long time behavior of solutions of gkdv equations abstract in this paper we consider the long time behavior of solutions of a version of gkdv equation on the real line the existence of the global attractor in h 2 r is proved when the force f belongs to h 2 r moreover we investigate the change of the attractor according to the change of f and the coefficient of damping term finally several numerical experiments confirm all the theoretical conclusions

prescribing the center of mass of a multi soliton solution for a perturbed semilinear wave equation abstract we construct a finite time blow up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension given any integer k 2 and 0 r we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times although the result is similar to the unperturbed case in its statement our method is new indeed our perturbed equation is not invariant under the lorentz transform and this requires new ideas in fact the main difficulty in this paper is to prescribe the center of mass 0 r we would like to mention that our method is valid also in the unperturbed case and simplifies the original proof by cote and zaag 9 as far as the center of mass prescription is concerned

an analysis of the time integration algorithms for the finite element solutions of incompressible navier stokes equations based on a stabilised formulation abstract this work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the navier stokes equations the stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes particular attention is given to the generalised method and the linear discontinuous in time finite element scheme the time integration schemes are first applied to two model problems represented by a first order differential equation in time and the one dimensional advection diffusion equation and subjected to a detailed mathematical analysis based on the fourier series expansion in order to establish the accuracy and efficiency of the time integration schemes for the navier stokes equations a detailed computational study is performed of two standard numerical examples unsteady flow around a cylinder and flow across a backward facing step it is concluded that the semi discrete generalised method provides a viable alternative to the more sophisticated and expensive space time methods for simulations of unsteady flows of incompressible fluids governed by the navier stokes equations

stability of the fractional step t scheme for the nonstationary navier stokes equations stability and convergence analysis of the fractional step theta scheme for the navier stokes equations coupled with the finite element approximations is presented the solutions of the underlying stokes and burgers problems are described in detail the asymptotic solution obtained by the numerical stabilization is compared with the analytical solution of the lid driven cavity problem

an analysis of the prothero robinson example for constructing new dirk and row methods this note analyses the order reduction phenomenon of diagonally implicit runge kutta methods dirk methods and rosenbrock wanner methods row methods applied on the prothero robinson example new order conditions to reduce order reduction are derived and a new third order dirk and row method is created the new schemes are applied to the prothero robinson example and on the semi discretised incompressible navier stokes equations numerical examples show that the new methods have better convergence properties than comparable methods

stability and asymptotic stability in the energy space of the sum of n solitons for subcritical gkdv equations we prove in this paper the stability and asymptotic stability in h1 of a decoupled sum of n solitons for the subcritical generalized kdv equations the proof of the stability result is based on energy arguments and monotonicity of the local l2 norm note that the result is new even for p 2 the kdv equation the asymptotic stability result then follows directly from a rigidity theorem in 16

implicit time discretization schemes for mixed least squares finite element formulations abstract this work is an extension of the ideas in averweg et al 2019 with the focus on a detailed investigation of implicit time discretization schemes to model instationary fluid flow based on the incompressible navier stokes equations and linear elastodynamic structural behavior the variational approaches for fluid and solid mechanics are based on a mixed least squares finite element method the l 2 norm minimization of the residuals of the constructed first order systems of the governing differential equations is based on two field stress velocity sv functionals for the time discretization of the sv fluid formulation four different types of implicit integration schemes are investigated namely the houbolt method the crank nicolson method and two explicit singly diagonally implicit runge kutta methods esdirk the sv formulation for the solid is discretized applying the houbolt method the presented time integration schemes are validated investigating an unsteady fluid flow and an elastodynamic structural benchmark since both fluid and solid sv formulations are discretized using conforming finite element spaces in h div and h 1 respectively the inherent fulfillment of coupling conditions when modeling fluid structure interaction problems is given a priori therefore the applicability is also examined by two simplified fsi problems for small deformations in order to represent the main characteristics of the presented approach

the exotic conformal galilei algebra and nonlinear partial differential equations abstract the conformal galilei algebra cga and the exotic conformal galilei algebra ecga are applied to construct partial differential equations pdes and systems of pdes which admit these algebras we show that there are no single second order pdes invariant under the cga but systems of pdes can admit this algebra moreover a wide class of nonlinear pdes exists which are conditionally invariant under cga it is further shown that there are systems of nonlinear pdes admitting ecga with the realisation obtained very recently in d martelli y tachikawa comments on galilei conformal field theories and their geometric realisation preprint arxiv 0903 5184v2 hep th 2009 moreover wide classes of nonlinear systems invariant under two different 10 dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented

continuous subgroups of the generalized schrodinger groups some general results on the subalgebras of the lie algebra asch n of the generalized schrodinger group sch n and on the subalgebras of the lie algebra asch n of the generalized extended schrodinger group sch n have been obtained the subalgebra structure of the algebras asch n and asch n are studied with respect to inner automorphisms of the groups sch n and sch n respectively the maximal abelian subalgebras and the one dimensional subalgebras of the algebras asch n and asch n have been explicitly found the full classification of the subalgebras of the algebras asch 3 asch 3 which are nonconjugate to the subalgebras of asch 2 asch 2 respectively has been carried out

construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension given a blow up solution with a characteristic point we refine the blow up behavior first derived by merle and zaag we also refine the geometry of the blow up set near a charac teristic point and show that except may be for one exceptional situation it is never symmetric with the respect to the characteristic point then we show that all blow up modalities predicted by those authors do occur more precisely given any integer k 2 and 0 we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times

description of two soliton collision for the quartic gkdv equation in this paper we give the rst description of the collision of two solitons for a nonintegrable equation in a special regime we consider solutions of the quartic gkdv equation tu x 2 xu u 4 0 which behave as t 1 like u t x qc1 x c1t qc2 x c2t t x where qc x ct is a soliton andk t kh1k qc2kh1k qc1kh1 the global behavior of u t is given by the following stability result for all t 2 r u t x qc1 t x y1 t qc2 t x y2 t t x where k t kh1k qc2kh1 and limt 1c1 t c limt 1c2 t c in the case where u t is a pure 2 soliton solution as t 1 i e limt 1 k t kh1 0 we obtain c c1 c 0 therefore in contrast with the integrable kdv equation or mkdv equation no global pure 2 soliton solution exists and the collision is inelastic a dierent notion of global 2 soliton is then proposed

adaptive time step control for the incompressible navier stokes equations abstract adaptive time stepping is an important tool in computational fluid dynamics for controlling the accuracy of simulations and for enhancing their efficiency this paper presents a systematic study of three classes of implicit and linearly implicit time stepping schemes with adaptive time step control applied to a 2d laminar flow around a cylinder schemes diagonal implicit runge kutta dirk methods and rosenbrock wanner row methods the time step is controlled using embedded methods it is shown that several row methods clearly outperform the more standard schemes and the dirk methods the results depend on a prescribed tolerance in the time step control algorithm whose appropriate choice varies from scheme to scheme

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

the galilean relativistic principle and nonlinear partial differential equations the second order partial differential equations invariant under transformations of galilei rotation scale and projection are described

adaptive time step control for higher order variational time discretizations applied to convection diffusion reaction equations abstract higher order variational time stepping schemes allow an efficient post processing for computing a higher order solution this paper presents an adaptive algorithm whose time step control utilizes the post processed solution the algorithm is applied to convection dominated convection diffusion reaction equations it is shown that the length of the time step properly reflects the dynamics of the solution with respect to the performance accuracy efficiency the variational time stepping schemes are compared with an adaptive crank nicolson scheme whose time step control relies on comparing two solutions computed with schemes of the same order

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

scattering for the quartic generalised korteweg de vries equation abstract we show that the quartic generalised kdv equation u t u x x x u 4 x 0 is globally well posed for data in the critical scale invariant space h x 1 6 r with small norm and locally well posed for large norm improving a result of grunrock a grunrock a bilinear airy estimate with application to gkdv 3 differential integral equations 18 12 2005 1333 1339 as an application we obtain scattering results in h x 1 r h x 1 6 r for the radiation component of a perturbed soliton for this equation improving the asymptotic stability results of martel and merle y martel f merle asymptotic stability of solitons for subcritical generalized kdv equations arch ration mech anal 157 3 2001 219 254

construction of solutions to the subcritical gkdv equations with a given asymptotical behavior we consider the subcritical generalized korteweg de vries equation ut uxx u4 x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v be in an adequate weighted sobolev space we construct a solution u t to the generalized korteweg de vries equation such that limt u t u t v j 1nrj t h1 0

asymptotic stability of high dimensional zakharov kuznetsov solitons we prove that solitons or solitary waves of the zakharov kuznetsov zk equation a physically relevant high dimensional generalization of the korteweg de vries kdv equation appearing in plasma physics and having mixed kdv and nonlinear schrodinger nls dynamics are strongly asymptotically stable in the energy space we also prove that the sum of well arranged solitons is stable in the same space orbital stability of zk solitons is well known since the work of de bouard proc r soc edinburgh 126 89 112 1996 our proofs follow the ideas of martel siam j math anal 157 759 781 2006 and martel and merle math ann 341 391 427 2008 applied for generalized kdv equations in one dimension in particular we extend to the high dimensional case several monotonicity properties for suitable half portions of mass and energy we also prove a new liouville type property that characterizes zk solitons and a key virial identity for the linear and nonlinear part of the zk dynamics obtained independently of the mixed kdv nls dynamics this last virial identity relies on a simple sign condition which is numerically tested for the two and three dimensional cases with no additional spectral assumptions required possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well posedness theory in the energy space and the verification of a corresponding sign condition

asymptotic stability of solitons for subcritical generalized kdv equations we prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized korteweg de vries equations in the subcritical case this includes in particular the kdv equation and the modified kdv equation this result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation which has its own interest the proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized kdv equation

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

a better consistency for low order stabilized finite element methods abstract the standard implementation of stabilized finite element methods with a piece wise function space of order lower than the highest derivative present in the partial differential equation often suffers from a weak consistency that can lead to reduced accuracy the popularity of these low order elements motivates the development of a new stabilization operator which globally reconstructs the derivatives not present in the local element function space this new method is seen to engender a stronger consistency leading to better convergence and improved accuracy applications to the navier stokes equations are given which illustrate the improvement at a negligible additional cost

improved traditional rosenbrock wanner methods for stiff odes and daes rosenbrock wanner methods usually have order reduction if they are applied to stiff ordinary differential or differential algebraic equations therefore in several papers further order conditions are derived to reduce this effect in rang 2014 the example of prothero and robinson 1974 is analysed to find further order conditions in this paper we consider traditional row methods such as ros3p lang and verwer 2001 ros3pw rang and angermann 2005 ros3pl lang and teleaga 2008 and rodasp steinebach 1995 and improve these methods such that these further order conditions are satisfied numerical examples show the advantages of the new methods

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

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

on the impact of the scheme for solving the higher dimensional equation in coupled population balance systems summary population balance systems are models for processes in nature and industry that lead to a coupled system of equations navier stokes equations transport equations etc where the equations are defined in domains with different dimensions this paper will study the impact of using different schemes for solving the three dimensional 3d equation of a precipitation process in a two dimensional flow domain the numerical schemes for the 3d equation are assessed with respect to the median of the volume fraction of the particle size distribution and the computational costs it turns out that in the case of a structured flow field with small variations in time all schemes give qualitatively the same results for a highly timedependent flow field the evolution of the median of the volume fraction differs considerably between first order and higher order schemes copyright q 2010 john wiley sons ltd

pressure corrected implicit θ schemes for the incompressible navier stokes equations abstract in this note the crank nicolson and the fractional step scheme are applied on the semi discretised incompressible navier stokes equations in a first step the formulation of the methods is modified such that the methods have optimal order for the pressure p i e the pressure p 0 is included in the scheme the main idea of this paper is that the fractional step scheme can be written as a diagonally implicit runge kutta method dirk method in this context it can easily be shown that the fractional step schemes have only stage order q 1 whereas the crank nicolson scheme has stage order q 2 hence the fractional step scheme may have order reduction if the method is applied on stiff odes and daes i e the semi discretised incompressible navier stokes equations some theoretical results and numerical examples illustrate this phenomena moreover it is shown that it is impossible to improve the fractional step method such that the scheme has the stage order q 2 and is strongly a stable or has the order p 3

a variational multiscale method for turbulent flow simulation with adaptive large scale space in turbulent flows it is only feasible to simulate large flow structures variational multiscale vms methods define these flow structures by projections onto appropriate function spaces this paper presents a finite element vms method which chooses the large scale projection space adaptively the adaption controls the influence of an eddy viscosity model and it is based on the size of the so called resolved small scales the adaptive procedure is described in detail numerical studies at a turbulent channel flow and a turbulent flow around a cylinder are presented it is shown that the method selects the large scale space in a reasonable way and that appropriately chosen parameters improve the results compared to the basic method which uses the same local large scale space in the whole domain and for all times

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

nonlinear galilei invariant pdes with infinite dimensional lie symmetry a class of nonlinear galilei invariant generalizations of the heat equation admitting infinite dimensional lie symmetry is presented lie symmetries and examples of exact solutions are found for some nonlinear equations generalizing the heat equation full sets of the absolute differential invariants for two non equivalent representations of the galilei algebra are constructed

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

instability of a class of dispersive solitary waves this paper studies the stability and instability properties of solitary wave solutions x ct of a general class of evolution equations of the form mu t f u x 0 which support weakly nonlinear dispersive waves it turns out that depending on their speed c and the relation between the dispersion i e the order of the pseudodifferential operator and the nonlinearity travelling waves may be stable or unstable sharp conditions to that effect are given

on integrable doebner goldin equations we integrate sub families of a family of nonlinear schrodinger equations proposed by doebner and goldin in the 1 1 dimensional case which have exceptional lie symmetries since the method of integration involves non local transformations of dependent and independent variables general solutions obtained include implicitly determined functions by properly specifying one of the arbitrary functions contained in these solutions we obtain broad classes of explicit square integrable solutions the physical significance and some analytical properties of the solutions obtained are briefly discussed

finite element les and vms methods on tetrahedral meshes finite element methods for problems given in complex domains are often based on tetrahedral meshes this paper demonstrates that the so called rational large eddy simulation model and a projection based variational multiscale method can be extended in a straightforward way to tetrahedral meshes numerical studies are performed with an inf sup stable second order pair of finite elements with discontinuous pressure approximation

an invariant set in energy space for supercritical nls in 1d we consider radial solutions of a mass supercritical monic nls and we prove the existence of a set which looks like a hypersurface in the space of finite energy functions invariant for the flow and formed by solutions which converge to ground states

a comparison of time discretization linearization approaches for the incompressible navier stokes equations abstract this paper presents a numerical study of two ways for discretizing and linearizing the time dependent incompressible navier stokes equations one approach consists in first applying a semi discretization in time by a fully implicit scheme then in each discrete time the equations are linearized by a fixed point iteration the number of iterations to reach a given stopping criterion is a priori unknown in this approach in the second approach rosenbrock schemes with s stages are used as temporal discretization the non linearity of the navier stokes equations is treated internally in the rosenbrock methods in each discrete time exactly s linear systems of equations have to be solved the numerical study considers five two dimensional problems with distinct features four implicit time stepping schemes and five rosenbrock methods are involved

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

asymptotic n soliton like solutions of the subcritical and critical generalized korteweg de vries equations we consider the generalized korteweg de vries equations ut uxx upx 0t x r inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in the subcritical and critical cases p 2 3 4 or 5 let r j t x qc j x c j t x j where j 1 n be n soliton solutions of this equation with corresponding speeds 0 c 1 c 2 c n in this paper we construct a solution u t of the generalized korteweg de vries equation such that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i this solution behaves asymptotically as t as the sum of n solitons without loss of mass by dispersion this is an exceptional behavior indeed being given the parameters c j 1 j n x j 1 j n we prove uniqueness of such a solution in the integrable cases p 2 and 3 such solutions are explicitly known and their properties were extensively studied in the literature they are called n soliton solutions therefore the existence result is new only for the nonintegrable cases the uniqueness result is new for all cases

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

intertwining operators for l conformal galilei algebras and hierarchy of invariant equations the l conformal galilei algebra denoted by is a non semisimple lie algebra specified by a pair of parameters d l the algebra is regarded as a nonrelativistic analogue of the conformal algebra we derive hierarchies of partial differential equations which have invariance of the group generated by with a central extension as kinematical symmetry this is done by developing a representation theory such as verma modules singular vectors of and vector field representations for d 1 2

a spectral deferred correction method for incompressible flow with variable viscosity abstract this paper presents a semi implicit spectral deferred correction sdc method for incompressible navier stokes problems with variable viscosity and time dependent boundary conditions the proposed method integrates elements of velocity and pressure correction schemes which yields a simpler pressure handling and a smaller splitting error than the sdpc method of minion and saye j comput phys 375 797 822 2018 combined with the discontinuous galerkin spectral element method for spatial discretization it can in theory reach arbitrary order of accuracy in time and space numerical experiments in three space dimensions demonstrate up to order 12 in time and 17 in space for constant as well as varying solution dependent viscosity compared to sdpc the present method yields a substantial improvement of accuracy and robustness against order reduction caused by time dependent boundary conditions

stability and convergence of the numerical solution for the species equation of a model for pemfcs abstract many mathematical models for fuel cells are presented in the literature however few studies of numerical analysis of these models are found the novelty of this work is the proof of stability and convergence of the equation of chemical species providing greater reliability to the mathematical model in this work a mathematical model is proposed that calculates the flow and concentration of species for proton exchange membrane fuel cells using the finite element method for spatial discretization and the crank nicolson method for temporal discretization the model takes into account the losses overpotentials at the anode and at the cathode the numerical results confirm the proof of convergence which is demonstrated in this work finally a numerical result is shown for the fuel cell power relative to current density and the result has good relation with the experimental data

construction of solutions to the l 2 critical kdv equation with a given asymptotic behaviour we consider the critical korteweg de vries equation ut uxx u x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v h be with sufficient decay on the right that is 1 x0 v l for some 0 0 we construct a solution u t to the critical korteweg de vries equation such that lim t u t u t v n x j 1 rj t h1 0

numerical studies of finite element variational multiscale methods for turbulent flow simulations different realizations of variational multiscale vms methods within the framework of finite element methods are studied in turbulent channel flow simulations one class of vms methods uses bubble functions to model resolved small scales whereas the other class contains the definition of the resolved small scales by an explicit projection in its set of equations all methods are employed with eddy viscosity models of smagorinsky type the simulations are performed on grids for which a direct numerical simulation blows up in finite time

large time asymptotics of solutions to the generalized korteweg de vries equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized korteweg de vries gkdv equationut u 1 u x 13uxxx 0 wherex t rwhen the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time more precisely we show that the solutionu t satisfies the decay estimate u t l c 1 t 1 3 1 1 for 4 uux t l ct 2 3 1 t 1 3and using these estimates we prove the existence of the scattering stateu l2such that u t u t u l2 ct 3 3for any small initial data belonging to the weighted sobolev spaceh1 1 f l2 1 x 2 1 2 1 2x 1 2 f l2 whereu t is the airy free evolution group

a local projection stabilization continuous galerkin petrov method for incompressible flow problems abstract a local projection stabilization lps method in space is considered to approximate the evolutionary oseen equations optimal error bounds with constants independent of the viscosity parameter are obtained in the continuous in time case for both the velocity and pressure approximation in addition the fully discrete case in combination with higher order continuous galerkin petrov cgp methods is studied error estimates of order k 1 are proved where k denotes the polynomial degree in time assuming that the convective term is time independent numerical results show that the predicted order is also achieved in the general case of time dependent convective terms

numerical methods for nonlinear second order hyperbolic partial differential equations ii rothe s techniques for 1 d problems abstract two families of rothe s methods that are based on the discretization of the time variable and keeping the spatial one continuous for the solution of second order hyperbolic equations with damping and nonlinear source terms are presented the first family is based on time integration and results in a volterra integro differential equation which upon approximating certain time integrals can be written as an ordinary differential equation in space upon discretizing the spatial derivatives by means of finite difference formula this family results in nonlinear algebraic equations which contain exponential terms that depend on the time step and its accuracy is shown to be lower than that of an explicit second order accurate discretization in both space and time of second order hyperbolic equations the second family of methods is based on the discretization of the time derivatives and time linearization and results in linear ordinary differential equations in space upon freezing the coefficients of these equations one can integrate analytically the resulting linear ordinary differential equations to obtain piecewise exponential solutions which are continuous in space and three point finite difference equations which depend exponentially on the time step spatial grid size and the diffusion relaxation damping and reaction times the finite difference equations are shown to result in non diagonally dominant matrices unless the time step is smaller than the characteristic relaxation diffusion damping and reaction times to avoid this problem two rothe s techniques that do not account for the jacobian of the reaction terms in the ordinary differential operator are developed and it is shown that the rothe s techniques belonging to the second family are as accurate as the linearly implicit methods presented in part i provided that the relaxation time is smaller than the critical time and the solution does not oscillate in space when the relaxation time is greater than the critical one rothe s methods have been found to be less accurate than the finite difference techniques of part i the rothe s techniques belonging to the second family are generalized to systems of nonlinear second order hyperbolic equations and mixed systems of parabolic and second order hyperbolic partial differential equations

a stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation abstract a stabilized finite element formulation for incompressible viscous flows is derived the starting point are the modified navier stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus fic procedure application of the standard finite element galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields the fic method also provides a natural explanation for the stabilization terms appearing in all equations for both the navier stokes and the simpler stokes equations transient solution schemes with enhanced stabilization properties are also proposed finally a procedure for computing the stabilization parameters is presented

blow up for the critical generalized korteweg de vries equation i dynamics near the soliton we consider the quintic generalized korteweg de vries equation gkdv u t u xx u 5 x 0 which is a canonical mass critical problem for initial data in h1 close to the soliton in earlier works on this problem finite or infinite time blow up was proved for non positive energy solutions and the solitary wave was shown to be the universal blow up profile see 16 26 and 20 for well localized initial data finite time blow up with an upper bound on blow up rate was obtained in 18