implicit high order time marching schemes for the linearized euler equations high order accurate implicit time marching methods are presented for discontinuous galerkin and spectral volume high order accurate spatial discretizations of the linearized euler equations that govern propagation of aeroacoustic disturbances it is found that despite the additional computational time that is required for the solution of the large sparse linear system for the degrees of freedom of high order spectral volume or discontinuous galerkin discretizations implicit methods offer an attractive alternative for practical aeroacoustic computations with these high order methods several explicit and implicit methods for time advancement are tested the advantages of implicit higher order accurate in time methods that involve multiple implicit steps are demonstrated the efficiency and accuracy of time marching methods is evaluated for test problems with exact solutions it is shown that complex domain aeroacoustic predictions can be obtained at a reduced computational cost with the use of high order accurate implicit methods

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

transient dynamics of three dimensional beam trusses using higher order kinematics spatial structures are often subjected to impulse loads which induce high frequency hf wave propagations despite some recent researches the characterization of the transient response to such loads remains an open problem the objective of this research is to develop a reliable model of the hf energy evolution within three dimensional beam trusses in order to predict for example their potential steady state behavior at late times or the energy paths the theory of micro local analysis of wave systems shows that the energy density associated with their hf solutions satisfies a liouville type transport equation a suitable hf transport model for beams is derived from the spectrum relations for lamb waves in the hf range at the interfaces between substructures the energy flow is partly reflected and partly transmitted the corresponding reflection transmission coefficients are also derived in this study numerical simulations are performed by a spectral discountinuous galerkin dg method for spatial

higher rank numerical ranges and compression problems we consider higher rank versions of the standard numerical range for matrices a central motivation for this investigation comes from quantum error correction we develop the basic structure theory for the higher rank numerical ranges and give a complete description in the hermitian case we also consider associated projection compression problems

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

insights on the coercivity of the esfr methods for elliptic problems abstract the flux reconstruction approach is a recent high order method which has been introduced for unsteady problems initial energy stability has been conducted for the advection problem leading to the well know energy stable flux reconstruction esfr scheme using the esfr scheme the energy stability proof has been extended for the advection diffusion using the local discontinuous galerkin ldg numerical flux recently stability conditions were derived for the compact interior penalty ip and bassi rebay ii br2 numerical fluxes here we apply esfr schemes to elliptic problems and derive the associated bilinear form for the poisson equation we show that for the compact ip and br2 numerical fluxes the bilinear form is independent of the auxiliary correction function finally we provide some insights on the coercivity of the esfr scheme

globally constraint preserving fr dg scheme for maxwell s equations at all orders abstract computational electrodynamics ced the numerical solution of maxwell s equations plays an incredibly important role in several problems in science and engineering high accuracy solutions are desired and the discontinuous galerkin dg method is one of the better ways of delivering high accuracy in numerical ced maxwell s equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable balsara and kappeli 2019 presented a von neumann stability analysis of globally constraint preserving dg schemes for ced up to fourth order that paper was focused on developing the theory and documenting the superior dissipation and dispersion of dgtd schemes in media with constant permittivity and permeability in this paper we present working dgtd schemes for ced that go up to fifth order of accuracy and analyze their performance when permittivity and permeability vary strongly in space our dgtd schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh our first finding is that at fourth and higher orders of accuracy one has to evolve some zone centered modes in addition to the face centered modes it is well known that the limiting step in dg schemes causes a reduction of the optimal accuracy of the scheme though the schemes still retain their formal order of accuracy with weno type limiters in this paper we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the dg scheme this very favorable second finding ensures that dgtd schemes retain optimal accuracy even in the presence of large spatial variations in permittivity and permeability we also study the conservation of electromagnetic energy in these problems our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space as long as the conductivity is zero

superconvergence of discontinuous galerkin and local discontinuous galerkin methods eigen structure analysis based on fourier approach various superconvergence properties of discontinuous galerkin dg and local dg ldg methods for linear hyperbolic and parabolic equations have been investigated in the past due to these superconvergence properties dg and ldg methods have been known to provide good wave resolution properties especially for long time integrations zhong and shu 2011 26 in this paper under the assumption of uniform mesh and via fourier approach we observe that the error of the dg or ldg solution can be decomposed into three parts 1 dissipation and dispersion errors of the physically relevant eigenvalue this part of error will grow linearly in time and is of order 2 k 1 for dg method and 2 k 2 for ldg method 2 projection error there exists a special projection of the exact solution such that the numerical solution is much closer to this special projection than the exact solution itself this part of error will not grow in time 3 the dissipation of non physically relevant eigenvectors this part of error will be damped exponentially fast with respect to the spatial mesh size x along this line we analyze the error for a fully discrete runge kutta rk dg scheme a collection of numerical examples for linear equations are presented to verify our observations above we also provide numerical examples based on non uniform mesh nonlinear burgers equation and high dimensional maxwell equations to explore superconvergence properties of dg methods in a more general setting

discontinuous galerkin spectral element approximations on moving meshes we derive and evaluate high order space arbitrary lagrangian eulerian ale methods to compute conservation laws on moving meshes to the same time order as on a static mesh we use a discontinuous galerkin spectral element method dgsem in space and one of a family of explicit time integrators such as adams bashforth or low storage explicit runge kutta the approximations preserve the discrete metric identities and the discrete geometric conservation law dgcl by construction we present time step refinement studies with moving meshes to validate the approximations the test problems include propagation of an electromagnetic gaussian plane wave a cylindrical pressure wave propagating in a subsonic flow and a vortex convecting in a uniform inviscid subsonic flow each problem is computed on a time deforming mesh with three methods used to calculate the mesh velocities from exact differentiation from the integration of an acceleration equation and from numerical differentiation of the mesh position

minimal and maximal operator spaces and operator systems in entanglement theory abstract we examine k minimal and k maximal operator spaces and operator systems and investigate their relationships with the separability problem in quantum information theory we show that the matrix norms that define the k minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k positive linear maps and bound entanglement similarly we investigate the k super minimal and k super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k block positive operators and unnormalized states with schmidt number no greater than k respectively we characterize a class of norms on the k super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the schmidt number of a quantum state that generalizes the recently developed separability criterion based on trace contractive maps

detecting quantum entanglement we review the criteria for separability and quantum entanglement both in a bipartite as well as a multipartite setting we discuss bell inequalities entanglement witnesses entropic inequalities bound entanglement and several features of multipartite entanglement we indicate how these criteria bear on the experimental detection of quantum entanglement

quantum gate fidelity in terms of choi matrices we provide new results for computing and comparing the quantum gate fidelity of quantum channels via their choi matrices we extend recent work that showed that there exist non dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels we use our characterization to show that when the dimension is 2 or 3 under slightly stronger hypotheses the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual a fact that has been shown to be false when the dimension is 4 or larger we also present a formula for the minimum gate fidelity of a channel in terms of a well studied norm on a compression of its choi matrix as a consequence several new ways of bounding and approximating the minimum gate fidelity follow including a simple semidefinite program to compute it for qubit channels

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

spatial and spectral superconvergence of discontinuous galerkin method for hyperbolic problems in this paper we analyze the spatial and spectral superconvergence properties of one dimensional hyperbolic conservation law by a discontinuous galerkin dg method the analyses combine classical mathematical arguments with matlab experiments some properties of the dg schemes are discovered using discrete fourier analyses superconvergence of the numerical wave numbers radau structure of the x spatial error

a symmetric trefftz dg formulation based on a local boundary element method for the solution of the helmholtz equation a general symmetric trefftz discontinuous galerkin method is built for solving the helmholtz equation with piecewise constant coefficients the construction of the corresponding local solutions to the helmholtz equation is based on a boundary element method a series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters and more importantly its outstanding ability to reduce the instabilities known as the pollution effect in the literature on numerical simulations of long range wave propagation

the generation of triangular meshes for nurbs enhanced fem summary this paper presents the first method that enables the fully automatic generation of triangular meshes suitable for the so called non uniform rational b spline nurbs enhanced finite element method nefem the meshes generated with the proposed approach account for the computer aided design boundary representation of the domain given by nurbs curves the characteristic element size is completely independent of the geometric complexity and of the presence of very small geometric features the proposed strategy allows to circumvent the time consuming process of de featuring complex geometric models before a finite element mesh suitable for the analysis can be produced a generalisation of the original definition of a nefem element is also proposed enabling to treat more complicated elements with an edge defined by several nurbs curves or more than one edge defined by different nurbs three examples of increasing difficulty demonstrate the applicability of the proposed approach and illustrate the advantages compared with those of traditional finite element mesh generators finally a simulation of an electromagnetic scattering problem is considered to show the applicability of the generated meshes for finite element analysis 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

bernstein bezier weight adjusted discontinuous galerkin methods for wave propagation in heterogeneous media abstract this paper presents an efficient discontinuous galerkin method to simulate wave propagation in heterogeneous media with sub cell variations this method is based on a weight adjusted discontinuous galerkin method wadg which achieves high order accuracy for arbitrary heterogeneous media 1 however the computational cost of wadg grows rapidly with the order of approximation in this work we propose a bernstein bezier weight adjusted discontinuous galerkin method bbwadg to address this cost by approximating sub cell heterogeneities by a fixed degree polynomial the main steps of wadg can be expressed as polynomial multiplication and l 2 projection which we carry out using fast bernstein algorithms the proposed approach reduces the overall computational complexity from o n 2 d to o n d 1 in d dimensions numerical experiments illustrate the accuracy of the proposed approach and computational experiments for a gpu implementation of bbwadg verify that this theoretical complexity is achieved in practice

spatial eigenanalysis of spectral hp continuous galerkin schemes and their stabilisation via dg mimicking spectral vanishing viscosity for high reynolds number flows abstract this study considers the spatial eigensolution analysis of spectral hp continuous galerkin cg schemes complementing a recent work by moura et al 2016 15 which addressed cg s temporal analysis while the latter assumes periodic boundary conditions the spatial approach presumes inflow outflow type conditions and therefore provides insights for a different class of problems the linear advection diffusion problem is considered for a wide range of peclet numbers allowing for viscous effects at different intensities the inviscid linear advection case receives particular attention owing to the manifestation of peculiar characteristics previously observed for discontinuous galerkin dg schemes in the limit of strong over upwinding these effects are discussed in detail due to their potential to negatively affect solution quality and numerical stability of under resolved simulations at high reynolds numbers the spectral vanishing viscosity svv technique is subsequently considered as a natural stabilization strategy in the context of linear advection an optimization procedure is employed to match svv diffusion levels to those of dg at appropriate polynomial orders the resulting cg svv discretisations are tested against under resolved computations of spatially developing vortex dominated flows and display excellent robustness at high reynolds numbers along with superior eddy resolving characteristics at higher polynomial orders this highlights the importance of appropriate stabilization techniques to improve the potential of spectral hp cg methods for high fidelity simulations of transitional and turbulent flows including implicit les under resolved dns approaches

efficient implementation of tam and auriault s time domain impedance boundary condition a new and efficient time domain implementation is proposed for the impedance model originally introduced by tam and auriault tam c k w and auriault l time domain impedance boundary conditions for computational aeroacoustics aiaa journal vol 34 no 5 may 1996 pp 917 923 in the frequency domain the pressure can be calculated from the product of the impedance with the normal velocity in the time domain a convolution is needed which may become a costly operation present solutions for this problem involve either a full convolution and storage of a time history of data or a translation of the frequency domain models to time domain formulations involving time derivatives the former approach is very costly in terms of memory and cpu time the latter makes the implementation discretization specific and vulnerable to stability issues a new formulation based on recursive convolution results in an efficient implementation that stores only one accumulator which can be updated at each time step with minimal addition and multiplication it allows for the exact representation of the impedance at a given frequency the formulation is validated by a comparison with the measurement data from the nasa langley grazing flow impedance tube the simulations are performed by introducing a plane wave at a single frequency in the tube and this is done for six frequencies and five different mean flow speeds

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

a family of norms with applications in quantum information theory we consider a family of vector and operator norms defined by the schmidt decomposition theorem for quantum states we use these norms to tackle two fundamental problems in quantum information theory the classification problem for k positive linear maps and entanglement witnesses and the existence problem for nonpositive partial transpose bound entangled states we begin with an analysis of the norms showing that the vector norms can be explicitly calculated and we derive several inequalities in order to bound the operator norms and compute them in special cases we then use the norms to establish what appears to be the most general spectral test for k positivity currently available showing how it implies several other known tests as well as some new ones building on this work we frame the nonpositive partial transpose bound entangled problem as a concrete problem on a specific limit specifically that a particular entangled werner state is bound entangled if and only if a certain norm inequality ho

on the valid frequency range of timoshenko beam theory abstract the frequency equation of timoshenko beam theory factorises for hinged hinged end conditions leading to a first and second spectrum of natural frequencies the latter is largely inaccurate and can be isolated and disregarded for the majority of other end conditions when the frequency equation does not factorise one may think in terms of pseudo second spectrum contributions arising when evanescent waves become propagating above the cut off frequency co ag i and it is conjectured that these may have a corrupting effect on the frequency predictions comparisons with measured and simulated frequencies lead to the conclusion that timoshenko predictions above the cut off frequency should be disregarded for those end conditions for which the frequency equation does not factorise

discontinuous galerkin methods for the vlasov maxwell equations discontinuous galerkin methods are developed for solving the vlasov maxwell system methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy such properties in general are hard to achieve within other numerical method frameworks for simulating the vlasov maxwell system the proposed scheme employs discontinuous galerkin discretizations for both the vlasov and the maxwell equations resulting in a consistent description of the distribution function and electromagnetic fields it is proven up to some boundary effects that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the maxwell equations and the underlying approximation spaces error estimates are established for several flux choices the scheme is tested on the streaming weibel instability the order of accuracy and conservation properties of the proposed method are verified

grid dispersion analysis of plane square biquadratic serendipity finite elements in transient elastodynamics summary the spatial discretisation of a continuum by the fem introduces dispersion errors to the numerical solution of stress wave propagation errors of phase and group velocities and the scatter of wave propagation are induced when these propagating phenomena are modelled by the fem the speed of a single harmonic wave depends on its frequency parasitic effects do not exist in an ideal unbounded continuum with higher order finite elements there are optical modes in the spectrum resulting in spurious oscillations of stress and velocity distributions near the sharp wavefronts in principle the dispersion errors and the spurious oscillations cannot be removed from these solutions but can be suppressed to a certain extent for reliable numerical solutions by the fem the dispersion errors should be analysed and determined the dispersion properties of a plane square biquadratic serendipity finite element are examined and compared with a bilinear one dispersion analysis is carried out for the consistent and lumped mass matrices a dispersive improved lumped mass matrix for biquadratic serendipity elements is proposed the paper closes with the recommendation of a choice of permissible dimensionless wavelengths for bilinear and biquadratic serendipity finite element meshes copyright 2013 john wiley sons ltd

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

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

high order discontinuous galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes abstract we propose and analyze a high order discontinuous galerkin finite element method for the approximate solution of wave propagation problems modeled by the elastodynamics equations on computational meshes made by polygonal and polyhedral elements we analyze the well posedness of the resulting formulation prove h p version error a priori estimates and present a dispersion analysis showing that polygonal meshes behave as classical simplicial quadrilateral grids in terms of dispersion properties the theoretical estimates are confirmed through various two dimensional numerical verifications

a free energy stable nodal discontinuous galerkin approximation with summation by parts property for the cahn hilliard equation abstract we present a nodal discontinuous galerkin dg scheme for the cahn hilliard equation that satisfies the summation by parts simultaneous approximation term sbp sat property the latter permits us to show that the discrete free energy is bounded and as a result the scheme is provably stable the scheme and the stability proof are presented for general curvilinear three dimensional hexahedral meshes we use the bassi rebay 1 br1 scheme to compute interface fluxes and a first order implicit explicit imex scheme to integrate in time we provide a semi discrete stability study and a fully discrete proof subject to a positivity condition on the solution lastly we test the theoretical findings using numerical cases that include two and three dimensional problems

spectral conditions for positive maps we provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions this construction generalizes the celebrated choi example of a map which is positive but not completely positive it is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements such maps provide basic tools to study quantum entanglement in multipartite systems

high order cut discontinuous galerkin methods with local time stepping for acoustics we propose a method to solve the acoustic wave equation on an immersed domain using the hybridizable discontinuous galerkin method for spatial discretization and the arbitrary derivative method wit

a gpu accelerated level set reinitialization for an adaptive discontinuous galerkin method gpu accelerated high order reconstruction of signed distance function of the level set method is studied the flow based reinitialization equation is discretized in space by using a nodal discontinuous galerkin method on adaptive unstructured grids artificial diffusion with a modal decay rate based regularity estimator is used to damp out high frequency solution components near kinks where mesh adaptivity is applied a two rate adams bashforth time integrator is developed to avoid time step restrictions resulting from artificial diffusion stabilization and local mesh refinement platform independence of the solver is achieved by using an extensible multi threading programming api that allows runtime selection of different computing devices gpu and cpu and threading interfaces cuda opencl and openmp overall a highly scalable numerical scheme that preserves the simplicity of the original level set method is obtained performance and accuracy of the method to construct signed distance function on highly disturbed initial data with smooth and non smooth interfaces are tested through distinct two and three dimensional problems

dispersion analysis of finite difference and discontinuous galerkin schemes for maxwell s equations in linear lorentz media abstract in this paper we consider maxwell s equations in linear dispersive media described by a single pole lorentz model for electronic polarization we study two classes of commonly used spatial discretizations finite difference methods fd with arbitrary even order accuracy in space and high spatial order discontinuous galerkin dg finite element methods both types of spatial discretizations are coupled with second order semi implicit leap frog and implicit trapezoidal temporal schemes by performing detailed dispersion analysis for the semi discrete and fully discrete schemes we obtain rigorous quantification of the dispersion error for lorentz dispersive dielectrics in particular comparisons of dispersion error can be made taking into account the model parameters and mesh sizes in the design of the two types of schemes this work is a continuation of our previous research on energy stable numerical schemes for nonlinear dispersive optical media 6 7 the results for the numerical dispersion analysis of the reduced linear model considered in the present paper can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models the numerical dispersion analysis of the fully discrete fd and dg schemes for the dispersive maxwell model considered in this paper clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations their order of accuracy mesh discretization parameters and model parameters the results obtained here cannot be arrived at by considering discretizations of maxwell s equations in free space in particular our results contrast the advantages and disadvantages of using high order fd or dg schemes and leap frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors finally we highlight the limitations of the second order accurate temporal discretizations considered

a discontinuous galerkin method for poroelastic wave propagation the two dimensional case abstract in this paper we consider a high order discontinuous galerkin dg method for modelling wave propagation in coupled poroelastic elastic media the upwind numerical flux is derived as an exact solution for the riemann problem including the poroelastic elastic interface attenuation mechanisms in both biot s low and high frequency regimes are considered the current implementation supports non uniform basis orders which can be used to control the numerical accuracy element by element in the numerical examples we study the convergence properties of the proposed dg scheme and provide experiments where the numerical accuracy of the scheme under consideration is compared to analytic and other numerical solutions

taming the cfl number for discontinuous galerkin methods on structured meshes the upwind discontinuous galerkin method is an attractive method for solving time dependent hyperbolic conservation laws it is possible to use high order explicit time stepping methods and high order spatial approximations without incurring heavy numerical linear algebra overheads however the courant friedrichs lewy cfl condition for these methods depends on the polynomial order used and there is a somewhat excessive cost for using very high order spatial approximation we discuss the impact of a covolume mesh based filter on the cfl number for these methods and present an algorithm which has a cfl number independent of the spatial order of approximation we present computational results for the advection equation and the wave equation on one dimensional meshes using up to tenth order in space and time

analysis of a spectral galerkin approximation to the helmholtz equation in exterior domains an error analysis is presented for the spectral galerkin method to the helmholtz equation in 2 and 3 dimensional exterior domains the problem in unbounded domains is first reduced to a problem on a bounded domain via the dirichlet to neumann operator and then a spectral galerkin method is employed to approximate the reduced problem the error analysis is based on exploring delicate asymptotic behaviors of the hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems explicit error bounds with respect to the wave number are derived and some illustrative numerical examples are also presented

dispersive behaviour of high order finite element schemes for the one way wave equation we study the ability of high order numerical methods to propagate discrete waves at the same speed as the physical waves in the case of the one way wave equation a detailed analysis of the finite element method is presented including an explicit form for the discrete dispersion relation and a complete characterisation of the numerical bloch waves admitted by the scheme a comparison is made with the spectral element method and the discontinuous galerkin method with centred fluxes it is shown that all schemes admit a spurious mode the spectral element method is always inferior to the finite element and discontinuous galerkin schemes a somewhat surprising result in view of the fact that in the case of the second order wave equation the spectral element method propagates waves with an accuracy superior to that of the finite element scheme the comparative behaviour of the finite element and discontinuous galerkin scheme is also somewhat surprising the accuracy of the finite element method is superior to that of the discontinuous galerkin method in the case of elements of odd order by two orders of accuracy but worse again by two orders of accuracy in the case of elements of even order

high order discontinuous galerkin schemes on general 2d manifolds applied to the shallow water equations an innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous galerkin method this new approach combines the advantages of the usual approaches described in the literature the vectorial fields are expressed in a unit non orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements while the convective flux functions are written is the usual 3d cartesian coordinate system the number of vectorial components is therefore minimum and the tangency constraint is naturally ensured while the method remains robust and general since not relying on a particular parametrization of the manifold the discontinuous galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition the possible discontinuities of this basis are then taken into account in the riemann solver on inter element interfaces the approach is validated on the sphere using the shallow water equations for computing standard atmospheric benchmarks in particular the williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error the propagation of gravity waves is eventually computed on non conventional irregular curved manifolds to illustrate the robustness and generality of the method

on consistency and rate of convergence of flux reconstruction for time dependent problems this study is directed at a rigorous characterization of the consistency and convergence of discontinuous finite element schemes formulated using flux reconstruction fr we show that the fr formulation is consistent for linear advection and converges to the exact solution for any scheme that is stable in the von neumann sense the numerical solution for a scheme of polynomial order p is composed of p 1p 1 eigenmodes of which one and exactly one is physical such that it exhibits the analytical dispersion behavior in the limit of asymptotic grid resolution the remaining p modes are spurious such that the fraction of energy received by them from the initial condition vanishes in the asymptotic limit on grid refinement the rate of convergence of the numerical solution is a function of time starting from a short time rate at t 0 t 0 associated with interpolation and asymptotically approaching a long time rate as t t associated with numerical differentiation both these rates can be inferred directly from the eigensystem of the numerical derivative operator we verify these analytical expectations using simple experiments in 1 d and 2 d

numerical resolution of discontinuous galerkin methods for time dependent wave equations abstract the discontinuous galerkin dg method is known to provide good wave resolution properties especially for long time simulation in this paper using fourier analysis we provide a quantitative error analysis for the semi discrete dg method applied to time dependent linear convection equations with periodic boundary conditions we apply the same technique to show that the error is of order k 2 superconvergent at radau points on each element and of order 2 k 1 superconvergent at the downwind point of each element when using piecewise polynomials of degree k an analysis of the fully discretized approximation is also provided we compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes numerical results are provided to verify our error analysis

multi patch discontinuous galerkin isogeometric analysis for wave propagation explicit time stepping and efficient mass matrix inversion abstract we present a class of spline finite element methods for time domain wave propagation which are particularly amenable to explicit time stepping the proposed methods utilize a discontinuous galerkin discretization to enforce continuity of the solution field across geometric patches in a multi patch setting which yields a mass matrix with convenient block diagonal structure over each patch we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight adjusted approximation of the mass matrix inverse this approximation restores a tensor product structure while retaining provable high order accuracy and semi discrete energy stability we also estimate the maximum stable timestep for spline based finite elements and show that the use of spline spaces results in less stringent cfl restrictions than equivalent c 0 or discontinuous finite element spaces finally we explore the use of optimal knot vectors based on l 2 n widths we show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep and present a simple heuristic method for approximating optimal knot positions numerical experiments confirm the accuracy and stability of the proposed methods

optimal numerical parameterization of discontinuous galerkin method applied to wave propagation problems this paper deals with the high order discontinuous galerkin dg method for solving wave propagation problems first we develop a one dimensional dg scheme and numerically compute dissipation and dispersion errors for various polynomial orders an optimal combination of time stepping scheme together with the high order dg spatial scheme is presented it is shown that using a time stepping scheme with the same formal accuracy as the dg scheme is too expensive for the range of wave numbers that is relevant for practical applications an efficient implementation of a high order dg method in three dimensions is presented using 1d convergence results we further show how to adequately choose elementary polynomial orders in order to equi distribute a priori the discretization error we also show a straightforward manner to allow variable polynomial orders in a dg scheme we finally propose some numerical examples in the field of aero acoustics

pressure forcing and dispersion analysis for discontinuous galerkin approximations to oceanic fluid flows this paper is part of an effort to examine the application of discontinuous galerkin dg methods to the numerical modeling of the general circulation of the ocean one step performed here is to develop an integral weak formulation of the lateral pressure forcing that is suitable for usage with a dg method and with a generalized vertical coordinate that includes level terrain fitted isopycnic and hybrid coordinates as examples this formulation is then tested in special cases with analyses of dispersion relations and numerical stability and with some computational experiments these results suggest that the advantages of dg methods may significantly outweigh their disadvantages in the settings tested here this paper also outlines some other issues that need to be addressed in future work

hp version discontinuous galerkin methods for hyperbolic conservation laws abstract the development of hp version discontinuous galerkin methods for hyperbolic conservation laws is presented in this work a priori error estimates are derived for a model class of linear hyperbolic conservation laws these estimates are obtained using a new mesh dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation the results generalize and extend previous results on mesh dependent norms to hp version discontinuous galerkin methods a posteriori error estimates which provide bounds on the actual error are also developed in this work numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

stability of energy stable flux reconstruction for the diffusion problem using the interior penalty and bassi and rebay ii numerical fluxes for linear triangular elements abstract the flux reconstruction fr method has gained popularity within the research community the approach has been demonstrated to recover high order methods such as the discontinuous galerkin dg method stability analyses have been conducted for a linear advection problem leading to the energy stable flux reconstruction esfr methods also named vincent castonguay jameson huynh vcjh methods esfr schemes can be viewed as dg schemes with modally filtered correction fields using this class of methods the linear advection diffusion problem has been shown to be stable using the local discontinuous galerkin scheme ldg to compute the viscous numerical flux this stability proof has been extended for linear triangular and tetrahedra elements although the ldg scheme is commonly used it requires on particular meshes a wide stencil which raises the computational cost as a consequence many prefer the compact interior penalty ip or the bassi and rebay ii br2 numerical fluxes this article for the first time derives for both schemes a condition on the penalty term to ensure the stability for the linear diffusion equation the derivations are based on gauss legendre flux points and on periodic boundary conditions moreover the article proposes that for both the ip and br2 numerical fluxes the stability of the esfr scheme is independent of the auxiliary correction field a von neumann analysis is conducted to study the maximal time step of various esfr methods

integral pinched gradient shrinking ρ einstein solitons abstract the gradient shrinking einstein soliton is a triple m n g f such that r i j f i j r g i j where m n g is a riemannian manifold 0 r 0 and f is the potential function on m n in this paper using algebraic curvature estimates and the yamabe sobolev inequality we prove some integral pinching rigidity results for compact gradient shrinking einstein solitons

dispersion analysis of plane wave discontinuous galerkin methods summary the plane wave dg pwdg method for the helmholtz equation was introduced and analyzed in gittelson c hiptmair r and perugia i plane wave discontinuous galerkin methods analysis of the h version math model numer anal 43 2009 297 331 as a generalization of the so called ultra weak variational formulation see o cessenat and b despres application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz equation siam j numer anal 35 1998 pp 255 299 the method relies on trefftz type local trial spaces spanned by plane waves of different directions and links cells of the mesh through numerical fluxes in the spirit of dg methods we conduct a partly empirical dispersion analysis of the method in a discrete translation invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves traveling in the same direction we find agreement of the wave numbers for directions represented in the local trial spaces for other directions the pwdg methods turn out to incur both phase and amplitude errors this manifests itself as a pollution effect haunting the h version of the method our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wave number and number of plane waves copyright 2014 john wiley sons ltd