local discontinuous galerkin methods for the stokes system in this paper we introduce and analyze local discontinuous galerkin methods for the stokes system for a class of shape regular meshes with hanging nodes we derive a priori estimates for the l2 norm of the errors in the velocities and the pressure we show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure for any k ge1 we also consider the case in which all the unknowns are approximated with polynomials of degree k and show that although the orders of convergence remain the same the method is more efficient numerical experiments verifying these facts are displayed

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

stability of a saddle node bifurcation under numerical approximations in this paper we show that the solution flows generated by a one parameter family of ordinary differential equations are stable under their numerical approximations in a vicinity of a saddle node our result sharpens the one in 1 and the proof is adapted from the method of sotomayor in 2 3

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

qualitative property between flows and numerical methods abstract in this paper we show that qualitative properties of axiom a flows with the strong transversality condition are preserved by its discretization mapping obtained via numerical methods more precisely if t is an axiom a flow with the strong transversality condition and n is a numerical method of order p 1 for t on a smooth compact manifold m then there exists t 1 such that for all sufficiently large n n there are a continuous onto function h n m m and a continuous function n m r such that n x h n x h n n t n n x for all x m

a hybrid reconstructed discontinuous galerkin and continuous galerkin finite element method for incompressible flows on unstructured grids a hybrid reconstructed discontinuous galerkin and continuous galerkin method based on an incremental pressure projection formulation termed rdg p n p m cg p n in this paper is developed for solving the unsteady incompressible navier stokes equations on unstructured grids in this method a reconstructed discontinuous galerkin method rdg p n p m is used to discretize the velocity and a standard continuous galerkin method cg p n is used to approximate the pressure the rdg p n p m cg p n method is designed to increase the accuracy of the hybrid dg p n cg p n method and yet still satisfy lady enskaja babuska brezzi lbb condition thus avoiding the pressure checkerboard instability an upwind method is used to discretize the nonlinear convective fluxes in the momentum equations in order to suppress spurious oscillations in the velocity field a number of incompressible flow problems for a variety of flow conditions are computed to numerically assess the spatial order of convergence of the rdg p n p m cg p n method the numerical experiments indicate that both rdg p 0 p 1 cg p 1 and rdg p 1 p 2 cg p 1 methods can attain the designed 2nd order and 3rd order accuracy in space for the velocity respectively moreover the 3rd order rdg p 1 p 2 cg p 1 method significantly outperforms its 2nd order rdg p 0 p 1 cg p 1 and rdg p 1 p 1 cg p 1 counterparts being able to not only increase the accuracy of the velocity by one order but also improve the accuracy of the pressure

a high order compact least squares reconstructed discontinuous galerkin method for the steady state compressible flows on hybrid grids abstract in this paper a class of new high order reconstructed dg rdg methods based on the compact least squares cls reconstruction 23 24 is developed for simulating the two dimensional steady state compressible flows on hybrid grids the proposed method combines the advantages of the dg discretization with the flexibility of the compact least squares reconstruction which exhibits its superior potential in enhancing the level of accuracy and reducing the computational cost compared to the underlying dg methods with respect to the same number of degrees of freedom to be specific a third order compact least squares rdg p 1 p 2 method and a fourth order compact least squares rdg p 2 p 3 method are developed and investigated in this work in this compact least squares rdg method the low order degrees of freedom are evolved through the underlying dg p 1 method and dg p 2 method respectively while the high order degrees of freedom are reconstructed through the compact least squares reconstruction in which the constitutive relations are built by requiring the reconstructed polynomial and its spatial derivatives on the target cell to conserve the cell averages and the corresponding spatial derivatives on the face neighboring cells the large sparse linear system resulted by the compact least squares reconstruction can be solved relatively efficient when it is coupled with the temporal discretization in the steady state simulations a number of test cases are presented to assess the performance of the high order compact least squares rdg methods which demonstrates their potential to be an alternative approach for the high order numerical simulations of steady state compressible flows

optimal error estimates of the direct discontinuous galerkin method for convection diffusion equations abstract in this paper we present the optimal l2 error estimate ofo hk 1 for polynomial elements of degree k of the semidiscrete direct discontinuous galerkin method for convection diffusion equations the main technical difficulty lies in the control of the inter element jump terms which arise because of the convection and the discontinuous nature of numerical solutions the main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so that trouble terms at the cell interfaces are eliminated or controlled in multi dimensional case the orders of k 1 hinge on a superconvergence estimate when tensor product polynomials of degree k are used on cartesian grids a collection of projection errors in both oneand multi dimensional cases is established

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

lower upper symmetric gauss seidel method for the euler and navier stokes equations on developpe un schema de relaxation multigrille application a des ecoulements transsoniques

structural stability of flows under numerics abstract in our previous paper siam j math anal 28 1997 381 388 we showed that the qualitative properties of a morse smale gradient like flow are preserved by its discretization mapping obtained via numerical methods in this paper we extend the result to flows which satisfy axiom a and the strong transversality condition we prove that if p 2 t is a c p 1 flow on a compact manifold satisfying axiom a and the strong transversality condition and n h is a numerical method of step size h and order p then for all sufficiently small h there are a homeomorphism h h and a continuous real valued function h on m such that h h h h h x x n h h h x and h h is o h p close to the identity map on m

a direct discontinuous galerkin method for the compressible navierstokes equations on arbitrary grids a direct discontinuous galerkin ddg method is developed for solving the compressible navierstokes equations on arbitrary grids in the framework of dg methods the ddg method originally introduced for scalar diffusion problems on structured grids is extended to discretize viscous and heat fluxes in the navierstokes equations two approaches of implementing the ddg method to compute numerical diffusive fluxes for the navierstokes equations are presented one is based on the conservative variables and the other is based on the primitive variables the importance of the characteristic cell size used in the ddg formulation on unstructured grids is examined the numerical fluxes on the boundary by the ddg method are discussed a number of test cases are presented to assess the performance of the ddg method for solving the compressible navierstokes equations based on our numerical results we observe that ddg method can achieve the designed order of accuracy and is able to deliver the same accuracy as the widely used br2 method at a significantly reduced cost clearly demonstrating that the ddg method provides an attractive alternative for solving the compressible navierstokes equations on arbitrary grids owning to its simplicity in implementation and its efficiency in computational cost

semi implicit discontinuous galerkin methods for the incompressible navier stokes equations on adaptive staggered cartesian grids in this paper a new high order semi implicit discontinuous galerkin method si dg is presented for the solution of the incompressible navier stokes equations on staggered space time adaptive cartesian grids amr in two and three space dimensions the pressure is written in the form of piecewise polynomials on the main grid which is dynamically adapted within a cell by cell amr framework according to the time dependent main grid different face based spatially staggered dual grids are defined for the piece wise polynomials of the respective velocity components although the resulting adaptive staggered grids are more complex than classical uniform cartesian meshes the numerical scheme can still be written in a rather compact form thanks to the use of a tensor product formulation for the definition of the nodal basis in the d dimensional space d 2 3 all the discrete operators can be efficiently written as a combination of linear one dimensional operators acting in the d space directions separately arbitrary high order of accuracy is achieved in space while a very simple semi implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation the real advantages of the staggered grid arise in the solution of the schur complement associated with the saddle point problem of the discretized incompressible navier stokes equations i e after substituting the discrete momentum equations into the discrete continuity equation this leads to a linear system for only one unknown the scalar pressure indeed the resulting linear pressure system is shown to be symmetric and positive definite in order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor an implicit discretization is also used for the diffusive terms in the momentum equation a particular feature of our staggered dg approach is that the viscous stress tensor is discretized on the dual mesh this corresponds to the use of a lifting operator but on the staggered grid both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix free conjugate gradient method for which fast convergence is observed moreover it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner due to the explicit discretization of the nonlinear convective terms the final algorithm is stable within the classical cfl type time step restriction for explicit dg methods based on the fluid velocity moreover by using a suitable local time stepping technique based on the ader dg framework each element can use its optimal local time step also within the amr context the new space time adaptive staggered dg scheme has been thoroughly verified for polynomial degrees up to n 9 for a large set of non trivial test problems in two and three space dimensions for which analytical numerical or experimental reference solutions exist the high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution to the knowledge of the authors this is the first staggered semi implicit dg scheme for the incompressible navier stokes equations on space time adaptive meshes in two and three space dimensions

a reconstructed discontinuous galerkin method based on a hierarchical weno reconstruction for compressible flows on tetrahedral grids a reconstructed discontinuous galerkin rdg method based on a hierarchical weno reconstruction termed hweno p 1p 2 in this paper designed not only to enhance the accuracy of discontinuous galerkin methods but also to ensure the nonlinear stability of the rdg method is presented for solving the compressible euler equations on tetrahedral grids in this hweno p 1p 2 method a quadratic polynomial solution p 2 is first reconstructed using a hermite weno reconstruction from the underlying linear polynomial p 1 discontinuous galerkin solution to ensure the linear stability of the rdg method and to improve the efficiency of the underlying dg method by taking advantage of handily available and yet invaluable information namely the derivatives in the dg formulation the stencils used in the reconstruction involve only von neumann neighborhood adjacent face neighboring cells and thus are compact the first derivatives of the quadratic polynomial solution are then reconstructed using a weno reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities thus ensuring the nonlinear stability of the rdg method the developed hweno p 1p 2 method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy robustness and non oscillatory property the numerical experiments indicate that the hweno p 1p 2 method is able to capture shock waves within one cell without any spurious oscillations and achieve the designed third order of accuracy one order accuracy higher than the underlying dg method

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

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

a reconstructed discontinuous galerkin method for incompressible flows on arbitrary grids abstract the discontinuous galerkin dg methods have attained increasing popularity for solving the incompressible navier stokes ins equations in recent years however the dg methods have their own weakness due to the high computational costs and storage requirements in this work we develop a high order hybrid reconstructed dg rdg method for solving the ins equations on arbitrary grids to be specific the inviscid term of the ins equations is discretized by applying the third order hybrid rdg p 1 p 2 method with a simplified artificial compressibility flux while the viscous term of the ins equations is discretized by using the simple direct dg ddg method a number of incompressible flow problems in both steady and unsteady forms for a variety flow conditions are computed to numerically assess the performance of the hybrid rdg p 1 p 2 method which confirm its ability to achieve the optimal third order of accuracy at a significantly reduced computational costs furthermore a detailed comparison of a variety of different reconstructed strategies is performed and presented numerical results demonstrate that the hybrid rdg p 1 p 2 method outperforms the rdg p 1 p 2 method based on either the original least squares reconstruction or the green gauss reconstruction for solving the ins equations

estimates in discretizing normally hyperbolic compact invariant manifolds of ordinary differential equations abstract discretization methods in the vicinity of normally hyperbolic compact invariant manifolds of ordinary differential equations are consisered the existence of a discretized normally hyperboic compact invariant manifold is proved and in various metric topologies the distance between the original manifold and its discretized counterpart is estimated proofs are based on reconsidering the hirsch 1 approach and supplying the necessary estimates

conjugacy in the discretized fold bifurcation abstract in this paper we construct a conjugacy between the time 1 map of the solution flow generated by an ordinary differential equation and its numerical approximation in a neighborhood of a fold bifurcation point our main result is that the conjugacy is o hp close to the identity on the center manifold where h is the step size and p is the order of the numerical method

on structural stability of ordinary differential equations with respect to discretization methods on compact m dimensional discs morse smale differential systems having no periodic orbits are considered the main result is that they are correctly reproduced by one step discretization methods for methods of order p and stepsize h sufficiently small the time h map of the induced local flow and the h discretized system are joined by a conjugacy o h p near to the identity the paper fits well in the rapidly growing list of results stating that hyperbolic transversal structures are preserved by discretization the proof relies heavily on techniques elaborated by robbin 1971 in establishing his structural stability theorem on self diffeomorphisms of compact c infty manifolds

equilibria stability and hamiltonian hopf bifurcation of a gyrostat in an incompressible ideal fluid abstract for a gyrostat in a incompressible ideal fluid by writing kirchhoff s equations as a lie poisson system and using a non canonical hamiltonian formulation we provide the expressions of the equilibria when the gyrostatic momentum is constant with the form l 0 0 l and present necessary and sufficient conditions for the stability of some of them via the energy casimir method and the study of the linearized equations of the motion finally using a recent geometric method introduced by hanssmann and van der meer we give a sufficient condition for the existence of a non degenerate hamiltonian hopf bifurcation at those equilibria when the gyrostat is symmetric

the local discontinuous galerkin method for the oseen equations we introduce and analyze the local discontinuous galerkin method for the oseen equations of incompressible fluid flow for a class of shape regular meshes with hanging nodes we derive optimal a priori estimates for the errors in the velocity and the pressure in l 2 and negative order norms numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of reynolds numbers

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

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