkin methods the proposed methods are illustrated first for linear constant coefficient equations they are successively extended to deal with nonlinear and multi dimensional problems numerical applications indicate their effectiveness

an efficient characteristic galerkin scheme for the advection equation in 3 d conventional galerkin based finite element algorithms perform poorly when modeling advective transport here we develop and test a characteristic galerkin scheme to solve the unsteady three dimensional advective transport equation the algorithm uses tracking of the gaussian quadrature points to project the information from the eulerian background grid to the lagrangian grid numerical experiments were carried out to investigate the performance of this scheme despite speculations to the contrary in the literature this scheme does not suffer from instability problems further it is conservative and shows good phase characteristics with slight numerical diffusion we conclude that the characteristic galerkin method is a viable and efficient scheme for solving advection problems in three dimensions

the class number one problem for some non abelian normal cm fields let n be a non abelian normal cm eld of degree 4p p any odd prime note that the galois group of n is either the dicyclic group of order 4p or the dihedral group of order 4p we prove that the relative class number of a dicyclic cm eld of degree 4p is always greater then one then we determine all the dihedral cm elds of degree 12 with class number one there are exactly nine such cm elds

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100