accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

global existence of harmonic maps in minkowski space on considere la construction d une classe d applications harmoniques classiques globales de l espace de minkowski m 4 dans une variete de riemann complete a n dimensions

accurate three dimensional lid driven cavity flow a chebyshev collocation method in space is introduced which allows an accurate calculation of three dimensional lid driven cavity flows the time integration is carried out by an adams bashforth backward euler scheme the accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flow field this way the most severe discontinuities in the boundary conditions which arise along the lines where moving and stationary walls meet are taken care of analytically and thus do not spoil the numerical part of the solution calculations are carried out for no slip boundary conditions at the cavity end walls as well as for periodic end wall conditions in general the spatial accuracy is better than fifth order for rigid end wall conditions the accuracy is reduced near the end walls to o n 3 2 but recovers in the bulk tabulated data are provided for the most interesting flow properties

assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows in this work we study the performance of some variational multiscale models vms in the large eddy simulation les of turbulent flows we consider vms models obtained by different subgrid scale approximations which include either static or dynamic subscales linear or nonlinear multiscale splitting and different choices of the subscale space after a brief review of these models we discuss some implementation aspects particularly relevant to the simulation of turbulent flows namely the use of a skew symmetric form of the convective term and the computation of projections when orthogonal subscales are used we analyze the energy conservation and numerical dissipation of the alternative vms formulations which is numerically evaluated in the numerical study we have considered three well known problems the decay of homogeneous isotropic turbulence the taylor green vortex problem and the turbulent flow in a channel we compare the results obtained using different vms models paying special attention to the effect of using orthogonal subscale spaces the vms results are also compared against classical les scheme based on filtering and the dynamic smagorinsky closure altogether our results show the tremendous potential of vms for the numerical simulation of turbulence further we study the sensitivity of vms to the algorithmic constants and analyze the behavior in the small time step limit we have also carried out a computational cost comparison of the different formulations out of these experiments we can state that the numerical results obtained with the different vms formulations as far as they converge are quite similar however some choices are prone to instabilities and the results obtained in terms of computational cost are certainly different the dynamic orthogonal subscales model turns out to be best in terms of efficiency and robustness

analysis of semi implicit dgfem for nonlinear convection diffusion problems on nonconforming meshes the paper deals with the numerical analysis of a scalar nonstationary nonlinear convection diffusion equation the space discretization is carried out by the discontinuous galerkin finite element method dgfem on general nonconforming meshes formed by possibly nonconvex elements with nonsymmetric treatment of stabilization terms and interior and boundary penalty the time discretization is carried out by a semi implicit euler scheme in which the diffusion and stabilization terms are treated implicitly whereas the nonlinear convective terms are treated explicitly we derive a priori asymptotic error estimates in the discrete l l2 l l2 norm l2 h1 l2 h1 seminorm and l h1 l h1 seminorm with respect to the mesh size h and time step numerical examples demonstrate the accuracy of the method and manifest the effect of nonconvexity of elements and nonconformity of the mesh

on integrability of the yang baxter σ model we prove that the recently introduced yang baxter model can be considered as an integrable deformation of the principal chiral model we find also an explicit one to one map transforming every solution of the principal chiral model into a solution of the deformed model with the help of this map the standard procedure of the dressing of the principal chiral solutions can be directly transferred into the deformed yang baxter context

an asymptotic preserving all speed scheme for the euler and navier stokes equations we present an asymptotic preserving all speed scheme for the simulation of compressible flows valid at all mach numbers ranging from very small to order unity the scheme is based on a semi implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly this discretization which is the key to the asymptotic preserving property provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime the divergence free condition on the velocity in the incompressible regime is respected and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations the implicit treatment of the acoustic part allows the time step to be independent of the mach number the scheme is conservative and applies to steady or unsteady flows and to general equations of state one and two dimensional numerical results provide a validation of the asymptotic preserving all speed properties

an implicit high order hybridizable discontinuous galerkin method for the incompressible navier stokes equations we present an implicit high order hybridizable discontinuous galerkin method for the steady state and time dependent incompressible navier stokes equations the method is devised by using the discontinuous galerkin discretization for a velocity gradient pressure velocity formulation of the incompressible navier stokes equations with a special choice of the numerical traces the method possesses several unique features which distinguish itself from other discontinuous galerkin methods first it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries thereby leading to a significant reduction in the degrees of freedom moreover if the augmented lagrangian method is used to solve the linearized system the globally coupled unknowns become the approximate trace of the velocity only second it provides for smooth viscous dominated problems approximations of the velocity pressure and velocity gradient which converge with the optimal order of k 1 in the l2 norm when polynomials of degree k 0 are used for all components of the approximate solution and third it displays superconvergence properties that allow us to use the above mentioned optimal convergence properties to define an element by element postprocessing scheme to compute a new and better approximate velocity indeed this new approximation is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 in the l2 norm moreover a novel and systematic way is proposed for imposing boundary conditions for the stress viscous stress vorticity and pressure which are not naturally associated with the weak formulation of the method this can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of reynolds numbers and for various polynomial degrees

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

a staggered semi implicit discontinuous galerkin method for the two dimensional incompressible navier stokes equations in this paper we propose a new spatially high order accurate semi implicit discontinuous galerkin dg method for the solution of the two dimensional incompressible navier stokes equations on staggered unstructured curved meshes while the discrete pressure is defined on the primal grid the discrete velocity vector field is defined on an edge based dual grid the flexibility of high order dg methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non zero blocks per element for only one scalar unknown namely the pressure the method is computationally efficient since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions furthermore all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage which leads to savings in terms of computational effort for the time evolution part in this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step the method is validated for polynomial degrees up to p 3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows ii efficient flux quadrature a new and efficient quadrature rule for the flux integrals arising in the space time discontinuous galerkin discretization of the euler equations in a moving and deforming space time domain is presented and analyzed the quadrature rule is a factor three more efficient than the commonly applied quadrature rule and does not affect the local truncation error and stability of the numerical scheme the local truncation error of the resulting numerical discretization is determined and is shown to be the same as when product gauss quadrature rules are used details of the approximation of the dissipation in the numerical flux are presented which render the scheme consistent and stable the method is successfully applied to the simulation of a three dimensional transonic flow over a deforming wing

semi implicit extension of a godunov type scheme based on low mach number asymptotics i one dimensional flow a single scale multiple space scale asymptotic analysis provides detailed insight into the low mach number limit behavior of solutions of the compressible euler equations we use the asymptotics as a guideline for developing a low mach number extension of an explicit higher order shock capturing scheme this semi implicit scheme involves multiple pressure variables large scale differencing and averaging procedures that are discretized versions of standard operations in multiple scales asymptotic analysis advection and acoustic wave propagation are discretized explicitly and upwind and only one scalar elliptic equation is to be solved implicitly at each time step this equation is a pressure correction equation for incompressible flows when the mach number is zero in the low mach number limit the time step is restricted by a courant number based essentially on the maximum flow velocity for moderate and large mach numbers the scheme reduces to the underlying explicit higher order shock capturing algorithm

on non integrability of classical strings in p brane backgrounds we investigate the question of possible integrability of classical string motion in curved p brane backgrounds for example the d3 brane metric interpolates between the flat and the ads5 s5 regions in which string propagation is integrable we find that while the point like string geodesic equations are integrable the equations describing an extended string in the complete d3 brane geometry are not the same conclusion is reached for similar brane intersection backgrounds interpolating between flat space and adsk sk we consider in particular the case of the ns 5 brane fundamental string background to demonstrate non integrability we make a special pulsating string ansatz for which the string equations reduce to an effective one dimensional system expanding near this simple solution leads to a linear differential equation for small fluctuations that cannot be solved in quadratures implying non integrability of the original set of string equations

picard vessiot theory and ziglin s theorem abstract given a hamiltonian system ziglin s theorem is one of the criteria used to study the integrability of the system for this we make use of the variational equations along a known solution in this paper we obtain some consequences of the application of picard vessiot differential galois theory to ziglin s theorem first we present the general results for the case of a finite arbitrary number of degrees of freedom then we pass to the case of two degrees of freedom where the results can be given with more detail under the hypothesis of ziglin s theorem and some additional technical assumptions the main results theorems 1 and 2 relate the integrability of the hamiltonian with the properties of the differential galois group of the picard vessiot extension associated to the normal reduced variational equations for two degrees of freedom it is possible to study also the resonant case theorems 4 and 5 give the full classification of the galois groups and the related extensions in the integrable case a couple of applications are made to recover ito s theorem and to study the lame equation

construction of lax connections by exponentiation we propose a method for constructing the lax connection of two dimensional relativistic integrable sigma models on coset spaces by means of exponentiation of a suitable operator we derive a simple quadratic relation that this operator must satisfy for an entire one parameter family of connections to be flat

a simple based discontinuous galerkin solver for steady incompressible flows in this paper we present how the well known simple algorithm can be extended to solve the steady incompressible navier stokes equations discretized by the discontinuous galerkin method the convective part is discretized by the local lax friedrichs fluxes and the viscous part by the symmetric interior penalty method within the simple algorithm the equations are solved in an iterative process the discretized equations are linearized and an equation for the pressure is derived on the discrete level the equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially leading to an efficient solution procedure the extension of the proposed scheme to the unsteady case is straightforward where fully implicit time schemes can be used various test cases are carried out the poiseuille flow the channel flow with constant transpiration the kovasznay flow the flow into a corner and the backward facing step flow using a mixed order formulation i e order k for the velocity and order k 1 for the pressure the scheme is numerically stable for all test cases convergence rates of k 1 and k in the l 2 norm are observed for velocity and pressure respectively a study of the convergence behavior of the simple algorithm shows that no under relaxation for the pressure is needed which is in strong contrast to the application of the simple algorithm in the context of the finite volume method or the continuous finite element method we conclude that the proposed scheme is efficient to solve the steady incompressible navier stokes equations in the context of the discontinuous galerkin method comprising hp accuracy

a projection hybrid finite volume element method for low mach number flows the purpose of this article is to introduce a projection hybrid finite volume element method for low mach number flows of viscous or inviscid fluids starting with a 3d tetrahedral finite element mesh of the computational domain the equation of the transport diffusion stage is discretized by a finite volume method associated with a dual mesh where the nodes of the volumes are the barycenters of the faces of the initial tetrahedra the transport diffusion stage is explicit upwinding of convective terms is done by classical riemann solvers as the q scheme of van leer or the rusanov scheme concerning the projection stage the pressure correction is computed by a piecewise linear finite element method associated with the initial tetrahedral mesh passing the information from one stage to the other is carefully made in order to get a stable global scheme numerical results for several test examples aiming at evaluating the convergence properties of the method are shown

foundations of the ads5 s5 superstring i we review the recent advances towards finding the spectrum of the ads5 s5 superstring we thoroughly explain the theoretical techniques which should be useful for the ultimate solution of the spectral problem in certain cases our exposition is original and cannot be found in the existing literature the present part i deals with foundations of classical string theory in ads5 s5 light cone perturbative quantization and the derivation of the exact light cone world sheet scattering matrix

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

stability accuracy and efficiency of a semi implicit method for three dimensional shallow water flow abstract the stability analysis the accuracy and the efficiency of a semi implicit finite difference scheme for the numerical solution of a three dimensional shallow water model are presented and discussed the governing equations are the three dimensional reynolds equations in which pressure is assumed to be hydrostatic the pressure gradient in the momentum equations and the velocities in the vertically integrated continuity equation are discretized with the method with being an implicitness parameter it is shown that the method is stable for 1 2 1 unstable for 1 2 and highest accuracy and efficiency is achieved when 1 2 the resulting algorithm is mass conservative and naturally allows for the simulation of flooding and drying of tidal flats

spatially periodic steady solutions to the three dimensional navier stokes equation with the abc force abstract for an appropriately scaled abc forcing the abc flow is a steady space periodic solution to the three dimensional navier stokes equation for an arbitrary reynolds number r we are investigating both numerically and analytically different branches of steady solutions in the case a b c when the equation has a group of symmetries isomorphic to the rotation group of the cube two families of steady flows each comprised of three mutually symmetric branches were detected numerically one was shown to persist for 7 9 r 2000 and another for 149 r 1000 branches of the first family intersect twice with the abc flow in a bifurcation generic to a system with symmetry group d3 other possible bifurcations of the abc flow and the possibility of existence of branches of steady flows with other symmetries are considered

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

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

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

general solutions of nonlinear equations in the geometric theory of the relativistic string general solutions for the system of nonlinear equations in the second order partial derivatives with two independent variables are obtained they determine the basic differential forms of the two dimensional minimal surface embedded inton dimensional pseudo euclidean space

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

a nested newton type algorithm for finite volume methods solving richards equation in mixed form a finite volume discretization of the mixed form of richards equation leads to a nonlinear numerical model which yields exact local and global mass conservation the resulting nonlinear system requires sophisticated numerical strategies especially in a variable saturated flow regime in this paper a nested newton type algorithm for the discretized richards equation is proposed and analyzed with a judicious choice of the initial guess the quadratic convergence rate is obtained for any time step size and for all flow regimes

modelling instability of abc flow using a mode interaction between steady and hopf bifurcations with rotational symmetries of the cube abstract we consider hydrodynamic stability of symmetric abc flow with a b c 1 regarded as a steady state of the three dimensional navier stokes equation with appropriate forcing numerical investigations have shown that its first instability on increasing the reynolds number is a hopf bifurcation at r 13 044 and that at this bifurcation the second dominant eigenvalue is real motivated by this we study generic interaction of steady state and hopf bifurcations in systems with rotational symmetry of the cube o with an eight dimensional normal form the generic branching and bifurcation behavior of the third order truncated normal form is investigated this is used to analyse a range of the bifurcations for particular values of the coefficients obtained by center manifold reduction from the hydrodynamic system the normal form system shows a sequence of bifurcations and attractors that closely follows the sequence observed for the original hydrodynamic system up to about r 13 91 this includes a torus breakdown to a chaotic attractor and a crisis of attractors that leads to a change in symmetry we discuss numerical simulations of the hydrodynamic system for larger values of r finally we present evidence that the system has robust heteroclinic cycles between fully symmetric abc flow and six steady states with broken symmetry for a range of parameter values near r 15

instantons in two and four dimensions it is shown that yang mills instantons in four dimensions can naturally be identified with the instantons of a two dimensional theory with values in the loop group

a space time discontinuous galerkin method for the incompressible navier stokes equations we introduce a space time discontinuous galerkin dg finite element method for the incompressible navier stokes equations our formulation can be made arbitrarily high order accurate in both space and time and can be directly applied to deforming domains different stabilizing approaches are discussed which ensure stability of the method a numerical study is performed to compare the effect of the stabilizing approaches to show the method s robustness on deforming domains and to investigate the behavior of the convergence rates of the solution recently we introduced a space time hybridizable dg hdg method for incompressible flows s rhebergen b cockburn a space time hybridizable discontinuous galerkin method for incompressible flows on deforming domains j comput phys 231 2012 4185 4204 we will compare numerical results of the space time dg and space time hdg methods this constitutes the first comparison between dg and hdg methods

a staggered semi implicit spectral discontinuous galerkin scheme for the shallow water equations a spatially arbitrary high order semi implicit spectral discontinuous galerkin dg scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed the free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis and as test functions each unknown is however defined on a different set of control volumes that are spatially staggered with respect to each other a semi implicit time integration yields a stable and efficient mass conservative algorithm the use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block penta diagonal system must be solved for the new free surface location which is a scalar quantity subsequently the new momentum components are directly obtained the staggered semi implicit approach makes the present dg scheme different from other published dg schemes the sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix free gmres algorithm furthermore for the shallow water equations the proposed scheme can be written as a quadrature free method where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors for the special case n 0 the proposed method reduces to a classical semi implicit finite difference scheme the proposed semi implicit scheme is particularly well suited for low froude number flows the method is validated on some typical academic benchmark problems using polynomial degrees of up to n 20

a space time hybridizable discontinuous galerkin method for incompressible flows on deforming domains we present the first space time hybridizable discontinuous galerkin hdg finite element method for the incompressible navier stokes and oseen equations major advantages of a space time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space time elements our formulation is related to the hdg formulation for incompressible flows introduced recently in e g n c nguyen j peraire b cockburn a hybridizable discontinuous galerkin method for stokes flow comput methods appl mech eng 199 2010 582 597 however ours is inspired in typical dg formulations for compressible flows which allow for a more straightforward implementation another difference is the use of polynomials of fixed total degree with space time hexahedral and quadrilateral elements instead of simplicial elements we present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time

iterative solutions of mildly nonlinear systems the correct numerical modelling of free surface hydrodynamics often requires the solution of diagonally nonlinear systems in doing this one may substantially enhance the model accuracy while fulfilling relevant physical constraints this is the case when a suitable semi implicit discretization is used e g to solve the one dimensional or the multi dimensional shallow water equations to model axially symmetric flows in compliant arterial systems to solve the boussinesq equation in confined unconfined aquifers or to solve the mixed form of the richards equation in this paper two nested iterative methods for solving a mildly nonlinear system of the form v h email protected b are proposed and analysed it is shown that the inner and the outer iterates are monotone and converge to the exact solution for a wide class of mildly nonlinear systems of applied interest a simple and yet non trivial test problem derived from the mathematical modelling of flows in porous media is formulated and solved with the proposed methods

solutions to yang mills equations that are not self dual abstract the yang mills functional for connections on principle su 2 bundles over s4 is studied critical points of the functional satisfy a system of second order partial differential equations the yang mills equations if in particular the critical point is a minimum it satisfies a first order system the self dual or anti self dual equations here we exhibit an infinite number of finite action nonminimal unstable critical points they are obtained by constructing a topologically nontrivial loop of connections to which min max theory is applied the construction exploits the fundamental relationship between certain invariant instantons on s4 and magnetic monopoles on h3 this result settles a question in gauge field theory that has been open for many years

on harmonic maps from r1 1 into hilbert loop groups motivated by the works of gu pure appl math 33 727 1980 ginibre and velo ann phys 142 393 1982 on classical solutions of a two dimensional model the global existence of harmonic maps from the minkowski plane r1 1 into hilbert loop groups is proved the construction of the harmonic maps strongly depends on some geometric properties of the loop groups

an artificial compressibility flux for the discontinuous galerkin solution of the incompressible navier stokes equations discontinuous galerkin dg methods have proved to be well suited for the construction of robust high order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems their application to the incompressible navier stokes ins equations has also been recently considered although the subject is far from being fully explored in this work we propose a new approach for the dg numerical solution of the ins equations written in conservation form the inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the exact solution of the riemann problem associated with a local artificial compressibility perturbation of the equations unlike in most of the existing methods artificial compressibility is here introduced only at the interface flux level therefore resulting in a consistent discretization of the ins equations irrespectively of the amount of artificial compressibility introduced the discretization of the viscous term follows the well established dg scheme named br2 the performance and the accuracy of the method are demonstrated by computing the kovasznay flow and the two dimensional lid driven cavity flow for a wide range of reynolds numbers and for various degrees of polynomial approximation

iterative solution of piecewise linear systems and applications to flows in porous media the correct numerical modeling of free surface hydrodynamic problems often requires to have the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself in doing so one may fulfill relevant physical constraints the existence the uniqueness and two constructive iterative methods to solve a piecewise linear system of the form max boldsymbol l min boldsymbol u mathbf x t mathbf x mathbf b are analyzed the methods are shown to have a finite termination property i e they converge to an exact solution in a finite number of steps and actually they converge very quickly as confirmed by a few numerical tests which are derived from the mathematical modeling of flows in porous media

space time discontinuous galerkin method for the compressible navier stokes equations a space time discontinuous galerkin finite element method for the compressible navier stokes equations is presented we explain the space time setting derive the weak formulation and discuss our choices for the numerical fluxes the resulting numerical method allows local grid adaptation as well as moving and deforming boundaries which we illustrate by computing the flow around a 3d delta wing on an adapted mesh and by simulating the dynamic stall phenomenon of a 2d airfoil in rapid pitch up maneuver

a high order semi implicit discontinuous galerkin method for the two dimensional shallow water equations on staggered unstructured meshes a well balanced spatially arbitrary high order accurate semi implicit discontinuous galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non orthogonal grids the semi implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime for p 0 the resulting scheme becomes a generalization of the classical semi implicit finite volume finite difference scheme of casulli and walters 2000 25 but with less conditions on the grid geometry the method proposed in this paper allows large time steps with respect to the surface wave speed gh and is thus particularly suitable for low froude number flows the approach is validated on some typical academic benchmark problems using polynomial degrees up to p 6

a space time discontinuous galerkin method for boussinesq type equations in this work we propose a new high order accurate fully implicit space time discontinuous galerkin dg method for advection diffusion dispersion equations i e for so called korteweg de vries type equations in particular we focus on boussinesq type models for free surface flows which are used for the modeling of water waves that travel in deep water where the classical shallow water equations are not valid any more our method follows the ideas of the local dg method ldg for dispersion equations proposed by yan and shu 2002 who used an explicit runge kutta method to integrate their scheme in time however such explicit time integrators applied to dispersive equations imply a very severe restriction on the time step which has to be taken proportional to the cube of the mesh spacing and which therefore can make even one dimensional computations prohibitively expensive on fine grids for the scalar case and with some simplifying assumptions the scheme presented in this paper can be proven to be unconditionally stable in l2 norm furthermore our method is based directly on a space time finite element formulation which also provides a natural way to discretize third order dispersive terms that contain a mixed space time derivative such terms appear frequently in the context of boussinesq type models for free surface flows we will show numerical convergence studies for linear scalar advection diffusion dispersion equations and furthermore we will also study the convergence of our method using solitary wave solutions of the nonlinear boussinesq type model of madsen murray and sorensen 1991

relativistic string model in a space time of a constant curvature the relativistic string model is investigated in a space time of a constant curvature de sitter universe the fundamental differential quadratic forms of the world surface of the string are considered as the dynamical variables the coefficients of these forms obey two nonlinear equations varphi 11 varphi 22 e varphi cos theta ke varphi theta 11 theta 22 e varphi sin theta the lax representation for this system is obtained

a semi implicit semi lagrangian p adaptive discontinuous galerkin method for the shallow water equations a semi implicit and semi lagrangian discontinuous galerkin method for the shallow water equations is proposed for applications to geophysical scale flows a non conservative formulation of the advection equation is employed in order to achieve a more treatable form of the linear system to be solved at each time step the method is equipped with a simple p adaptivity criterion that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution numerical results show that the method captures well the main features of gravity and inertial gravity waves as well as reproducing correct solutions in nonlinear test cases with analytic solutions the accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high courant numbers and with automatic choice of the local approximation degree

non stationary helical flows for incompressible 3d navier stokes equations in fluid mechanics a lot of authors have been executing their researches to obtain the analytical solutions of navier stokes equations even for 3d case of compressible gas flow but there is an essential deficiency of non stationary solutions indeed in our derivation we explore the case of non stationary helical flow of the navier stokes equations for incompressible fluids at any given initial conditions for velocity fields it means an open choice for the space part of a solution such a non stationary helical flow is proved to be decreasing exponentially in regard to the time parameter the extent of time dependent exponential component is given by the coefficient of kinematic viscosity multiplied by the square of the coefficient of proportionality between the vorticity and velocity field

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

an optimal order interior penalty discontinuous galerkin discretization of the compressible navier stokes equations in this article we propose a new symmetric version of the interior penalty discontinuous galerkin finite element method for the numerical approximation of the compressible navier stokes equations here particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest such as the lift and drag coefficients of a body immersed in a viscous fluid with this in mind the key ingredients in the construction of the method include i an adjoint consistent imposition of the boundary conditions ii an adjoint consistent reformulation of the underlying target functional of practical interest iii design of appropriate interior penalty stabilization terms numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the l 2 norm as well as for certain target functionals computational comparisons with other discontinuous galerkin schemes proposed in the literature including the second scheme of bassi and rebay cf f bassi s rebay gmres discontinuous galerkin solution of the compressible navier stokes equations in b cockburn g karniadakis c w shu eds discontinuous galerkin methods lecture notes in comput sci engrg vol 11 springer berlin 2000 pp 197 208 f bassi s rebay numerical evaluation of two discontinuous galerkin methods for the compressible navier stokes equations int j numer methods fluids 40 2002 197 207 the standard sipg method outlined in r hartmann p houston symmetric interior penalty dg methods for the compressible navier stokes equations i method formulation int j numer anal model 3 1 2006 1 20 and an nipg variant of the new scheme will be undertaken

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

a reconstructed discontinuous galerkin method for the compressible navier stokes equations on arbitrary grids a reconstruction based discontinuous galerkin rdg method is presented for the solution of the compressible navier stokes equations on arbitrary grids the rdg method originally developed for the compressible euler equations is extended to discretize viscous and heat fluxes in the navier stokes equations using a so called inter cell reconstruction where a smooth solution is locally reconstructed using a least squares method from the underlying discontinuous dg solution similar to the recovery based dg rdg methods this reconstructed dg method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional dg methods unlike rdg methods this rdg method does not need to judiciously choose a proper form of a recovered polynomial thus is simple flexible and robust and can be used on arbitrary grids the developed rdg method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy efficiency robustness and versatility the numerical results indicate that this rdg method is able to deliver the same accuracy as the well known bassi rebay ii scheme at a half of its computing costs for the discretization of the viscous fluxes in the navier stokes equations clearly demonstrating its superior performance over the existing dg methods for solving the compressible navier stokes equations

asymptotic single and multiple scale expansions in the low mach number limit an asymptotic analysis of the euler equations in the limit of vanishing mach number is presented which can be employed to extend the validity of numerical methods from the compressible to the low mach number regime r klein j comput phys 121 1995 pp 213 237 l hoffmann and a meister in proceedings of the fourth eccomas computational fluid dynamics conference vol 1 c hirsch et al eds athens 1998 pp 567 572 the work is based on the analysis recently proposed by klein whereby the paper is devoted to a rigorous mathematical justification of this asymptotic investigation in order to give reliable statements concerning the assumptions under which the results hold in contrast to klein a single scale as well as multiple scale expansion is used depending on the spatial domain under consideration

on harmonic maps into gauge groups we continue the work of the paper j math phys 37 4076 1996 on the existence of globally defined harmonic maps from the minkowski plane r1 1 into an infinite dimensional hilbert lie group we prove that the cauchy problem for harmonic maps from r1 1 into hilbert loop groups hs lg can be solved globally for all remaining cases 12 s 34 and we obtain similar results for harmonic maps from r1 1 into certain hilbert lie gauge groups hs sn g n 2 these answer the two questions remaining from the above paper

a high order discontinuous galerkin method for the unsteady incompressible navier stokes equations we present a high order discontinuous galerkin discretization of the unsteady incompressible navier stokes equations in convection dominated flows using triangular and tetrahedral meshes the scheme is based on a semi explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the stokes operator the nonlinear term is discretized in divergence form by using the local lax friedrichs fluxes thus local conservativity is inherent spatial discretization of the stokes operator has employed both equal order p k p k and mixed order p k p k 1 velocity and pressure approximations a second order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic helmholtz equation for each component of the velocity and a consistent poisson equation for the pressure the consistent poisson operator is replaced by an equivalent in stability and convergence operator namely that arising from the interior penalty discretization of the standard poisson operator with appropriate boundary conditions this yields a simpler and more efficient method characterized by a compact stencil size we show the temporal and spatial behavior of the method by solving some popular benchmarking tests for an unsteady stokes problem second order temporal convergence is obtained while for the taylor vortex test problem on both semi structured and fully unstructured triangular meshes spectral convergence with respect to the polynomial degree k is obtained by studying the orr sommerfeld stability problem we demonstrate that the p k p k method yields a stable solution while the p k p k 1 formulation leads to unphysical instability the good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder we conclude that the proposed discontinuous galerkin method with the p k p k formulation is an efficient scheme for accurate and stable solution of the unsteady navier stokes equations in convection dominated flows

the geometry of loop groups l espace g des boucles basees sur un groupe de lie compact admet une metrique de kahler on exprime sa courbure en termes d operateurs de toeplitz et on definit des classes de chern par analogie avec la theorie de chern weil aux dimensions finies il y a une structure de fredhohn naturelle sur g induite a partir de la famille des operateurs de toeplitz on obtient des formules explicites pour su n

integrability and the ads cft correspondence the description of gauge theories at strong coupling is one of the long standing problems in theoretical physics the idea of a relation between strongly coupled gauge theories and string theory was pioneered by t hooft wilson and polyakov a decade ago maldacena made this relation explicit by conjecturing the exact equivalence of a conformally invariant theory in four dimensions the maximally supersymmetric yang mills theory with string theory in the ads5 s5 background other examples of correspondence between a conformally invariant theory and string theory in an ads background were discovered recently the comparison of the two sides of the correspondence requires the use of non perturbative methods the discovery of integrable structures in gauge theory and string theory led to the conjecture that the two theories are integrable for any value of the coupling constant and that they share the same integrable structure defined non perturbatively the last 8 years brought remarkable progress in identifying this solvable model and in explicitly solving the problem of computing the spectrum of conformal dimensions of the theory the progress came from the identification of the dilatation operator with an integrable spin chain and from the study of the string sigma model in this review i present the evolution of the concept of integrability in the framework of the ads cft correspondence and the main results obtained using this approach

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

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