advection diffusion reaction equations hyperbolization and high order ader discretizations the purpose of this paper is twofold first we extend the applicability of cattaneo s relaxation approach one of the currently known relaxation approaches to reformulate time dependent advection

pollutant transport by shallow water equations on unstructured meshes hyperbolization of the model and numerical solution via a novel flux splitting scheme abstract the purpose of this paper is twofold first using the cattaneo s relaxation approach we reformulate the system of governing equations for the pollutant transport by shallow water flows over non flat topography and anisotropic diffusion as hyperbolic balance laws with stiff source terms the proposed relaxation system circumvents the infinite wave speed paradox which is inherent in standard advection diffusion models this turns out to give a larger stability range for the choice of the time step second following a flux splitting approach we derive a novel numerical method to discretise the resulting problem in particular we propose a new flux splitting and study the associated two systems of differential equations called the hydrodynamic and the relaxed diffusive system respectively for the presented splitting we analyse the resulting two systems of differential equations and propose two discretisation schemes of the godunov type these schemes are simple to implement robust accurate and fast when compared with existing methods the resulting method is implemented on unstructured meshes and is systematically assessed for accuracy robustness and efficiency on a carefully selected suite of test problems including non flat topography and wetting and drying problems formal second order accuracy is assessed through convergence rates studies

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

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

finite element solution of two dimensional inverse elastic problems using spatial smoothing a finite element method fem is presented for calculating the surface tractions on a body from internal measurements of displacement at discrete sensor locations the solution algorithm employs a sensitivity analysis which minimizes the difference between the calculated and measured displacements at each sensor location spatial regularization is one technique employed to stabilize the minimization process by imposing various degrees of smoothness on the solution it also allows the problem to be solved with fewer sensors than traction boundary nodes as an alternative to spatial regularization a method based on keynodes is introduced which assumes that each component of the boundary traction distribution can be described by a polynomial of specified order the methods are applied to several two dimensional examples including a rolling contact problem the effects of parameters such as the number of sensors the location of the sensors and the error in the sensor displacements are discussed

an arbitrary order runge kutta discontinuous galerkin approach to reinitialization for banded conservative level sets abstract we present a runge kutta discontinuous galerkin method for solving conservative reinitialization in the context of the conservative level set method 18 19 this represents an extension of the method recently proposed by owkes and desjardins 21 by solving the level set equations on the refined level set grid 9 and projecting all spatially dependent variables into the full basis used by the discontinuous galerkin discretization by doing so we achieve the full k 1 order convergence rate in the l 1 norm of the level set field predicted for rkdg methods given kth degree basis functions when the level set profile thickness is held constant with grid refinement shape and volume errors for the 0 5 contour of the level set on the other hand are found to converge between first and second order we show a variety of test results including the method of manufactured solutions reinitialization of a circle and sphere zalesak s disk and deforming columns and spheres all showing substantial improvements over the high order finite difference traditional level set method studied for example by herrmann 9 we also demonstrate the need for kth order accurate normal vectors as lower order normals are found to degrade the convergence rate of the method

reformulations for general advection diffusion reaction equations and locally implicit ader schemes following cattaneo s original idea in this article we first present two relaxation formulations for time dependent non linear systems of advection diffusion reaction equations such formulations yield time dependent non linear hyperbolic balance laws with stiff source terms then we present a locally implicit version of the ader method to solve these stiff systems to high accuracy the new ingredient of the numerical methodology is a locally implicit solution of the generalised riemann problem we illustrate the formulations and the resulting numerical approach by solving the compressible navier stokes equations

the compact discontinuous galerkin cdg method for elliptic problems we present a compact discontinuous galerkin cdg method for an elliptic model problem the problem is first cast as a system of first order equations by introducing the gradient of the primal unknown or flux as an additional variable a standard discontinuous galerkin dg method is then applied to the resulting system of equations the numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level thus resulting in a global system that involves only the original unknown variable the proposed method is closely related to the local discontinuous galerkin ldg method b cockburn and c w shu siam j numer anal 35 1998 pp 2440 2463 but unlike the ldg method the sparsity pattern of the cdg method involves only nearest neighbors also unlike the ldg method the cdg method works without stabilization for an arbitrary orientation of the element interfaces the computation of the numerical interface fluxes for the cdg method is slightly more involved than for the ldg method but this additional complication is clearly offset by increased compactness and flexibility compared to the br2 f bassi and s rebay j comput phys 131 1997 pp 267 279 and ip j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 lecture notes in phys 58 springer berlin 1976 pp 207 216 methods which are known to be compact the present method produces fewer nonzero elements in the matrix and is computationally more efficient

on the computation of steady state compressible flows using a discontinuous galerkin method computation of compressible steady state flows using a high order discontinuous galerkin finite element method is presented in this paper an accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities a physics based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave a recently developed fast low storage p multigrid method is used for solving the governing compressible euler equations to obtain steady state solutions the method is applied to compute a variety of compressible flow problems on unstructured grids numerical experiments for a wide range of flow conditions in both 2d and 3d configurations are presented to demonstrate the accuracy of the developed discontinuous galerkin method for computing compressible steady state flows

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 global mathematical model for the simulation of stenoses and bypass placement in the human arterial system this paper is concerned with a global mathematical model for the human circulation we describe its construction its validation and its application to study the haemodynamical effect of stenoses and bypass placement in the arterial system a geometric multiscale approach is adopted combining one dimensional 1d descriptions to represent 55 major arteries and zero dimensional 0d compartmental descriptions for the heart lungs the venous system and the microcirculation modern non linear numerical methods are implemented for solving the 1d systems of hyperbolic partial differential equations and high order rungekutta methods are used to solve the systems of ordinary differential equations resulting from the compartmental models the complete global mathematical model is then validated for healthy and stenotic cases comparing numerical predictions from the model against in vivo measurements results show overall satisfactory agreement in addition a sensitivity analysis is conducted in order to assess the influence of the most important parameters on some haemodynamical quantities of interest we then apply the model to simulate the haemodynamics for healthy controls and subjects with arterial stenosis we study the haemodynamical effect of bypass placement considering different locations in the arterial system different degrees of severity for both single and multiple arterial stenoses haemodynamical quantities of interest in the study are local pressure and flow rate we observe that large stenotic plaques near the heart cause abnormally strong pressure waves in the circulatory system and two adjacent occlusions worsen the phenomenon we find that the mathematical model is capable of predicting the most convenient bypass location and graft dimensions consistent with normal haemodynamics

estimation of sums of random variables examples and information bounds this paper concerns the estimation of sums of functions of observable and unobservable variables lower bounds for the asymptotic variance and a convolution theorem are derived in general finite and infinite dimensional models an explicit relationship is established between efficient influence functions for the estimation of sums of variables and the estimation of their means certain plug in estimators are proved to be asymptotically efficient in finite dimensional models while u v estimators of robbins are proved to be efficient in infinite dimensional mixture models examples include certain species network and data confidentiality problems

implicit semi analytical solution of the generalized riemann problem for stiff hyperbolic balance laws we present a semi analytical implicit solution to the generalized riemann problem grp for non linear systems of hyperbolic balance laws with stiff source terms the solution method is based on an implicit time taylor series expansion and the cauchy kowalewskaya procedure along with the solution of a sequence of classical riemann problems our new grp solver is then used to construct locally implicit ader methods of arbitrary accuracy in space and time for solving the general initial boundary value problem for non linear systems of hyperbolic balance laws with stiff source terms analysis of the method for model problems is carried out and empirical convergence rate studies for suitable tests problems are performed confirming the theoretically expected high order of accuracy

quadrature free implementation of discontinuous galerkin method for hyperbolic equations a discontinuous galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions this approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent in the discontinuous galerkin method test problems include the linear and nonlinear one dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity and two dimensional linear euler solutions on unstructured grids

low dissipation centred schemes for hyperbolic equations in conservative and non conservative form abstract we propose a one parameter family of low dissipation centred numerical schemes for solving hyperbolic equations in conservative or non conservative form using finite volume or discontinuous galerkin finite element methods the new schemes spring out from the multi dimensional force method and are determined by a single parameter 1 given an increasing sequence of real numbers 1 1 2 k there corresponds a sequence of numerical schemes with stability restriction associated to a decreasing sequence of courant numbers 1 c 1 c 2 c k 0 and a decreasing sequence of corresponding numerical viscosity functions d 1 d 2 d k for a given courant number c k 1 there is a real number k 0 and a corresponding stable scheme with minimal numerical viscosity d k the proposed schemes suit very well the family of high order discontinuous galerkin finite element methods and the recently proposed class of ader tr schemes whose orders of accuracy define decreasing sequences of courant numbers as the order of accuracy increases the centred methods of this paper are stable in 2d and 3d in the frame of simultaneous updating formulae unlike other centred methods such as 1d force which are not furthermore the schemes are highly accurate for slowly moving waves which is precisely the kind of waves that traditional centred methods smear disastrously additional features of the proposed schemes include ease of implementation and applicability to any hyperbolic system either in conservative or non conservative form here the proposed schemes are analysed and computationally assessed through a suite of test problems for a linear model system for the euler equations in one and two space dimensions and for the baer nunziato equations for compressible two phase flow

a p multigrid discontinuous galerkin method for the euler equations on unstructured grids a p multigrid p polynomial degree discontinuous galerkin method is presented for the solution of the compressible euler equations on unstructured grids the method operates on a sequence of solution approximations of different polynomial orders a distinct feature of this p multigrid method is to use different time integration schemes on different approximation levels resulting in an accurate fast and low memory method that can be used to accelerate the convergence of the euler equations to a steady state for discontinuous galerkin methods the developed method is used to compute the compressible flows for a variety of test problems on unstructured grids the numerical results obtained strongly indicate the order independent property of this p multigrid method an overall speed up factor more than one order of magnitude for both second and third order solutions of all test cases in comparison with the explicit method is demonstrated

fast p multigrid discontinuous galerkin method for compressible flows at all speeds ap multigrid wherep is the polynomial degree discontinuous galerkin method is presented for the solution of the compressible euler equations on unstructured grids the method operates on a sequence of solution approximations of different polynomial orders a distinct feature ofthisp multigrid method is the application of an explicit smoother on the higher level approximations p 0 and an implicit smoother on the lowest level approximation p 0 resulting in a fast and low storage method that can be efficiently used to accelerate the convergence to a steady state solution furthermore this p multigrid method can be naturally applied to compute the flows with discontinuities in which a monotonic limiting procedure is usually required for discontinuous galerkin methods an accurate representation of the boundary normals based on the definition of the geometries is used for imposing slip boundary conditions for curved geometries krivodonova l and berger m high order implementation of solid wall boundary conditions in curved geometries journal of computational physics vol 211 no 2 2006 pp 492 512 and luo h baum j d and lohner r on the computation of steady state compressible flows using a discontinuous galerkin method international journal for numerical methods in engineering vol 73 no 5 2008 pp 597 623 a variety of compressible flow problems for a wide range of flow conditions from low mach number to supersonic in both two dimensional and three dimensional configurations are computed to demonstrate the performance ofthisp multigrid method the numerical results obtained strongly indicate the order independent property of this p multigrid method and demonstrate that this method is orders of magnitude faster than its explicit counterpart the performance comparison with a finite volume method shows that using this p multigrid method the discontinuous galerkin method provides a viable attractive competitive and probably even superior alternative to the finite volume method for computing compressible flows at all speeds

a new spatial regularization scheme for the identification of the geometric shape of an inclusion in a finite body this paper presents a system identification scheme to determine the geometric shape of an inclusion in a finite body the proposed algorithm is based on the minimization of the least squared errors between the measured displacement field and calculated displacement field by the finite element model the domain parameterization technique is adopted to manipulate the shape variation of an inclusion to stabilize the optimization process a new regularization function defined by the length of the boundary curve of an inclusion is added to the error function a variable regularization factor scheme is proposed for a consistent regularization effect the modified newton method with the active set method is adopted for optimization copyright 1999 john wiley sons ltd

a pressure correction scheme for convection dominated incompressible flows with discontinuous velocity and continuous pressure in this work we present a pressure correction scheme for the incompressible navier stokes equations combining a discontinuous galerkin approximation for the velocity and a standard continuous galerkin approximation for the pressure the main interest of pressure correction algorithms is the reduced computational cost compared to monolithic strategies in this work we show how a proper discretization of the decoupled momentum equation can render this method suitable to simulate high reynolds regimes the proposed spatial velocity pressure approximation is lbb stable for equal polynomial orders and it allows adaptive p refinement for velocity and global p refinement for pressure the method is validated against a large set of classical two and three dimensional test cases covering a wide range of reynolds numbers in which it proves effective both in terms of accuracy and computational cost

robust implicit direct discontinuous galerkin method for simulating the compressible turbulent flows this paper proposes a robust and accurate implicit direct discontinuous galerkin ddg method for simulating laminar and turbulent flows the reynolds averaged navier stokes rans equations couple

an inverse method for determining material properties of a multi layer medium by boundary element method abstract an optimization system which combines a particular boundary element method and a modified levenberg marqardt method and which can be used to determine material properties of a multi layer medium from the specified or observed displacements is presented in this paper

a hermite weno reconstruction based discontinuous galerkin method for the euler equations on tetrahedral grids a hermite weno reconstruction based discontinuous galerkin method rdg p 1p 2 designed not only to enhance the accuracy of discontinuous galerkin method but also to ensure linear stability of the rdg method is presented for solving the compressible euler equations on tetrahedral grids in this rdg p 1p 2 method a quadratic polynomial solution p 2 is first reconstructed using a least squares method from the underlying linear polynomial p 1 discontinuous galerkin solution 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 and consistent with the underlying dg method the final quadratic polynomial solution is then obtained using a weno reconstruction which is necessary to ensure linear stability of the rdg method the developed rdg method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy efficiency robustness and versatility the numerical experiments demonstrate that the developed rdg p 1p 2 method is able to maintain the linear stability achieve the designed third order of accuracy one order accuracy higher than the underlying dg method without significant increase in computing costs and storage requirements

estimation and inference of quantile regression for survival data under biased sampling abstractbiased sampling occurs frequently in economics epidemiology and medical studies either by design or due to data collecting mechanism failing to take into account the sampling bias usually leads to incorrect inference we propose a unified estimation procedure and a computationally fast resampling method to make statistical inference for quantile regression with survival data under general biased sampling schemes including but not limited to the length biased sampling the case cohort design and variants thereof we establish the uniform consistency and weak convergence of the proposed estimator as a process of the quantile level we also investigate more efficient estimation using the generalized method of moments and derive the asymptotic normality we further propose a new resampling method for inference which differs from alternative procedures in that it does not require to repeatedly solve estimating equations it is proved that the resampling method consistently estimates the asymptoti

a boundary element formulation for the inverse elastostatics problem iesp of flaw detection a boundary integral formulation is presented for the detection of flaws in planar structural members from the displacement measurements given at some boundary locations and the applied loading such inverse problems usually start with an initial guess for the flaw location and size and proceed towards the final configuration in a sequence of iterative steps a finite element formulation will require a remeshing of the object corresponding to the revised flaw configuration in each iteration making the procedure computationally expensive and cumbersome no such remeshing is required for the boundary element approach the inverse problem is written as an optimization problem with the objective function being the sum of the squares of the differences between the measured displacements and the computed displacements for the assumed flaw configuration the geometric condition that the flaw lies within the domain of the object is imposed using the internal penalty function approach in which the objective function is augmented by the constraint using a penalty parameter a first order regularization procedure is also implemented to modify the objective function in order to minimize the numerical fluctuations that may be caused in the numerical procedure due to errors in the experimental measurements for displacements the flaw configuration is defined in terms of geometric parameters and the sensitivities with respect to these parameters are obtained in the boundary element framework using the implicit differentiation approach a series of numerical examples involving the detection of circular and elliptical flaws of various sizes and orientations are solved using the present approach good predictions of the flaw shape and location are obtained

junction generalized riemann problem for stiff hyperbolic balance laws in networks in this paper we design a new implicit solver for the junction generalized riemann problem j grp which is based on a recently proposed implicit method for solving the generalized riemann problem grp for systems of hyperbolic balance laws we use the new j grp solver to construct an ader scheme that is globally explicit locally implicit and with no theoretical accuracy barrier in both space and time the resulting ader scheme is able to deal with stiff source terms and can be applied to non linear systems of hyperbolic balance laws in domains consisting on networks of one dimensional sub domains in this paper we specifically apply the numerical techniques to networks of blood vessels we report on a test problem with exact solution for a simplified network of three vessels meeting at a single junction which is then used to carry out a systematic convergence rate study of the proposed high order numerical methods schemes up to fifth order of accuracy in space and time are implemented and tested we then show the ability of the ader scheme to deal with stiff sources through a numerical simulation in a network of vessels an application to a physical test problem consisting of a network of 37 compliant silicon tubes arteries and 21 junctions reveals that it is imperative to use high order methods at junctions in order to preserve the desired high order of accuracy in the full computational domain for example it is demonstrated that a second order method throughout gives comparable results to a method that is fourth order in the interior of the domain and first order at junctions we are concerned with stiff hyperbolic balance laws in networks an implicit solver for the junction generalized riemann problem is proposed ader schemes of arbitrary accuracy in space and time for networks are constructed convergence rates studies of schemes up to fifth order are carried out it is necessary to match the accuracy at junctions to that of the rest of the domain

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

a convex entropy for a hyperbolic system with relaxation abstract we explicitly construct a convex entropy function for a hyperbolic system with relaxation this entropy is defined in the sense of chen levermore and liu which resembles boltzman s h theorem for kinetic equations this construction follows the idea of suliciu s energy function for a rate type mixed hyperbolic elliptic system such an entropy will be useful in proving the entropy property of the relaxation schemes introduced by jin and xin for conservation laws

identification of geometric shapes and material properties of inclusions in two dimensional finite bodies by boundary parameterization a system identification scheme based on the boundary element method bem is proposed to determine geometric shape and elastic material properties of an inclusion in a finite body the proposed algorithm is based on the minimization of least squared errors between measured displacement and calculated displacement by a boundary element model a regularization function that consists of the geometric term and material property term is added to the error function to overcome ill posedness of inverse problems to deal with the shape variation of an inclusion during the optimization process the boundary parameterization technique is applied to the bem the recursive quadratic programming rqp technique with line search is used for optimization

estimating a multiplicative treatment effect under biased allocation suppose that given x is poisson with mean while given y is poisson with mean or according as treatment 1 or treatment 2 is used moreover suppose that treatment 1 is used if x

solution of the generalized riemann problem for advection reaction equations we present a method for solving the generalized riemann problem for partial differential equations of the advection reaction type the generalization of the riemann problem here is twofold firstly the governing equations include nonlinear advection as well as reaction terms and secondly the initial condition consists of two arbitrary but infinitely differentiable functions an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws the solution procedure local and valid for sufficiently small times reduces the solution of the generalized riemann problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional riemann problems for homogeneous advection equations for spatial derivatives of the initial conditions we illustrate the approach via the model advection reaction equation the inhomogeneous burgers equation and the nonlinear shallow water equations with variable bed elevation

an explicit update scheme for inverse parameter and interface estimation of piecewise constant coefficients in linear elliptic pdes we introduce a general and efficient method to recover piecewise constant coefficients occurring in elliptic partial differential equations as well as the interface where these coefficients have jump discontinuities for this purpose we use an output least squares approach with level set and augmented lagrangian methods our formulation incorporates the inherent nature of the piecewise constant coefficients which eliminates the need for a complicated nonlinear solve at every iteration instead we obtain an explicit update formula and therefore vastly speed up computation we employ our approach to the example problems of poisson s equation and linear elasticity and provide the combination of simultaneously recovering coefficients and interface

a bem based pattern search solution for a class of inverse elastostatic problems a boundary element based solution employing the hooke jeeves pattern search method is developed to determine internal cavity geometries in the inverse elastostatics problem in contrast to existing methods which employ gradient based solutions the developed method does not require gradient information and consequently results in a simpler solution procedure a boundary difference function bdf that provides a guide for the initial guess is introduced the unknown cavity geometry is first assumed to be circular and the algorithm is run until the general location of the cavity is determined subsequent refinement of the detected cavity geometry is affected using a periodic cubic spline discretization numerical results are presented to illustrate the effectiveness of the boundary difference function and to demonstrate the successful detection of the cavity the effects of simulated experimental inputs with prescribed error are also presented

estimating a treatment effect under biased sampling methods are presented for the estimation of a treatment effect based on before and after treatment values where for ethical reasons all and only those patients are treated whose before treatment values exceed a given constant

robust geometrically based automatic two dimensional mesh generation a technical description of the algorithms employed in the modified quadtree mesh generator is given although the basis of the mesh generator is the same as the original version developed by yerry and shephard 1 2 the actual algorithms on which it is built have been entirely changed for the purpose of ensuring the robustness of the technique as demonstrated in the paper the algorithmic changes made do ensure the robustness of the approach but introduce additional algorithmic difficulties the solutions of which are also presented in addition to examples showing the capability of the mesh generator the linear computational growth rate of the mesh generator is demonstrated

a ghost cell immersed boundary method for flow in complex geometry an efficient ghost cell immersed boundary method gcibm for simulating turbulent flows in complex geometries is presented a boundary condition is enforced through a ghost cell method the reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second order accuracy of the base solver both dirichlet and neumann boundary conditions can be treated the current ghost cell treatment is both suitable for staggered and non staggered cartesian grids the accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface numerical results are compared with experimental data and boundary fitted grid results the method is further extended to an existing ocean model mitgcm to simulate geophysical flow over a three dimensional bump the method is easily implemented as evidenced by our use of several existing codes

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

a finite element formulation for a convection diffusion equation based on cattaneo s law a number of contributions have been made during the last decades to model pure diusive transport problems by using the so called hyperbolic diusion equa on problems based on a cattaneo type law the finite element solution of the proposed equations is addressed the performance of the algorithm is verified by solving some 2d test cases some interesting features of the proposed model can be observed from these examples and we conclude that the proposal is a feasible alternative to the parabolic model for real engineering simulations

high order ader schemes for a unified first order hyperbolic formulation of newtonian continuum mechanics coupled with electro dynamics abstract in this paper we propose a new unified first order hyperbolic model of newtonian continuum mechanics coupled with electro dynamics the model is able to describe the behavior of moving elasto plastic dielectric solids as well as viscous and inviscid fluids in the presence of electro magnetic fields it is actually a very peculiar feature of the proposed pde system that viscous fluids are treated just as a special case of elasto plastic solids this is achieved by introducing a strain relaxation mechanism in the evolution equations of the distortion matrix a which in the case of purely elastic solids maps the current configuration to the reference configuration the model also contains a hyperbolic formulation of heat conduction as well as a dissipative source term in the evolution equations for the electric field given by ohm s law via formal asymptotic analysis we show that in the stiff limit the governing first order hyperbolic pde system with relaxation source terms tends asymptotically to the well known viscous and resistive magnetohydrodynamics mhd equations furthermore a rigorous derivation of the model from variational principles is presented together with the transformation of the euler lagrange differential equations associated with the underlying variational problem from lagrangian coordinates to eulerian coordinates in a fixed laboratory frame the present paper hence extends the unified first order hyperbolic model of newtonian continuum mechanics recently proposed in 110 42 to the more general case where the continuum is coupled with electro magnetic fields the governing pde system is symmetric hyperbolic and satisfies the first and second principle of thermodynamics hence it belongs to the so called class of symmetric hyperbolic thermodynamically compatible systems shtc which have been studied for the first time by godunov in 1961 61 and later in a series of papers by godunov and romenski 67 69 119 an important feature of the proposed model is that the propagation speeds of all physical processes including dissipative processes are finite the model is discretized using high order accurate ader discontinuous galerkin dg finite element schemes with a posteriori subcell finite volume limiter and using high order ader weno finite volume schemes we show numerical test problems that explore a rather large parameter space of the model ranging from ideal mhd viscous and resistive mhd over pure electro dynamics to moving dielectric elastic solids in a magnetic field

a cascade optimization methodology for automatic parameter identification and shape process optimization in metal forming simulation computer simulations of metal forming processes using the finite element method fem are today well established this form of simulation uses an increasing number of sophisticated geometrical and material models relying on a certain number of input data which are not always readily available the aim of inverse problems which will be considered here is to determine one or more of the input data relating to these forming process simulations thereby leading to a desired result in this paper we will focus on two categories of such inverse problems the first category consists of parameter identification inverse problems these involve evaluating the material parameters for material constitutive models that would lead to the most accurate results with respect to physical experiments i e minimizing the difference between experimental results and fem simulations the second category consists of shape process optimization inverse problems these involve determining the initial geometry of the specimen and or the shape of the forming tools as well as some parameters of the process itself in order to provide the desired final geometry after the forming process these two categories of inverse problems can be formulated as optimization problems in a similar way i e by using identical optimization algorithms in this paper we intend firstly to solve these two types of optimization problems by using different non linear gradient based optimization methods and secondly to compare their efficiency and robustness in a variety of numerical applications

material constant sensitivity boundary integral equation for anisotropic solids in this paper the material constant sensitivity boundary integral equation is presented and its numerical solution proposed based on boundary element techniques the formulation deals with plane problems with general rectilinear anisotropy expressions for the computation of sensitivities for displacements tractions strains and stresses are derived both for boundary and interior points the sensitivities can be computed with respect to the bulk material properties or to the properties of part of the domain inclusions coatings etc to assess the accuracy of the proposed approach the computed results are compared to analytical ones derived from exact solutions obtained by complex potential theory when possible or finite difference derivatives otherwise copyright 2005 john wiley sons ltd

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 runge kutta discontinuous galerkin method for viscous flow equations this paper presents a runge kutta discontinuous galerkin rkdg method for viscous flow computation the construction of the rkdg method is based on a gas kinetic formulation which not only couples the convective and dissipative terms together but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function due to the intrinsic connection between the gas kinetic bgk model and the navier stokes equations the navier stokes flux is automatically obtained by the present method numerical examples for both one dimensional 1d and two dimensional 2d compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current rkdg method

estimating the non homogeneous elastic modulus distribution from surface deformations abstract we present a novel approach to solve the inverse problem in finite elasticity for the non homogeneous shear modulus distribution solely from known surface deformation fields the inverse problem is posed as a constrained optimization problem under regularization and solved utilizing the adjoint equations hypothetical measured surface displacement fields are created by inducing indentations on the exterior of the specimen these surface displacement fields are used to test the inverse strategy on a problem domain consisting of a stiff circular inclusion in a softer homogeneous background we observe that the shear modulus reconstruction as well as the shape of the circular inclusion improves with an increasing number of surface displacement fields furthermore with increasing noise level in the surface displacement field the contrast of the reconstructions decreases

improved accuracy for time splitting methods for the numerical solution of parabolic equations in this work we study time splitting strategies for the numerical approximation of evolutionary reaction diffusion problems in particular we formulate a family of domain decomposition splitting methods that overcomes some typical limitations of classical alternating direction implicit adi schemes the splitting error associated with such methods is observed to be o 2 in the time step in order to decrease the size of this splitting error to o 3 we add a correction term to the right hand side of the original formulation this procedure is based on the improved initialization technique proposed by douglas and kim in the framework of adi methods the resulting non iterative schemes reduce the global system to a collection of uncoupled subdomain problems that can be solved in parallel computational results comparing the newly derived algorithms with the crank nicolson scheme and certain adi methods are presented

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

an inverse approach to determining myocardial material properties passive myocardial material properties have been measured previously by subjecting test samples of myocardium to in vitro load deformation analysis or in the intact heart by pressure volume relationships a new method for determining passive material properties described in this paper couples a p version finite element model of the heart a nonlinear optimization algorithm and a dense set of transmural measured strains that could be obtained in the intact heart by magnetic resonance imaging mri radiofrequency tissue tagging unknown material parameters for a nonlinear nonhomogeneous material law are determined by solving an inverse boundary value problem an objective function relating the least squares difference of model predicted and measured strains is minimized with respect to the unknown material parameters using a novel optimization algorithm that utilizes forward finite element solutions to calculate derivatives of model predicted strains with respect to the material parameters test cases incorporating several salient features of the inverse material identification problem for the heart are formulated to test the performance of the inverse algorithm in typical experimental conditions known true material parameters can be determined to within a small tolerance and random noise is shown not to affect the stability of the inverse solution appreciably on the basis of these validation experiments we conclude that the inverse material identification problem for the heart can be extended to solve for unknown material parameters that describe in vivo myocardial material behavior

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

a reconstructed discontinuous galerkin method for magnetohydrodynamics on arbitrary grids a reconstructed discontinuous galerkin rdg method designed not only to enhance the accuracy of dg methods but also to ensure the nonlinear stability of the rdg method is developed for solving the magnetohydrodynamics mhd equations on arbitrary grids in this rdg p1p2 method a quadratic polynomial solution p2 is first obtained using a hermite weighted essentially non oscillatory weno reconstruction from the underlying linear polynomial p1 discontinuous galerkin solution to ensure linear stability of the rdg method and to improves efficiency of the underlying dg method by taking advantage of handily available and yet invaluable information namely the first derivatives in the dg formulation the stencils used in 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 hlld riemann solver introduced in the literature for one dimensional mhd problems is adopted in normal direction to compute numerical fluxes the divergence free constraint is satisfied using the locally divergence free ldf approach the developed rdg method is used to compute a variety of 2d and 3d mhd problems on arbitrary grids to demonstrate its accuracy robustness and non oscillatory property our numerical experiments indicate that the rdg p1p2 method is able to capture shock waves sharply essentially without any spurious oscillations and achieve the designed third order of accuracy one order accuracy higher than the underlying dg method

an inverse method for determining elastic material properties and a material interface the problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix

a five equation model for the simulation of miscible and viscous compressible fluids abstract typical multispecies compressible navier stokes computations employ conservative equations for mass fraction transport upwind discretisations of these governing equations produce spurious pressure oscillations at diffuse contact surfaces between gases of differing ratio of specific heat capacities which degrade the convergence rate of the algorithm adding quasi conservative equations for volume fraction can solve this error however this approach has been derived only for immiscible fluids here a five equation quasi conservative model is proposed that includes the effects of species diffusion viscosity and thermal conductivity the derivation of the model is presented along with a numerical method to solve the governing equations at second order accuracy in space and time formal convergence studies demonstrate the expected order of accuracy is achieved for three benchmark problems cross validated against two standard mass fraction models in these test cases the new model has between 2 and 10 times lower error for a given grid size simulations of a two dimensional air sf6 richtmyer meshkov instability demonstrate that the new model converges to the solution with four times fewer points in each direction when compared to the mass fraction model in an identical numerical framework this represents an 40 times lower computational cost for an equivalent error in two dimensional computations the proposed model is thus very suitable for direct numerical simulation and large eddy simulation of compressible mixing

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

brain venous haemodynamics neurological diseases and mathematical modelling a review behind medicine m is physiology p behind physiology is physics p and behind physics is always mathematics m for which i expect that the symmetry of the quadruplet mppm will be compatible with the characteristic bias of hyperbolic partial differential equations a theme of this paper i start with a description of several idiopathic brain pathologies that appear to have a strong vascular dimension of which the most prominent example considered here is multiple sclerosis the most common neurodegenerative disabling disease in young adults other pathologies surveyed here include retinal abnormalities transient global amnesia transient monocular blindness meniere s disease and idiopathic parkinson s disease it is the hypothesised vascular aspect of these conditions that links medicine to mathematics through fluid mechanics in very complex networks of moving boundary blood vessels the second part of this paper is about mathematical modelling of the human cardiovascular system with particular reference to the venous system and the brain a review of a recently proposed multi scale mathematical model then follows consisting of a one dimensional hyperbolic description of blood flow in major arteries and veins coupled to a lumped parameter description of the remaining main components of the human circulation derivation and analysis of the hyperbolic equations is carried out for blood vessels admitting variable material properties and with emphasis on the venous system a much neglected aspect of cardiovascular mathematics veins unlike their arterial counterparts are highly deformable even collapsible under mild physiological conditions we address mathematical and numerical challenges regarding the numerical analysis of the hyperbolic pdes we deploy a modern non linear finite volume method of arbitrarily high order of accuracy in both space and time the ader methodology in vivo validation examples and brain haemodynamics computations are shown we also point out two preliminary but important new findings through the use of mathematical models namely that extracranial venous strictures produce chronic intracranial venous hypertension and that augmented pressure increases the blood vessel wall permeability

efficiency of the u v method of estimation given a pool of motorists how do we estimate the total intensity of those who had a prespecified number of traffic accidents in the past year we previously have proposed the u v method as a solution to estimation problems of this type in this paper we prove that the u v method provides asymptotically efficient estimators in an important special case

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

reconstructed discontinuous galerkin methods for linear advection diffusion equations based on first order hyperbolic system abstract newly developed reconstructed discontinuous galerkin rdg methods are presented for solving linear advection diffusion equations on hybrid unstructured grids based on a first order hyperbolic system fohs formulation benefiting from both fohs and rdg methods the developed hyperbolic rdg methods are reliable accurate efficient and robust achieving higher orders of accuracy than conventional dg methods for the same number of degrees of freedom superior accuracy is achieved by reconstruction of higher order terms in the solution polynomial via gradient variables introduced to form a hyperbolic diffusion system and least squares variational reconstruction unsteady capability is demonstrated by an l stable implicit time integration scheme a number of advection diffusion test cases with a wide range of reynolds numbers including boundary layer type problems and unsteady cases are presented to assess accuracy and performance of the newly developed hyperbolic rdg methods numerical experiments demonstrate that the hyperbolic rdg methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular irregular and heterogeneous grids indicating that the developed hyperbolic rdg methods provide an attractive and probably an even superior alternative for solving the linear advection diffusion 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

a birkhoff contraction formula with applications to riccati equations in this paper we show that the symplectic hamiltonian operators on a hilbert space give rise to linear fractional transformations on the open convex cone of positive definite operators that contract a natural invariant finsler metric the thompson or part metric on the convex cone more precisely the constants of contraction for the hamiltonian operators satisfy the classical birkhoff formula the lipschitz constant for the corresponding linear fractional transformations on the cone of positive definite operators is equal to the hyperbolic tangent of one fourth the diameter of the image by means of the close connections between hamilitonian operators and riccati equations this result and the associated machinery are applied to obtain convergence results for discrete algebraic riccati equations and riccati differential equations

the behavior of hyperbolic heat equations solutions near their parabolic limits standard energy methods are used to study the relation between the solutions of one parameter families of hyperbolic systems of equations describing heat propagation near their parabolic limits which for these cases are the usual diffusive heat equation in the linear case it is proven that given any solution to the hyperbolic equations there is always a solution to the diffusion equation which after a short time stays very close to it for all times the separation between these solutions depends on the square of the ratio between the assumed very short decay time appearing in cattaneo s relation and the usual characteristic smoothing time initial data dependent of the limiting diffusive equation the techniques used in the linear case can be readily used for nonlinear equations as an example we consider the theories of heat propagation introduced by coleman fabrizio and owen and prove that near a solution to the limiting diffusive equation there is always a solution to the nonlinear hyperbolic equations for a time which usually is much longer than the decay time of the corresponding cattaneo relation an alternative derivation of the heat theories of divergence type which are consistent with thermodynamic principles is given as an appendix