on the identification of symmetric quadrature rules for finite element methods in this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals triangles tetrahedra prisms pyramids and hexahedra the methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points we also present polyquad which is an implementation of our methodology using polyquad v1 0 we proceed to derive a complete set of symmetric rules on the aforementioned domains all rules possess purely positive weights and have all points inside the domain many of the rules appear to be new and an improvement over those tabulated in the literature

the component model for elementary landscapes and partial neighborhoods local search algorithms exploit moves on an adjacency graph of the search space an elementary landscape exists if the objective function f is an eigenfunction of the laplacian of the graph induced by the neighborhood operator this allows various statistics about the neighborhood to be computed in closed form a new component based model makes it relatively simple to prove that certain types of landscapes are elementary the traveling salesperson problem weighted graph vertex coloring and the minimum graph bisection problem yield elementary landscapes under commonly used local search operators the component model is then used to efficiently compute the mean objective function value over partial neighborhoods for these same problems for a traveling salesperson problem over n cities the 2 opt neighborhood can be decomposed into n email protected partial neighborhoods for graph coloring and the minimum graph bisection problem partial neighborhoods can be used to focus search on those moves that are capable of producing a solution with a strictly improving objective function value

on the landscape ruggedness of the quadratic assignment problem local search based heuristics have been demonstrated to give very good results to approximately solve the quadratic assignment problem qap in this paper following the works of weinberger and stadler we introduce a parameter called the ruggedness coeffcient which measures the ruggedness of the qap landscape which is the union of a cost function and a neighborhood we give an exact expression and a sharp lower bound for this parameter we are able toderive from it that the landscape of the qap is rather flat and so it gives a theoretical justification of the effectiveness of local search based heuristics for this problem experimental results with simulated annealing are presented which con8rm this conclusion and also the influence of the ruggedness coe5cient on the quality of results obtained

a generalized framework for nodal first derivative summation by parts operators a generalized framework is presented that extends the classical theory of finite difference summation by parts sbp operators to include a wide range of operators where the main extensions are i non repeating interior point operators ii nonuniform nodal distribution in the computational domain iii operators that do not include one or both boundary nodes necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the sbp property it is proven that the positive definite norm matrix of each sbp operator must be associated with a quadrature rule moreover given a quadrature rule there exists a corresponding sbp operator where for diagonal norm sbp operators the weights of the quadrature rule must be positive the generalized framework gives a straightforward means of posing many known approximations to the first derivative as sbp operators several are surveyed such as discontinuous galerkin discretizations based on the legendre gauss quadrature points and shown to be sbp operators moreover the new framework provides a method for constructing sbp operators by starting from quadrature rules this is illustrated by constructing novel sbp operators from known quadrature rules to demonstrate the utility of the generalization the legendre gauss and legendre gauss radau quadrature points are used to construct sbp operators that do not include one or both boundary nodes

on pseudo spectral time discretizations in summation by parts form abstract fully implicit discrete formulations in summation by parts form for initial boundary value problems must be invertible in order to provide well functioning procedures we prove that under mild assumptions pseudo spectral collocation methods for the time derivative lead to invertible discrete systems when energy stable spatial discretizations are used

extended skew symmetric form for summation by parts operators and varying jacobians abstract a generalised analytical notion of summation by parts sbp methods is proposed extending the concept of sbp operators in the correction procedure via reconstruction cpr a framework of high order methods for conservation laws for the first time sbp operators with dense norms and not including boundary points are used to get an entropy stable split form of burgers equation moreover overcoming limitations of the finite difference framework stability for curvilinear grids and dense norms is obtained for sbp cpr methods by using a suitable way to compute the jacobian

energy conservative sbp discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids abstract we develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner the use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a cartesian basis decomposition on rotated grids the discretization is based on high order summation by parts sbp operators and preserves both symmetry and positive definiteness of the contravariant metric tensor to improve accuracy and decrease computational cost we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization bounds are derived that yield a point wise condition that can be evaluated to check for stability of the modified discretization this condition shows that the interpolation operators should be constructed such that their norm is close to one

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

a comparison of the dispersion and dissipation errors of gauss and gauss lobatto discontinuous galerkin spectral element methods we examine the dispersion and dissipation properties of the gauss and gauss lobatto discontinuous galerkin spectral element methods dgsems for linear wave propagation problems we show that the inherent underintegration in the gauss lobatto variant can be interpreted as a modal filtering of the highest polynomial mode this in turn has a drastic impact on the dispersion and dissipation relations of the gauss lobatto dgsem compared to the gauss variant we show that the gauss dgsem is typically more accurate than the gauss lobatto variant needing fewer points per wavelength for a given accuracy while on the other hand being more restricted in the explicit time step choice we show that the spectra of the dgsem operators depend on the boundary conditions applied and that the ratio of the time step restrictions of the two schemes depends on the choice of boundary conditions

autocorrelation coefficient for the graph bipartitioning problem abstract local search and its variants simulated annealing and tabu search are widely used heuristics to approximately solve np hard optimization problems to use local search one simply has to specify a neighborhood structure and a cost function which has to be optimized however from a theoretical point of view many questions remain unanswered and one of the most important is which neighborhood structure will provide the best quality solutions the aim of this paper is to theoretically justify some results previously reported by johnson et al 1989 1991 in their extended empirical study concerning simulated annealing and the graph bipartitioning problem and to sharply tune the best landscape among the two reported in that study experimental results perfectly agree with the theoretical predictions

spectral direction splitting methods for two dimensional space fractional diffusion equations a numerical method for a kind of time dependent two dimensional two sided space fractional diffusion equations is developed in this paper the proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization the direction splitting approach renders the underlying two dimensional equation into a set of one dimensional space fractional diffusion equations at each time step then these one dimensional equations are solved by using the spectral method based on weak formulations a time error estimate is derived for the semi discrete solution and the unconditional stability of the fully discretized scheme is proved some numerical examples are presented to validate the proposed method

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

well posed and stable transmission problems we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time well posedness and stability is analysed for continuous and d

an energy stable discontinuous galerkin spectral element discretization for variable coefficient advection problems we develop a stable split form discontinuous galerkin spectral element method for the solution of variable coefficient linear hyperbolic systems of conservation laws in our presentation we start with the simplest problem and introduce complexity as we progress we begin with the approximation of scalar conservation laws in one space dimension we then extend the derivation to hyperbolic systems finally we approximate the two dimensional problem numerical experiments are performed to compare the approximations

a general framework to construct schemes satisfying additional conservation relations application to entropy conservative and entropy dissipative schemes abstract we are interested in the approximation of a steady hyperbolic problem in some cases the solution can satisfy an additional conservation relation at least when it is smooth this is the case of an entropy in this paper we show starting from the discretization of the original pde how to construct a scheme that is consistent with the original pde and the additional conservation relation since one interesting example is given by the systems endowed by an entropy we provide one explicit solution and show that the accuracy of the new scheme is at most degraded by one order in the case of a discontinuous galerkin scheme and a residual distribution scheme we show how not to degrade the accuracy this improves the recent results obtained in 1 2 3 4 in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved we study the behavior of the method on a non linear scalar problem however the method is not restricted to scalar problems

generalised summation by parts operators and variable coefficients abstract high order methods for conservation laws can be highly efficient if their stability is ensured a suitable means mimicking estimates of the continuous level is provided by summation by parts sbp operators and the weak enforcement of boundary conditions recently there has been an increasing interest in generalised sbp operators both in the finite difference and the discontinuous galerkin spectral element framework however if generalised sbp operators are used the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely interpolating the product of two functions will in general result in a value different from the product of the interpolations thus desired properties such as conservation and stability are more difficult to obtain here new formulations are proposed allowing the creation of discretisations using general sbp operators that are both conservative and stable thus several shortcomings that might be attributed to generalised sbp operators are overcome cf nordstrom and ruggiu 2017 38 and manzanero et al 2017 39

elementary landscape decomposition of the frequency assignment problem the frequency assignment problem fap is an important problem that arises in the design of radio networks when a channel has to be assigned to each transceiver of the network this problem is a generalization of the graph coloring problem in this paper we study a general version of the fap that can include adjacent frequency constraints using concepts from landscapes theory we prove that this general fap can be expressed as a sum of two elementary landscapes further analysis also shows that some subclasses of the problem correspond to a single elementary landscape this allows us to compute the kind of neighborhood information that is normally associated with elementary landscapes we also provide a closed form formula for computing the autocorrelation coefficient for the general fap which can be useful as an a priori indicator of the performance of a local search method

summation by parts operators for correction procedure via reconstruction the correction procedure via reconstruction cpr formerly known as flux reconstruction is a framework of high order methods for conservation laws unifying some discontinuous galerkin spectral difference and spectral volume methods linearly stable schemes were presented by vincent et al 2011 2015 but proofs of non linear entropy stability in this framework have not been published yet to the knowledge of the authors we reformulate cpr methods using summation by parts sbp operators with simultaneous approximation terms sats a framework popular for finite difference methods extending the results obtained by gassner 2013 for a special discontinuous galerkin spectral element method this reformulation leads to proofs of conservation and stability in discrete norms associated with the method recovering the linearly stable cpr schemes of vincent et al 2011 2015 additionally extending the skew symmetric formulation of conservation laws by additional correction terms entropy stability for burgers equation is proved for general sbp cpr methods not including boundary nodes

summation by parts operators and high order quadrature summation by parts sbp operators are finite difference operators that mimic integration by parts the sbp operator definition includes a weight matrix that is used formally for discrete integration however the accuracy of the weight matrix as a quadrature rule is not explicitly part of the sbp definition we show that sbp weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes for diagonal weight matrices the accuracy of sbp quadrature extends to curvilinear domains provided the jacobian is approximated with the same sbp operator used for the quadrature this quadrature has significant implications for sbp based discretizations in particular the diagonal norm accurately approximates the l 2 norm for functions and multi dimensional sbp discretizations accurately approximate the divergence theorem

connections between the filtered discontinuous galerkin method and the flux reconstruction approach to high order discretizations the purpose of this paper is to provide new insights on the connections that exist between the discontinuous galerkin method dg the flux reconstruction method fr and the recently identified energy stable flux reconstruction method esfr when solving time dependent conservation laws all these schemes appear to be quite similar and it is important to understand how they are related in this paper we first review results on the stability of the discontinuous galerkin method and extend it to the filtered discontinuous galerkin method we then consider the flux reconstruction approach and show its connections with dg in particular we show how the energy stable flux reconstruction method introduced by vincent et al is equivalent to a filtered dg method hence giving a new proof of its stability also it allows the use of the method without having to know the special form of the flux correction polynomials finally we underline some fundamental differences that exist between fr and dg

the spatial operator in the incompressible navier stokes oseen and stokes equations abstract we investigate the spatial operator in the incompressible navier stokes oseen and stokes equations and show how to avoid singularities associated with null spaces by choosing specific boundary conditions the theoretical results are derived for a general form of energy stable boundary conditions and applied to a few commonly used ones the analysis is done on a system that simultaneously covers the nonlinear incompressible navier stokes the oseen and the stokes equations when the spectrum of the spatial operator is investigated we restrict the analysis to the oseen and stokes equations the continuous analysis carries over to the discrete setting by using the summation by parts framework

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed