a class of logarithmically completely monotonic functions and the best bounds in the second kershaw s double inequality in the article the sufficient and necessary conditions such that a class of functions which involve the psi function j and the ratio c x t c x s are logarithmically completely monotonic are established the best bounds for the ratio c x t c x s are given and some comparisons with known results are carried out where s and t are two real numbers and x min s t

finite element solution of compressible viscous flows using conservative variables abstract a finite element method for solving the compressible navier stokes equations is presented these equations are solved in conservation form and using conservative variables appropriate finite element approximations are discussed when a galerkin variational formulation is used for high speed flows the formulation is stabilized using the stream line upwind petrov galerkin method for which we propose an analytical expression for the matrix a new discontinuity capturing operator is also proposed to accurately solve flow problems exhibiting sharp gradients the non linear systems of equations arising from the discretization are solved using an iterative strategy based on the generalized minimal residual gmres algorithm numerical results for transonic and supersonic flows are presented that demonstrate the effectiveness of the proposed finite element method

convex entropies and hyperbolicity for general euler equations the compressible euler equations possess a family of generalized entropy densities of the form rho f sigma where rho is the mass density sigma is the specific entropy and f is an arbitrary function entropy inequalities associated with convex entropy densities characterize physically admissible shocks for polytropic gases harten has determined which rho f sigma are strictly convex in this paper we extend this determination to gases with an arbitrary equation of state moreover we show that at every state where the sound speed is positive i e where the euler equations are hyperbolic there exist rho f sigma that are strictly convex thereby establishing the converse of the general fact that the existence of a strictly convex entropy density implies hyperbolicity

on the convergence of a finite element method for a nonlinear hyperbolic conservation law we consider a space time finite element discretization of a time dependent nonlinear hyperbolic conservation law in one space dimension burgers equation the finite element method is higher order accurate and is a petrov galerkin method based on the so called streamline diffusion modification of the test functions giving added stability we first prove that if a sequence of finite element solutions converges boundedly almost everywhere as the mesh size tends to zero to a function u then u is an entropy solution of the conservation law this result may be extended to systems of conservation laws with convex entropy in several dimensions we then prove using a compensated compactness result of murat tartar that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of burgers equation we also consider a further modification of the test functions giving a method with improved shock capturing finally we present the results of some numerical experiments

a hierarchical duality approach to bounds for the outputs of partial differential equations abstract we present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to finiteelement discretizations of symmetric or nonsymmetric coercive linear partial differential equations the method is based upon the construction of an augmented lagrangian which integrates i a quadratic energy reformulation of the desired output as the objective to be minimized with ii the finite element equilibrium equations and conforming hybridized intersubdomain continuity conditions as the constraints to be satisfied the bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate lagrange multipliers generated by a less expensive approximation such as a low dimensional finite element discretization as in many a posteriori error estimation techniques the bound calculation requires only the solution of subdomain local symmetric neumann problems on the refined truth mesh the technique is presented and illustrated for the case of the one dimensional convection diffusion equation

bddc preconditioning for high order galerkin least squares methods using inexact solvers a high order galerkin least squares gls finite element discretization is combined with a balancing domain decomposition by constraints bddc preconditioner and inexact local solvers to provide an efficient solution technique for large scale convection dominated problems the algorithm is applied to the linear system arising from the discretization of the two dimensional advection diffusion equation and euler equations for compressible inviscid flow a robin robin interface condition is extended to the euler equations using entropy symmetrized variables the bddc method maintains scalability for the high order discretization of the diffusion dominated flows and achieves low iteration count in the advection dominated regime the bddc method based on inexact local solvers with incomplete factorization and p 1 coarse correction maintains the performance of the exact counterpart for the wide range of the peclet numbers considered

constructing an entropy stable upwind scheme for compressible fluid flow computations a relationship between entropy processes and numerical upwinding is examined in the context of computational gasdynamics a discretized form of the entropy inequality is constructed at the integration point where convection diffusion modeling occurs in finite volume methods conventional upwinding schemes may violate the local form of the second law of thermodynamics but a modified upwinding scheme uses additional momentum constraints and pressure terms to provide a positive entropy production rate the second law is seen as an important quantitative measure of nonphysical numerical results as well as a sound basis for error analysis applications to converging diverging nozzle and blunt body flow problems demonstrate the promising performance of the overall numerical algorithm

invariant region preserving dg methods for multi dimensional hyperbolic conservation law systems with an application to compressible euler equations abstract an invariant region preserving irp limiter for multi dimensional hyperbolic conservation law systems is introduced as long as the system admits a global invariant region which is a convex set in the phase space it is shown that the order of approximation accuracy is not destroyed by the irp limiter provided the cell average is away from the boundary of the convex set moreover this limiter is explicit and easy for computer implementation a generic algorithm incorporating the irp limiter is presented for high order finite volume type schemes for arbitrarily high order discontinuous galerkin dg schemes to hyperbolic conservation law systems sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one dimensional system shares the same invariant region as the full multi dimensional hyperbolic system does the general results are then applied to both one and two dimensional compressible euler equations so to obtain high order irp dg schemes numerical experiments are provided to validate the proven properties of the irp limiter and the performance of irp dg schemes for compressible euler equations

on discontinuous galerkin approximations of boltzmann moment systems with levermore closure abstract this work considers the discontinuous galerkin dg finite element discretization of first order systems of conservation laws derivable as moments of the kinetic boltzmann equation with levermore c d levermore moment closure hierarchies for kinetic theories j statist phys 83 5 6 1996 1021 1065 closure using standard energy analysis techniques a new class of energy stable numerical flux functions are devised for the dg discretization of boltzmann moment systems simplified energy stable numerical fluxes are then constructed which replace exact state space integration in the numerical flux with gauss lobatto quadrature numerical results for supersonic flow over a cylinder geometry in the continuum and transitional regimes using 5 and 10 moment approximations are presented using the newly devised dg discretizations

on the stability and convergence of a galerkin reduced order model rom of compressible flow with solid wall and far field boundary treatment a reduced order model rom based on the proper orthogonal decomposition pod galerkin projection method is proposed as an alternative discretization of the linearized compressible euler equations it is shown that the numerical stability of the rom is intimately tied to the choice of inner product used to define the galerkin projection for the linearized compressible euler equations a symmetry transformation motivates the construction of a weighted l 2 inner product that guarantees certain stability bounds satisfied by the rom sufficient conditions for well posedness and stability of the present galerkin projection method applied to a general linear hyperbolic initial boundary value problem ibvp are stated and proven well posed and stable far field and solid wall boundary conditions are formulated for the linearized compressible euler rom using these more general results a convergence analysis employing a stable penalty like formulation of the boundary conditions reveals that the rom solution converges to the exact solution with refinement of both the numerical solution used to generate the rom and of the pod basis an a priori error estimate for the computed rom solution is derived and examined using a numerical test case published in 2010 by john wiley sons ltd

a preconditioning mass matrix to accelerate the convergence to the steady euler solutions using explicit schemes when explicit time marching algorithms are used to reach the steady state of problems governed by the euler equations the rate of convergence is strongly impaired both in the zones with low mach number and in the zones with transonic flow e g mach and mach 1 with 0 2 the rate of convergence becomes slower as diminishes we show in this paper with analytical and numerical results how the use of a preconditioning mass matrix accelerates the convergence in the aforementioned ranges of mach numbers the preconditioning mass matrix pmm we advocate in this paper can be applied to any fem fvm that uses an explicit time marching scheme to find the steady state the method s rate of convergence to the steady state is studied and results for the one and two dimiensional cases are presented in sections 1 3 using the one dimensional euler equations we first explain why there exists a slow rate of convergence when the plain lumping of mass is used then the convergence rate to steady solutions is analysed from its two constituents that is convergence by absorption at the boundaries and by damping in the domain next we give the natural solution to this problem and with several examples we show the effectiveness of the proposed mass matrix when compared with the plain scheme in sections 4 8 we give the multidimensional version of the preconditioning mass matrix we make a stability analysis and compare the group velocities and damping with and without the new mass matrix to finish we show the velocity of convergence for a common test problem

global existence for systems of parabolic conservation laws in several space variables abstract we prove the global existence of solutions of the cauchy problem for certain systems of conservation laws with artificial viscosity terms added the system is assumed to admit a quadratic entropy which is consistent with the viscosity matrix and the initial data is assumed to be close to a constant in l2 l in particular our result applies to the equations of compressible fluid flow in two and three space variables

a minimum entropy principle of high order schemes for gas dynamics equations the entropy solutions of the compressible euler equations satisfy a minimum principle for the specific entropy tadmor in appl numer math 2 211 219 1986 first order schemes such as godunov type and lax friedrichs schemes and the second order kinetic schemes khobalatte and perthame in math comput 62 119 131 1994 also satisfy a discrete minimum entropy principle in this paper we show an extension of the positivity preserving high order schemes for the compressible euler equations in zhang and shu j comput phys 229 8918 8934 2010 and zhang et al j scientific comput in press to enforce the minimum entropy principle for high order finite volume and discontinuous galerkin dg schemes

algebraic multigrid for stabilized finite element discretizations of the navier stokes equations abstract an algebraic multigrid method for the solution of stabilized finite element discretizations of the euler and navier stokes equations on generalized unstructured grids is described the method is based on an elemental agglomeration multigrid strategy employing a semi coarsening scheme designed to reduce grid anisotropy the viscous terms are discretized in a consistent manner on coarse grids using an algebraic galerkin coarse grid approximation in which higher order grid transfer operators are constructed from the underlying triangulation however the combination of higher order transfer operators and galerkin rediscretization diminishes the stability of stabilized inviscid operators on coarse grids and a modification is proposed to alleviate this problem a generalized line implicit relaxation scheme is also described where the lines are constructed to follow the direction of strongest coupling applications are demonstrated for convection diffusion euler and laminar navier stokes the results show that the convergence rate is largely unaffected by mesh size over a wide range of reynolds peclet numbers

time implicit high order discontinuous galerkin method with reduced evaluation cost an efficient and robust time integration procedure for a high order discontinuous galerkin method is introduced for solving nonlinear second order partial differential equations the time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term the procedure uses an iterative algorithm with reduced evaluation cost the size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low order and high order degrees of freedom numerical examples are presented for the nonlinear convection diffusion equation in one and two dimensions including steady and unsteady flow problems the performance of the present method is investigated in terms of cpu time and compared to a fully implicit method a von neumann stability analysis is carried out in order to determine the stability and damping properties of the method besides a fairly reduced cpu effort numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method

staggered grid entropy stable multidimensional summation by parts discretizations on curvilinear coordinates abstract the entropy conservative stable staggered grid tensor product algorithm of parsani et al 1 is extended to multidimensional sbp discretizations the required sbp preserving interpolation operators are proven to exist under mild restrictions and the resulting algorithm is proven to be entropy conservative stable as well as elementwise conservative for 2 dimensional simplex elements the staggered grid algorithm is shown to be more accurate and have a larger maximum time step restriction as compared to the collocated algorithm the staggered algorithm significantly reduces the number of computationally expensive two point flux evaluations which is potentially important for both explicit and implicit time marching schemes furthermore the staggered algorithm requires fewer degrees of freedom for comparable accuracy which has favorable implications for implicit time marching schemes

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

a posteriori finite element bounds for output functionals of discontinuous galerkin discretizations of parabolic problems we present a neumann subproblem a posteriori finite element procedure for the efficient calculation of constant free sharp lower and upper estimators for linear functional outputs of parabolic equations discretized by a discontinuous galerkin method in time in space a global coarse mesh and a decoupled fine mesh are used to compute the estimators which are shown to converge to the value of the output obtained for a global coupled fine mesh we first formulate the bound procedure with particular emphasis on the proof of the bounding properties we then provide an illustrative numerical example a problem of heat conduction in a composite material

stable boundary conditions and difference schemas for navier stokes equations the navier stokes equations at high reynolds numbers can be viewed as an incompletely elliptic perturbation of the euler equations by using the entropy function for the euler equations as a measure of energy for the navier stokes equations we are able to obtain nonlinear energy estimates for the mixed initial boundary value problem these estimates are used to derive boundary conditions that guarantee l 2 boundedness even when the reynolds number tends to infinity finally we propose a new difference scheme for modeling the navier stokes equations in multidimensions for which we are able to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation

stabilized residual distribution for shallow water simulations we propose a stabilized residual distribution rd scheme for the simulation of shallow water flows the final discretization is obtained combining the stabilized rd approach proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 with the conservative formulation already used in ricchiuto et al j comp phys 222 2007 to simulate shallow water flows the scheme proposed is a nonlinear variant of a lax friedrichs type discretization it is well balanced it actually yields second order of accuracy in smooth areas and it preserves the positivity of the height of the water in presence of dry areas this is made possible by the residual character of the discretization by properly adapting the stabilization operators proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 and thanks to the positivity preserving character of the underlying lax friedrichs scheme we demonstrate the properties of the discretization proposed on a wide variety of tests

a new lower bound in the second kershaw s double inequality in the paper a new and elegant lower bound in the second kershaw s double inequality is established some alternative simple and polished proofs are given several deduced functions involving the gamma and psi functions are proved to be decreasingly monotonic and logarithmically completely monotonic and some remarks and comparisons are stated

entropy viscosity method for the single material euler equations in lagrangian frame a new finite element method for solving the euler equations in lagrangian coordinates is proposed the method is stabilized by adding artificial diffusion terms compatible with positivity of mass and internal energy a minimum principle on the specific entropy and all generalized entropy inequalities two options of first order artificial diffusion are considered one is in the spirit of the eulerian based method guermond et al 2011 23 22 zingan et al 2013 and the other is similar to existing viscosity stabilizations in lagrangian frame e g dobrev et al 2012 the method is verified to be high order for smooth solutions even with active viscosity terms this is achieved by using high order finite element spaces and an entropy based viscosity stabilization that degenerates the first order viscous terms this stabilization automatically distinguishes smooth and singular regions the formal accuracy and convergence properties of the proposed methods are tested on a series of benchmark problems this is the first result extending the entropy viscosity methodology to the lagrangian hydrodynamics

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

entropy stable high order discontinuous galerkin methods with suitable quadrature rules for hyperbolic conservation laws abstract it is well known that semi discrete high order discontinuous galerkin dg methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws jiang and shu 1994 39 and symmetric hyperbolic systems hou and liu 2007 36 in any space dimension and for any triangulations however this property holds only for the square entropy and the integrations in the dg methods must be exact it is significantly more difficult to design dg methods to satisfy entropy inequalities for a non square convex entropy and or when the integration is approximated by a numerical quadrature in this paper we develop a unified framework for designing high order dg methods which will satisfy entropy inequalities for any given single convex entropy through suitable numerical quadrature which is specific to this given entropy our framework applies from one dimensional scalar cases all the way to multi dimensional systems of conservation laws for the one dimensional case our numerical quadrature is based on the methodology established in carpenter et al 2014 5 and gassner 2013 19 the main ingredients are summation by parts sbp operators derived from legendre gauss lobatto quadrature the entropy conservative flux within elements and the entropy stable flux at element interfaces we then generalize the scheme to two dimensional triangular meshes by constructing sbp operators on triangles based on a special quadrature rule a local discontinuous galerkin ldg type treatment is also incorporated to achieve the generalization to convection diffusion equations extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable dg methods

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

integral representations of bivariate complex geometric mean and their applications abstract in the paper the authors survey integral representations including the levy khintchine representations and applications of some bivariate means including the logarithmic mean the identric mean stolarsky s mean the harmonic mean the weighted geometric means and their reciprocals and the toader qi mean and the multivariate weighted geometric means and their reciprocals derive integral representations of bivariate complex geometric mean and its reciprocal and apply these newly derived integral representations to establish integral representations of heronian mean of power 2 and its reciprocal

improving the convergence rate of the petrov galerkin techniques for the solution of transonic and supersonic flows this paper report progress on a technique to accelerate the convergence to steady solutions when the streamline upwind petrov galerkin supg technique is used both the description of a supg formulation and the documentation of the development of a code for the finite element solution of transonic and supersonic flows are reported the aim of this work is to present a formulation to be able to treat domains of any configuration and to use the appropriate physical boundary conditions which are the major stumbling blocks of the finite difference schemes together with an appropriate convergence rate to the steady solution the implemented code has the following features the hughes supg type formulation with an oscillation free shock capturing operator adaptive refinement explicit integration with local time step and hourglassing control an automatic scheme for dealing with slip boundary conditions and a boundary augmented lumped mass matrix for speeding up convergence it is shown that the velocities at which the error is absorbed in and ejected from the domain that is damping and group velocities respectively are strongly affected by the time step used and that damping gives an o n2 algorithm contrasting with the o n one given by absorption at the boundaries nonetheless the absorbing effect is very low when very different eigenvalues are present such as in the transonic case because the stability condition imposes a too slow group velocity for the smaller eigenvalues to overcome this drawback we present a new mass matrix that provides us with a scheme having the highest group velocity attainable in all the components in section 1 we will describe briefly the theoretical background of the supg formulation in section 2 it is described how the foregoing formulation was used in the finite element code and which are the appropriate boundary conditions to be used finally in section 3 we will show some results obtained with this code

higher entropy conservation and numerical stability of compressible turbulence simulations we present a numerical formulation for the treatment of nonlinear instabilities in shock free compressible turbulence simulations the formulation is high order and contains no artificial dissipation numerical stability is enhanced through semi discrete satisfaction of global conservation properties stemming from the second law of thermodynamics and the entropy equation the numerical implementation is achieved using a conservative skew symmetric splitting of the nonlinear terms the robustness of the method is demonstrated by performing unresolved numerical simulations and large eddy simulations of compressible isotropic turbulence at a very high reynolds number results show the scheme is capable of capturing the statistical equilibrium of low mach number compressible turbulent fluctuations at infinite reynolds number comparisons with the entropy splitting technique j comput phys 162 2000 33 j comput phys 178 2002 307 staggered method j comput phys 191 2 2003 392 and skew symmetric like schemes j comput phys 161 2000 114 confirm the superiority of the current approach we also discuss a flaw in the skew symmetric splitting implemented in the literature very good results are obtained based on the proper splitting

wendel s and gautschi s inequalities refinements extensions and a class of logarithmically completely monotonic functions abstract in the article sufficient and necessary conditions that a class of functions involving ratio of euler s gamma functions and originating from wendel s and gautschi s inequalities are logarithmically completely monotonic are presented from this wendel s gautschi s kershaw s laforgia s bustoz ismail s merkle s and elezovic giordano pecaric s inequalities are refined extended and sharpened and a double inequality on the divided differences of the psi and polygamma functions is deduced straightforwardly

h adaptive stabilized finite element solver for calculation of generalized aerodynamic forces a finite element method based on the streamline upwind petrov galerkin stabilization technique is developed to solve the compressible euler equations the method is extended for computing unsteady small disturbances in the flow domain as a result of boundary motion which is applied through the linearized solid wall boundary condition the method is able to calculate the frequency dependent generalized aerodynamic forces without any need for deformation of mesh inside the flow domain error estimates are used to drive automated h refinement of mesh for steady state and frequency domain calculations steady state solutions and generalized aerodynamic forces are compared to benchmark data for transonic and supersonic mach numbers the results show that the presence of shocks in the flow makes it difficult to use a single mesh for all computations including the linearized flow the h refinement procedure is shown to be an effective way of ensuring reliable computations for the nonlinear and linearized solve

entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts framework are used to derive entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations a semi discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator the data at the boundary are weakly imposed using a penalty flux approach and a simultaneous approximation term penalty technique although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy the new boundary conditions are compatible with any diagonal norm summation by parts spatial operator including finite element finite difference finite volume discontinuous galerkin and flux reconstruction correction procedure via reconstruction schemes the proposed boundary treatment is tested for three dimensional subsonic and supersonic flows the numerical computations corroborate the non linear stability entropy stability and accuracy of the boundary conditions

a new upper bound in the second kershaw s double inequality and its generalizations in the paper a new upper bound in the second kershaw s double inequality involving ratio of gamma functions is established and as generalizations of the second kershaw s double inequality the divided differences of the psi and polygamma functions are bounded

application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes we consider the numerical solution of the shallow water equations on unstructured grids we focus on flows over wet areas the extension to the case of dry bed will be reported elsewhere the shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding in this case with the total energy our aim is to show the application of a particular class of conservative residual distribution rd schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties we give a review of conservative rd schemes showing relations between different approaches previously published and recall l stability and accuracy criteria characterizing the schemes in particular the accuracy of the rd method in presence of source terms is analyzed and conditions to construct rth order discretizations on irregular triangular grids are proved it is shown that the rd approach gives a natural way of obtaining high order discretizations which moreover preserves exactly the steady lake at rest solution independently on mesh topology nature of the variation of the bottom and polynomial order of interpolation used for the unknowns we also consider more general analytical solutions which are less investigated from the numerical view point on irregular grids linearity preserving rd schemes yield a truly second order approximation we also sketch a strategy to achieve discretizations which preserve exactly some of these solutions numerical results on steady and time dependent problems involving smooth and non smooth variations of the bottom topology show very promising features of the approach

a posteriori finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

on discretely entropy stable weight adjusted discontinuous galerkin methods curvilinear meshes abstract we construct entropy conservative and entropy stable discontinuous galerkin dg discretizations for time dependent nonlinear hyperbolic conservation laws on curvilinear meshes the resulting schemes preserve a semi discrete quadrature approximation of a continuous global entropy inequality the proof requires the satisfaction of a discrete geometric conservation law which we enforce through an appropriate polynomial approximation we extend the construction of entropy conservative and entropy stable dg schemes to the case when high order curvilinear mass matrices are approximated using low storage weight adjusted approximations and describe how to retain global conservation properties under such an approximation for certain types of curvilinear meshes these weight adjusted approximations deliver optimal rates of convergence finally the high order accuracy local conservation and discrete conservation or dissipation of entropy for under resolved solutions are verified through numerical experiments for the compressible euler equations on triangular and tetrahedral meshes

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

on discretely entropy conservative and entropy stable discontinuous galerkin methods abstract high order methods based on diagonal norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic pdes 1 2 these methods can also be interpreted as nodal discontinuous galerkin methods with diagonal mass matrices 3 4 5 6 in this work we describe how use flux differencing quadrature based projections and sbp like operators to construct discretely entropy conservative schemes for dg methods under more arbitrary choices of volume and surface quadrature rules the resulting methods are semi discretely entropy conservative or entropy stable with respect to the volume quadrature rule used numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible euler equations in one and two dimensions

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

kinetic symmetrizations and pressure laws for the euler equations abstract the symmetrizations of the compressible euler equations written as kinetic mean are studied some of their properties are pointed out we emphasize for numerical purpose the case of real rarefied gases with pressure laws p f flux splitting formulae are derived for such pressure laws

bounds of functional outputs for parabolic problems part ii bounds of the exact solution the paper introduces a methodology to compute upper and lower bounds for linear functional outputs of the exact solutions of parabolic problems in this second part the bounds account for the error both in space and time the assumption stating that the error introduced by the time marching scheme is negligible used in the first part is removed here the bounds are computed starting from an approximation of the exact solution associated with a spatial mesh and a time grid nevertheless the bounds are guaranteed with respect to the exact solution with no reference to any mesh or time discretization

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

bounds of functional outputs for parabolic problems part i exact bounds of the discontinuous galerkin time discretization classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest consequently the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh exact bounds that is bounds guaranteed with respect to the exact solution are needed to properly certify the accuracy of the results especially if the meshes are coarse the paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems in this first part the bounds account for the error associated with the spatial discretization the error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error the time discretization is performed using the discontinuous galerkin method both for the primal and adjoint problems in the error estimation procedure equilibrated fluxes at interelement edges are calculated using hybridization techniques

discretely conservative finite difference formulations for nonlinear conservation laws in split form theory and boundary conditions the lax wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities the discrete operator however need not be derived from the divergence form of the continuous equations indeed conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal norm skew symmetric summation by parts sbp spatial operator yield discrete operators that are conservative furthermore split form discretely conservation operators can be derived for periodic or finite domain sbp spatial operators of any order examples are presented of a fourth order sbp finite difference operator with second order boundary closures sixth and eighth order constructions are derived and are supplied in an accompanying text file

a complete monotonicity property of the gamma function abstract a logarithmically completely monotonic function is completely monotonic the function 1 ln x 1 x ln x 1 is strictly completely monotonic on 0 the function x 1 x x is strictly logarithmically completely monotonic on 0

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

construction of stabilization operators for galerkin least squares discretizations of compressible and incompressible flows the design and analysis of a class of stabilization operators suitable for space time galerkin least squares finite element discretizations of the symmetrized compressible navier stokes equations is discussed the obtained stabilization matrix is well defined in the incompressible limit and reduces to the matrix described in m polner j j w van der vegt and r m j van damme analysis of stabilization operators for galerkin least squares discretizations of the incompressible navier stokes equations comput meth appl mech eng 195 2006 982 1006 the stabilization matrix is investigated to give necessary and sufficient conditions for its positive definiteness under these conditions on the stabilization matrix the galerkin least squares method for the symmetrized compressible navier stokes equations satisfies the entropy condition which ensures global nonlinear stability of the discretization

a multidomain spectral method for supersonic reactive flows this paper has a dual purpose it presents a multidomain chebyshev method for the solution of the two dimensional reactive compressible navier stokes equations and it reports the results of the application of this code to the numerical simulations of high mach number reactive flows in recessed cavity the computational method utilizes a newly derived interface boundary conditions as well as an adaptive filtering technique to stabilize the computations the results of the simulations are relevant to recessed cavity flame holders

discontinuous galerkin method for navier stokes equations using kinetic flux vector splitting kinetic schemes for compressible flow of gases are constructed by exploiting the connection between boltzmann equation and the navier stokes equations this connection allows us to construct a flux splitting for the navier stokes equations based on the direction of molecular motion from which a numerical flux can be obtained the naive use of such a numerical flux function in a discontinuous galerkin dg discretization leads to an unstable scheme in the viscous dominated case stable schemes are constructed by adding additional terms either in a symmetric or non symmetric manner which are motivated by the dg schemes for elliptic equations the novelty of the present scheme is the use of kinetic fluxes to construct the stabilization terms in the symmetric case interior penalty terms have to be added for stability and the resulting schemes give optimal convergence rates in numerical experiments the non symmetric schemes lead to a cell energy entropy inequality but exhibit sub optimal convergence rates these properties are studied by applying the schemes to a scalar convection diffusion equation and the 1 d compressible navier stokes equations in the case of navier stokes equations entropy variables are used to construct stable schemes

a class of logarithmically completely monotonic functions and application to the best bounds in the second gautschi kershaw s inequality in this article the logarithmically complete monotonicity of the function c x b c x a 1 a b exp j x c are discussed where a b c are real numbers and c is the classical euler s gamma function from this the best upper and lower bounds for walls ratio c x 1 c x s are established which refine the second gautschi kershaw s inequality

upwind schemes and boundary conditions with applications to euler equations in general geometries abstract a unified treatment of several upwind shock capturing algorithms is presented each algorithm has a riemann initial value problem as its basis the treatment of boundaries involves solving the associated riemann initial boundary value problem the first author s algorithm applied to multidimensional euler equations in general geometries is then presented its worth is verified by various calculations which include mach 8 supersonic flow past a circular cylinder

skew selfadjoint form for systems of conservation laws abstract hyperbolic systems of conservation laws augumented with an entropy inequality are studied it is shown that such systems can be written in a quasilinear skew selfadjoint form centered differencing of such a form under the smooth regime ends up with a systematic recipe for constructing quasiconservative schemes where the global entropy conservation is recovered employing the above formulation in bounded regions under the nonsmooth regime as well a local entropy decay estimate is also concluded examples of the shallow water and the full gasdynamics equations are explicitly treated

non weak strong solutions in gas dynamics a c11 p version stlsfef in lagrangian frame of reference using ρ u p primitive variables in this paper we present computations of the non weak solutions of class c11 of transient navier stokes equations for compressible flow in lagrangian frame of reference using space time least squares finite element formulation with primitive variables u p for high speed compressible flows the solutions reported here possess the same orders of continuity as the governing differential equations the role of diffusion i e viscosity physical or artificial and thermal conductivity on shock structure is demonstrated compression of air in a rigid cylinder by a rigid massless and frictionless piston is used as a model problem true time evolutions of class c11 are reported beginning with the first time step until steady shock conditions are achieved comparisons with analytical solutions are presented when possible copyright 2001 john wiley sons ltd

a stable galerkin reduced order model for coupled fluid structure interaction problems summary a stable reduced order model rom of a linear fluid structure interaction fsi problem involving linearized compressible inviscid flow over a flat linear von karman plate is developed separate stable roms for each of the fluid and the structure equations are derived both roms are built using the continuous galerkin projection approach in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product the mode shapes for the structure rom are the eigenmodes of the governing linear plate equation the fluid rom basis is constructed via the proper orthogonal decomposition for the linearized compressible euler fluid equations a symmetry transformation is required to obtain a stable formulation of the galerkin projection step in the model reduction procedure stability of the galerkin projection of the structure model in the standard l2 inner product is shown the fluid and structure roms are coupled through solid wall boundary conditions at the interface plate boundary an a priori energy linear stability analysis of the coupled fluid structure system is performed it is shown that under some physical assumptions about the flow field the fsi rom is linearly stable a priori if a stabilization term is added to the fluid pressure loading on the plate the stability of the coupled rom is studied in the context of a test problem of inviscid supersonic flow past a thin square elastic rectangular panel that will undergo flutter once the non dimensional pressure parameter exceeds a certain threshold this a posteriori stability analysis reveals that the fsi rom can be numerically stable even without the addition of the aforementioned stabilization term moreover the rom constructed for this problem properly predicts the maintenance of stability below the flutter boundary and gives a reasonable prediction for the instability growth rate above the flutter boundary copyright 2013 john wiley sons ltd

a finite element formulation for the hydrodynamic semiconductor device equations abstract a new formulation employing the galerkin least squares finite element method is presented for the simulation of the hydrodynamic model of semiconductor devices numerical simulations are performed on the coupled poisson and hydrodynamic equations for one carrier devices the hydrodynamic equations for a single carrier i e for the electrons or holes resemble the compressible navier stokes equations with the addition of highly nonlinear source terms and without the viscous terms the governing equations are nondimensionalized to improve the conditioning on the resulting system of equations and the efficiency of the numerical algorithms furthermore to establish the stability of the discrete solution the system of hydrodynamic equations is symmetrized by considering generalized entropy functions a staggered solution strategy is employed to treat the coupled hydrodynamic and poisson equations numerical results are presented for one dimensional and two dimensional one carrier n n n devices the presence of velocity overshoot has been observed and it is recognized that the heat flux term plays an important role in the simulation of semiconductor devices employing the hydrodynamic model

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

simultaneous approximation term based boundary discretization for moment equations of rarefied gas dynamics abstract previous works have developed boundary conditions that lead to the l 2 boundedness of solutions to the linearised moment equations here we present a spatial discretization that preserves the l 2 stability by recovering integration by parts over the discretized domain and by imposing boundary conditions using a simultaneous approximation term sat we develop three different forms of the sat using i characteristic splitting of moment equation s boundary conditions ii decoupling of moments in moment equations and iii characteristic splitting of boltzmann equation s boundary conditions we discuss how the first two forms differ in terms of their usage and implementation we show that the third form is equivalent to using an upwind kinetic numerical flux along the boundary and we argue that even though it provides stability it prescribes the incorrect number of boundary conditions using benchmark problems we compare the accuracy of moment solutions computed using different sats our numerical experiments also provide new insights into the convergence of moment approximations to the boltzmann equation s solution

viscous regularization of the euler equations and entropy principles this paper investigates a general class of viscous regularizations of the compressible euler equations a unique regularization is identified that is compatible with all the generalized entropies la harten et al siam j numer anal 35 1998 pp 2117 2127 and satisfies the minimum entropy principle a connection with a recently proposed phenomenological model by h brenner phys a 370 2006 pp 190 224 is made

a positivity preserving finite element finite volume solver for the spalart allmaras turbulence model we propose a stable numerical method preserving the positivity of the turbulent viscosity in the spalart allmaras model focusing in particular on the stiffness of the source term the goal is to apply this model for turbulent flows using an effective numerical method based on a mixed fe fv method a mathematical analysis of stability and positivity of the continuous and numerical models is provided numerical tests on the 3d boundary layer over a flat plate show the convergence and robustness of the proposed method confirming the theoretical analysis 2006 elsevier b v all rights reserved

some extensions of w gautschi s inequalities for the gamma function it has been shown by w gautschi that if 0 i xi s f x exp i s x 1 the following closer bounds are proved exp i s 4 x 12 f exp i s x s i f x s 2 and x 2 t x 2 s 4 these are compared with each other and with inequalities given by t erber and j d keckic and p m vasic

entropy splitting for high order numerical simulation of compressible turbulence a stable high order numerical scheme for direct numerical simulation dns of shock free compressible turbulence is presented the method is applicable to general geometries it contains no upwinding artificial dissipation or filtering instead the method relies on the stabilizing mechanisms of an appropriate conditioning of the governing equations and the use of compatible spatial difference operators for the interior points interior scheme as well as the boundary points boundary scheme an entropy splitting approach splits the inviscid flux derivatives into conservative and nonconservative portions the spatial difference operators satisfy a summation by parts condition leading to a stable scheme combined interior and boundary schemes for the initial boundary value problem using a generalized energy estimate a laplacian formulation of the viscous and heat conduction terms on the right hand side of the navier stokes equations is used to ensure that any tendency to odd even decoupling associated with central schemes can be countered by the fluid viscosity the resulting methods are able to minimize the spurious high frequency oscillations associated with pure central schemes especially for long time integration applications such as dns for validation purposes the methods are tested in a dns of compressible turbulent plane channel flow at low values of friction mach number where reference turbulence data bases exist it is demonstrated that the methods are robust in terms of grid resolution and in good agreement with published channel data accurate turbulence statistics can be obtained with moderate grid sizes stability limits on the range of the splitting parameter are determined from numerical tests

entropy stable discontinuous interfaces coupling for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts sbp framework are used to derive entropy stable interior interface coupling for the semi discretized three dimensional 3d compressible navier stokes equations a complete semidiscrete entropy estimate for the interior domain is achieved combining a discontinuous entropy conservative operator of any order 1 2 with an entropy stable coupling condition for the inviscid terms and a local discontinuous galerkin ldg approach with an interior penalty ip procedure for the viscous terms the viscous penalty contributions scale with the inverse of the reynolds number re so that for re their contributions vanish and only the entropy stable inviscid interface penalty term is recovered this paper extends the interface couplings presented 1 2 and provides a simple and automatic way to compute the magnitude of the viscous ip term the approach presented herein is compatible with any diagonal norm summation by parts sbp spatial operator including finite element finite volume finite difference schemes and the class of high order accurate methods which include the large family of discontinuous galerkin discretizations and flux reconstruction schemes this note relies on the formalism introduced in 1 3 and complements the new class of interior entropy stable sbp operators of any order for the 3d compressible navier stokes equations on unstructured grids that was proposed in 1 2 to keep the notation as simple as possible a uniform cartesian grid is considered in the derivation however the extension to generalized curvilinear coordinates and unstructured grids follows immediately if the transformation from computational to physical space preserves the semi discrete geometric conservation 4 the proposed interface coupling technique has been successfully combined with a high order entropy stable discretization for the simulation of two dimensional 2d and 3d viscous subsonic and supersonic flows presented in 1 5 6

analysis of stabilization operators for galerkin least squares discretizations of the incompressible navier stokes equations in this paper we discuss the design and analysis of a class of stabilization operators for space time galerkin least squares finite element discretizations suitable for the incompressible limit of the symmetrized navier stokes equations given in g hauke t j r hughes a comparative study of different sets of variables for solving compressible and incompressible flows comput methods appl mech engrg 153 1998 1 44 this set of equations consists of the incompressible navier stokes equations in combination with the heat equation the analysis results in stabilization operators which are positive definite and dimensionally consistent in addition a detailed proof is given that the space time galerkin least squares discretization together with the proposed stabilization operators satisfies a coercivity condition for the linearized form of the equations this ensures that necessary conditions for uniqueness and stability of the numerical solution are satisfied by the finite element discretization

some inequalities and monotonicity properties associated with the gamma and psi functions abstract in this paper we present new completely monotonic functions and bernstein functions relating to the gamma and psi functions as consequences we establish new upper and lower bounds for the gamma function in terms of function also we present some estimates for stirling s formula remainder

a posteriori finite element output bounds for the incompressible navier stokes equations application to a natural convection problem abstract we present a new neumann subproblem a posteriori finite element procedure for the efficient calculation of rigorous constant free sharp lower and upper bounds for linear and nonlinear functional outputs of the incompressible navier stokes equations we first formulate the bound procedure we derive and discuss a bound error expression and we then demonstrate the capabilities of the method with numerical results obtained for natural convection problems we also implement an optimal adaptive refinement strategy based on a local elemental decomposition of the bound gap

fully discrete explicit locally entropy stable schemes for the compressible euler and navier stokes equations abstract recently relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation here we generalize this approach to guarantee local entropy inequalities for finitely many convex functionals entropies and apply the resulting methods to the compressible euler and navier stokes equations based on the unstructured h p adaptive ssdc framework of entropy conservative or dissipative semidiscretizations using summation by parts and simultaneous approximation term operators we develop the first discretizations for compressible computational fluid dynamics that are primary conservative locally entropy stable in the fully discrete sense under a usual cfl condition explicit except for the parallelizable solution of a single scalar equation per element and arbitrarily high order accurate in space and time we demonstrate the accuracy and the robustness of the fully discrete explicit locally entropy stable solver for a set of test cases of increasing complexity

optimal time splitting for two and three dimensional navier stokes equations with mixed derivatives a new explicit time splitting algorithm has been developed for finite difference modelling of the full two and three dimensional time dependent compressible viscous navier stokes equations of fluid mechanics the scheme is optimal in the sense that the split operators achieve their maximum allowable time step i e the corresponding courant number the algorithm allows a conservation form formulation stability is proven analytically and verified numerically in proving stability it was shown that all nine matrix coefficients of the navier stokes equations are simultaneously symmetrizable by a similarity transformation two such transformations and their resulting symmetric matrix coefficients are presented explicitly

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations

an entropy stable nodal discontinuous galerkin method for the resistive mhd equations part i theory and numerical verification abstract the first paper of this series presents a discretely entropy stable discontinuous galerkin dg method for the resistive magnetohydrodynamics mhd equations on three dimensional curvilinear unstructured hexahedral meshes compared to other fluid dynamics systems such as the shallow water equations or the compressible navier stokes equations the resistive mhd equations need special considerations because of the divergence free constraint on the magnetic field for instance it is well known that for the symmetrization of the ideal mhd system as well as the continuous entropy analysis a non conservative term proportional to the divergence of the magnetic field typically referred to as the powell term must be included as a consequence the mimicry of the continuous entropy analysis in the discrete sense demands a suitable dg approximation of the non conservative terms in addition to the ideal mhd terms this paper focuses on the resistive mhd equations our first contribution is a proof that the resistive terms are symmetric and positive definite when formulated in entropy space as gradients of the entropy variables which enables us to show that the entropy inequality holds for the resistive mhd equations this continuous analysis is the key for our dg discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality typically termed entropy stability our second contribution is a detailed derivation and analysis of the discretization on three dimensional curvilinear meshes the discrete analysis relies on the summation by parts property which is satisfied by the dg spectral element method dgsem with legendre gauss lobatto lgl nodes although the divergence free constraint is included in the non conservative terms the resulting method has no particular treatment of the magnetic field divergence errors which might pollute the solution quality our final contribution is the extension of the standard resistive mhd equations and our dg approximation with a divergence cleaning mechanism that is based on a generalized lagrange multiplier glm as a conclusion to the first part of this series we provide detailed numerical validations of our dgsem method that underline our theoretical derivations in addition we show a numerical example where the entropy stable dgsem demonstrates increased robustness compared to the standard dgsem