existence results for the flow of viscoelastic fluids with a differential constitutive law where 2 is the symmetric extra stress tensor the total stress is given by g pl i where p is the hydrodynamic pressure 4 is the rate of deformation tensor g u vu vu u being the velocity field a is the relaxation time a2 the retardation time 0 i a a1 and q is the fluid viscosity the symbol a d t denotes an objective frame indifferent derivative i 181 more precisely

a pseudo arclength continuation method for nonlinear eigenvalue problems a variant of the classical pseudo arclength continuation method is proposed basically the method can be viewed as pseudo arclength continuation in r lambda space where r is a functional of the solution another difference is a three parameter predictor instead of the standard euler step this predictor as well as the newton corrector iteration are justified and some numerical results for reaction diffusion equations are presented the method provides a simple algebraic check for symmetry breaking bifurcation the most common type of secondary bifurcation in physical examples

analysis of a defect correction method for viscoelastic fluid flow we study a defect correction method for a viscoelastic fluid flow obeying an oldroyd btype constitutive law in the defect step the constitutive equation is computed with the artificially reduced weissenberg parameter for stability and the residual is taken care of in the correction step we show the convergence of the defect correction method and derive an error estimate

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

construction of solution curves for large two dimensional problems of steady state flows of incompressible fluids this work represents a step toward advancing classical methods of bifurcation analysis in conjunction with very large scale scientific computing needed to model realistically processes which involve laminar and steady flows of incompressible fluids a robust method of analysis based on pseudoarclength continuation newton s method and direct solution of linear systems is proposed and applied to the analysis of a representative system of fluid mechanics namely the tilted lid driven cavity accurate solution curves possessing simple singular points were computed for reynolds numbers varying from 0 to 10 000 and for different angles of tilt the results demonstrate that two dimensional models with up to 1 000 000 algebraic equations can be studied feasibly using the methods described here with state of the art vector supercomputers

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds we study a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectivelyp 1 discontinuous p 2 continuous p 1 continuous we use the method of lesaint raviart for the convection of the extra stress tensor we suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution we show by a fixed point method that the approximate problem has a solution and we give an error bound

weakly singular gronwall inequalities and applications to fractional differential equations abstract we obtain some new gronwall type inequalities which are applicable to some weakly singular volterra integral equations similar to the ones first studied by d henry the main interest is that we consider cases with a double singularity and we obtain explicit l bounds rather than l 1 bounds furthermore our bounds involve the exponential function and not the mittag leffler function as in some previous works we give applications to some volterra integral equations with a doubly singular kernel that arise from caputo fractional differential equations where as opposed to previous papers we have a singularity in the nonlinearity

predictor and steplength selection in continuation methods for the navier stokes equations abstract continuation methods are in common use for solving the nonlinear equations resulting from discretizations of the navier stokes equations an imoportant ingrediet in these algorithms is the selection of the step size in the continuation parameter for certain values of the reynolds number it is shown that the step size in the reynolds number may be chosen independently of the particular predictor step used within the continuation algorithm estimates for the path derivatives are derived and used to show that the allowable step sizes are substantially greater than would be predicted by theories that apply to general nonlinear mappings

a stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

condition estimates for pseudo arclength continuation we bound the condition number of the jacobian in pseudo arclength continuation problems and we quantify the effect of this condition number on the linear system solution in a newton gmres solve pseudo arclength continuation solves parameter dependent nonlinear equations g u lambda 0 by introducing a new parameter s which approximates arclength and viewing the vector x u lambda as a function of s in this way simple fold singularities can be computed directly by solving a larger system f x s 0 by simple continuation in the new parameter s it is known that the jacobian f x of f with respect to x u lambda is nonsingular if the path contains only regular points and simple fold singularities we introduce a new characterization of simple folds in terms of the singular value decomposition and we use it to derive a new bound for the norm of f x 1 we also show that the convergence rate of gmres in a newton step for f x s 0 is essentially the same as that of the original problem g u lambda 0 in particular we prove that the bounds on the degrees of the minimal polynomials of the jacobians f x and g u differ by at most 2 we illustrate the effectiveness of our bounds with an example from radiative transfer theory

first second and third order fractional step methods for the three field viscoelastic flow problem in this paper three different fractional step methods are designed for the three field viscoelastic flow problem whose variables are velocity pressure and elastic stress the starting point of our methods is the same as for classical pressure segregation algorithms used in the newtonian incompressible navier stokes problem these methods can be understood as an inexact lu block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level the final schemes allow one to solve the problem in a fully decoupled form where each equation for velocity pressure and elastic stress is solved separately the first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation whereas the key point for the second order scheme is a first order extrapolation of these variables the third order fractional step method relies on yosida s scheme referring to the spatial discretization either the galerkin method or a stabilized finite element formulation can be used we describe the fractional step methods first assuming the former and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work this discretization in space shows very good stability permitting in particular the use of equal interpolation for all variables

computation of viscoelastic fluid flows using continuation methods the numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter the weissenberg number increases specifically at a weissenberg number larger than a critical value the iterative nonlinear solver fails to converge a phenomenon known as the high weissenberg number problem in this work we describe the application and implementation of continuation methods to the nonlinear johnson segalman model for steady state viscoelastic flows simple natural and pseudo arclength continuation approaches in weissenberg number are investigated for a discontinuous galerkin finite element discretization of the equations computations are performed for a benchmark contraction flow and several aspects of the performance of the continuation methods including high weissenberg number limits are discussed

resonance phenomenon for the galerkin truncated burgers and euler equations it is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial fourier modes having wave numbers in excess of a threshold k mathrm g exhibit unexpected features the study is carried out for both the one dimensional burgers equation and the two dimensional incompressible euler equation for large k mathrm g and smooth initial conditions the first symptom of truncation a localized short wavelength oscillation which we call a tyger is caused by a resonant interaction between fluid particle motion and truncation waves generated by small scale features shocks layers with strong vorticity gradients etc these tygers appear when complex space singularities come within one galerkin wavelength ensuremath lambda mathrm g 2 ensuremath pi k mathrm g from the real domain and typically arise far away from preexisting small scale structures at locations whose velocities match that of such structures tygers are weak and strongly localized at first in the burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to k mathrm g ensuremath 2 3 and k mathrm g ensuremath 1 3 respectively but grow and eventually invade the whole flow they are thus the first manifestations of the thermalization predicted by t d lee q j appl math 10 69 1952 the sudden dissipative anomaly the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t ensuremath star which is well known for the burgers equation and sometimes conjectured for the three dimensional euler equation has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit k mathrm g ensuremath rightarrow ensuremath infty this leads to reynolds stresses acting on scales larger than the galerkin wavelength and thus prevents the flow from converging to the inviscid limit solution there are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid limit behavior

a two parameter defect correction method for computation of steady state viscoelastic fluid flow the numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter the weissenberg number increases specifically at a weissenberg number larger than a critical value the iterative nonlinear solver fails to converge in this paper a two parameter defect correction method for viscoelastic fluid flow is presented and analyzed in the defect step the weissenberg number is artificially reduced to solve a stable nonlinear problem the approximation is then improved in the correction step using a linearized correction iteration numerical experiments support the theoretical results and demonstrate the effectiveness of the method

continuation techniques for a penalty approximation of the navier stokes equations continuation techniques are used to solve a penalty finite element approximation of the navier stokes equations sufficient conditions are given for convergence of the euler newton continuation method in a reynolds parameter to an isolated solution of the finite element problem numerical results are presented for the driven cavity problem the approach is extended to arc length continuation to treat problems with singular points on the continuation path

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

interplay between the beale kato majda theorem and the analyticity strip method to investigate numerically the incompressible euler singularity problem decrease to zero in order to sustain a finite time singularity consistent with the bkm theorem a simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the bkm theorem with the analyticity strip method it is shown that a finite time blowup can exist only if t vanishes sufficiently fast at the singularity time in particular if a power law is assumed for t then its exponent must be greater than some critical value thus providing a new test that is applied to our 4096 3 taylor green numerical simulation our main conclusion is that the numerical results are not inconsistent with a singularity but that higher resolution studies are needed to extend the time interval on which a well resolved power law behavior of t takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay