fully discrete entropy conservative schemes of arbitrary order we consider weak solutions of hyperbolic or hyperbolic elliptic systems of conservation laws in one space dimension and their approximation by finite difference schemes in conservative form the systems under consideration are endowed with an entropy entropy flux pair we introduce a general approach to construct second and third order accurate fully discrete in both space and time entropy conservative schemes in general these schemes are fully nonlinear implicit but in some important cases can be explicit or linear implicit furthermore semidiscrete entropy conservative schemes of arbitrary order are presented the entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization sensitive shock waves finally specific examples are investigated and tested numerically our approach extends the results and techniques by tadmor in numerical methods for compressible flows finite difference element and volume techniques asme new york 1986 pp 149 158 lefloch and rohde siam j numer anal 37 2000 pp 2023 2060

forecasting with the age period cohort model and the extended chain ladder model we consider forecasting from age period cohort models as well as from the extended chain ladder model the parameters of these models are known only to be identified up to linear trends forecasts from such models may therefore depend on arbitrary linear trends a condition for invariant forecasts is proposed a number of standard forecast models are analysed

asymptotics for in sample density forecasting this paper generalizes recent proposals of density forecasting models and it develops theory for this class of models in density forecasting the density of observations is estimated in regions where the density is not observed identification of the density in such regions is guaranteed by structural assumptions on the density that allows exact extrapolation in this paper the structural assumption is made that the density is a product of one dimensional functions the theory is quite general in assuming the shape of the region where the density is observed such models naturally arise when the time point of an observation can be written as the sum of two terms e g onset and incubation period of a disease the developed theory also allows for a multiplicative factor of seasonal effects seasonal effects are present in many actuarial biostatistical econometric and statistical studies smoothing estimators are proposed that are based on backfitting full asymptotic theory is derived for them a practical example from the insurance business is given producing a within year budget of reported insurance claims a small sample study supports the theoretical results

three dimensional vibrations of thick circular and annular plates abstract the ritz method is applied in a three dimensional 3 d analysis to obtain accurate frequencies for thick circular and annular plates the method is formulated in a manner which allows one to have any combination of free or fixed plate boundaries admissible functions for the three displacement components are chosen as trigonometric functions in the circumferential co ordinate and algebraic polynomials in the radial and axial co ordinates upper bound convergence of the non dimensional frequencies to at least four significant figures is demonstrated comparisons of results are made with ones obtained by others using 2 d mindlin thick plate theory and with other 3 d solutions extensive and accurate four significant figure frequencies are presented for completely free circular plates having thickness to diameter ratios of 0 2 0 3 0 4 and 0 5 for poisson s ratios 0 0 3 and 0 499 frequencies are also given for thick annular plates having a thickness to outer diameter of 0 2 inside to outside diameter ratios of 0 1 0 5 and 0 9 and 0 3 all 3 d modes are included in the analyses e g flexural thickness shear in plane stretching and torsional the circular and annular plate frequency data given is exact to at least four digits thus being benchmark data against which results from 2 d thick plate theories or other approximate methods e g finite elements may be compared

algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws abstract in this work we modify a continuous galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property preserving low order scheme the addition of a second order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains validity of local maximum principles and entropy stability the new methodology combines the advantages of modern entropy stable entropy conservative schemes and their local extremum diminishing counterparts the process of algebraic flux correction is based on inequality constraints which provably provide the desired properties no free parameters are involved the proposed algebraic fixes are readily applicable to unstructured meshes finite element methods general time discretizations and steady state residuals numerical studies of explicit entropy constrained schemes are performed for linear and nonlinear test problems

arbitrarily high order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws we design arbitrarily high order accurate entropy stable schemes for systems of conservation laws the schemes termed tecno schemes are based on two main ingredients i high order accurate entropy conservative fluxes and ii suitable numerical diffusion operators involving eno reconstructed cell interface values of scaled entropy variables numerical experiments in one and two space dimensions are presented to illustrate the robust numerical performance of the tecno schemes

over dispersed age period cohort models summarywe consider inference and forecasting for aggregate data organised in a two way table with age and cohort as indices but without measures of exposure this is modelled using a poisson likelihood with an age period cohort structure for the mean while allowing for over dispersion we propose a repetitive structure that keeps the dimension of table fixed while increasing the latent exposure for this we use a class of infinitely divisible distributions which include a variety of compound poisson models and poisson mixture models this results in asymptotic f inference and t forecast distributions

out of plane vibrations of thick rings abstract depending on its cross sectional geometry a ring can be modelled by at least three different theories kirkhope developed a simple ring theory which is applicable to out of plane vibrations of rings with small cross sectional aspect ratio i e compact rings and shell like rings for plate like rings the present analysis uses a levinson type plate theory to analyze the out of plane vibrations also a three dimensional finite element analysis using nastran shows better agreement than results of kirkhope s ring theory and mindlin s thick plate theory

entropy stable high order discontinuous galerkin methods for ideal compressible mhd on structured meshes abstract we present a discontinuous galerkin dg scheme with suitable quadrature rules 15 for ideal compressible magnetohydrodynamic mhd equations on structural meshes the semi discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by godunov 32 the entropy stable dg framework with suitable quadrature rules 15 the entropy conservative flux in 14 inside each cell and the entropy dissipative approximate godunov type numerical flux at cell interfaces to make the scheme entropy stable the main difficulty in the generalization of the results in 15 is the appearance of the non conservative source terms added in the modified mhd model introduced by godunov 32 which do not exist in the general hyperbolic system studied in 15 special care must be taken to discretize these source terms adequately so that the resulting dg scheme satisfies entropy stability total variation diminishing bounded tvd tvb limiters and bound preserving limiters are applied to control spurious oscillations we demonstrate the accuracy and robustness of this new scheme on standard mhd examples

three dimensional vibration analysis of cantilevered and completely free isosceles triangular plates the free vibration of cantilevered and completely free isosceles triangular plates based on exact three dimensional elasticity theory is investigated the actual plate domain is first mapped onto a basic cubic domain then the ritz method is applied to derive the eigenfrequency equation from the strain energy and the kinetic energy of the plate a set of chebyshev polynomial series multiplied by a boundary function chosen to satisfy the essential geometry boundary conditions of the plate is developed as the admissible functions of each displacement component the convergence and comparison studies show that rather accurate results can be obtained by using this approach the effects of thickness to width ratio and apex angle on eigenfrequencies of the plates are studied in detail sets of valuable results are presented which may serve as the benchmark values for future numerical techniques in thick plate vibration analysis data for the completely free isosceles triangular plates are presented for the first time

stability analysis and a priori error estimates of the third order explicit runge kutta discontinuous galerkin method for scalar conservation laws in this paper we present an analysis of the runge kutta discontinuous galerkin method for solving scalar conservation laws where the time discretization is the third order explicit total variation diminishing runge kutta method we use an energy technique to prove the mathrm l 2 norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws quasi optimal order is obtained for general numerical fluxes and optimal order is given for upwind fluxes the theoretical results are obtained for piecewise polynomials with any degree k geq1 under the standard temporal spatial cfl condition tau leq gamma h where h and tau are the element length and time step respectively and the positive constant gamma is independent of h and tau

analysis and optimization of trimorph ring transducers abstract the kirchoff love plate theory and electroelasticity theory are combined to simulate the dynamic behaviors of the trimorph ring transducers under different boundary conditions the transducer consists of an isotropic elastic ring laminated between two identical piezoelectric rings their electric current response resonant frequencies antiresonant frequencies and electromechanical coupling coefficients emccs are theoretically formulated and studied by numerical simulation also the resonant frequencies and their corresponding mode shape are simulated by the finite element modelling to verify the theoretical results finally to obtain the maximum energy conversion efficiency the dynamic emcc is optimized by varying the proportion of piezoelectric and elastic parts it is shown that the dynamic emcc depends on geometric thickness and radii ratios optimum settings for a particular transducer to reach the maximum dynamic emcc are found for different boundary conditions the trimorph ring transducer for the fixed inner and free outer surfaces boundary condition has slightly lower resonant and anti resonant frequencies and larger emccs than that for the free inner and fixed outer surfaces boundary condition does

three dimensional vibration analysis of a torus with circular cross section the free vibration characteristics of a torus with a circular cross section are studied by using the three dimensional small strain elasticity theory a set of three dimensional orthogonal coordinates system comprising the polar coordinate r at each circular cross section and the circumferential coordinate around the ring is developed each of the displacement components ur and w in the r and directions respectively is taken as a product of the chebyshev polynomials in the r direction and the trigonometric functions in the and directions eigenfrequencies and vibration mode shapes have been obtained via a three dimensional displacement based extremum energy principle upper bound convergence of the first seven eigenfrequencies accurate to at least six significant figures is obtained by using only a few terms of the admissible functions the eigenfrequency responses due to variation of the ratio of the radius of the ring centroidal axis to the cross sectional radius are investig

constrained estimators and consistency of a regression model on a lexis diagram this article considers a regression model on a lexis diagram of an a p table with a single response in each cell following a distribution in the exponential family a regression model on the fixed effects of a rows p columns and a p 1 diagonals induces a singular design matrix and yields multiple estimators leading to parameter identifiability problem in age period cohort analysis in social sciences demography and epidemiology where assessment of secular trend in age period and birth cohort of social events e g violence and diseases e g cancer is of interest similar problems also exist in other settings such as in supersaturated designs in this article we study the finite sample properties of the multiple estimators propose a penalized profile likelihood method to study the consistency and asymptotic bias and demonstrate the results through simulations and data analysis as a by product the identifiability problem is addressed with consistent estimation for model parameters an

an energy conserving ultra weak discontinuous galerkin method for the generalized korteweg de vries equation abstract we propose an energy conserving ultra weak discontinuous galerkin dg method for the generalized korteweg de vries kdv equation in one dimension optimal a priori error estimate of order k 1 is obtained for the semi discrete scheme for the kdv equation without the convection term on general nonuniform meshes when polynomials of degree k 2 are used we also numerically observe optimal convergence of the method for the kdv equation with linear or nonlinear convection terms it is numerically observed for the new method to have a superior performance for long time simulations over existing dg methods

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

free vibration of laminated circular piezoelectric plates and discs the free vibration characteristics of laminated circular piezoelectric plates and discs are considered using a discrete layer model of the weak form of the equations of periodic motion through thickness approximations are used for the three displacement components and the electrostatic potential resulting in an accurate representation of the discontinuity in gradients of these quantities from the mismatch in material properties at a dissimilar interface in the radial and circumferential co ordinates several different approximation functions are used that depend upon the type of problem being considered specifically the boundary conditions at the outer edge of the solid representative cases are studied both for thin plates and for thick discs excellent agreement is found with results of previous studies and several new results are presented

on positivity preserving high order discontinuous galerkin schemes for compressible navier stokes equations we construct a local lax friedrichs type positivity preserving flux for compressible navier stokes equations which can be easily extended to multiple dimensions for generic forms of equations of state shear stress tensor and heat flux with this positivity preserving flux any finite volume type schemes including discontinuous galerkin dg schemes with strong stability preserving runge kutta time discretizations satisfy a weak positivity property with a simple and efficient positivity preserving limiter high order explicit runge kutta dg schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy numerical tests suggest that the positivity preserving flux and the positivity preserving limiter do not induce excessive artificial viscosity and the high order positivity preserving dg schemes without other limiters can produce satisfying non oscillatory solutions when the nonlinear diffusion in compressible navier stokes equations is accurately resolved

entropy stable shock capturing space time discontinuous galerkin schemes for systems of conservation laws we present a streamline diffusion shock capturing spacetime discontinuous galerkin dg method to approximate nonlinear systems of conservation laws in several space dimensions the degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes we show entropy stability of the formally arbitrarily high order accurate method for a general system of conservation laws furthermore we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown numerical experiments are presented to illustrate the robustness of the proposed schemes

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

3 d vibration analysis of circular rings with sectorial cross sections the free vibration characteristics of circular rings with sectorial cross section are studied based on the three dimensional 3 d small strain linear elasticity theory the complete vibration spectrum has been obtained by using the ritz method a set of three dimensional orthogonal coordinates composing of the polar coordinates r at the origin of the sectorial cross section and circumferential coordinate around the ring is developed to describe the variables in the analysis each of the displacement components is taken as a triplicate series two chebyshev polynomial series respectively about the r and coordinates and a trigonometric series about the coordinate frequency parameters and vibration mode shapes are computed by means of the displacement based extremum energy principle upper bound convergence of the first eight frequency parameters accurate to at least five significant figures is presented the effect of radius ratio subtended angle and initial slope angle on frequency parameters is investigated in detail all major modes such as flexural modes thickness shear modes stretching modes and torsional modes etc are presented in the paper the present results may serve as a benchmark reference to validate the accuracy of various approximate theories and other computational techniques for the vibration of circular rings

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

a discontinuous galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials abstract we consider a class of time dependent second order partial differential equations governed by a decaying entropy the solution usually corresponds to a density distribution hence positivity non negativity is expected this class of problems covers important cases such as fokker planck type equations and aggregation models which have been studied intensively in the past decades in this paper we design a high order discontinuous galerkin method for such problems if the interaction potential is not involved or the interaction is defined by a smooth kernel our semi discrete scheme admits an entropy inequality on the discrete level furthermore by applying the positivity preserving limiter our fully discretized scheme produces non negative solutions for all cases under a time step constraint our method also applies to two dimensional problems on cartesian meshes numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics

free vibration of circular and annular plates with variable thickness and different combinations of boundary conditions in this paper the first nine frequency parameters of circular and annular plates with variable thickness and combined boundary conditions are computed for different thickness to radius ratios several combined boundary conditions are considered for inner and outer edges these are free soft and hard simply supported and clamped boundary conditions three dimensional elasticity theory is used results of this paper are compared with works of other authors and the results of finite element method analysis which are in agreement data for thick circular and annular plates with linear and parabolic thickness variation and different combinations are presented for the first time

optimization of multidimensional diagonal norm summation by parts operators on simplices abstract this paper investigates the construction and optimization of multidimensional summation by parts sbp operators the optimization of free parameters in the cubature rule the interpolation extrapolation operator r and in the skew symmetric matrix s is investigated and a methodology to optimize the operators is presented it is demonstrated that increasing the degree of the cubature rule from 2 p 1 to 2p where p is the degree of the operator is beneficial to minimizing the solution error the optimization method developed is applied to construct novel multidimensional sbp operators on triangular and tetrahedral elements for example new operators with collocated element and facet cubature nodes are constructed with fewer element nodes and with lower solution errors for instance the p 2 operator has 10 fewer element nodes and a solution error 44 to 68 lower for the same computational cost than the previous best operator of the same degree and family

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

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

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

free vibration analysis of stepped circular mindlin plates this paper presents the first known investigation on exact vibration of circular mindlin plates with multiple step wise thickness variations a stepped circular plate is divided into multiple annular and one circular segments along the locations of the step variations the governing differential equations for harmonic vibration of annual and circular segments are derived and an analytical method based on the domain decomposition technique is developed to solve the plate vibration problem exact vibration solutions are presented for circular mindlin plates of different edge support conditions and various combinations of step wise thickness variations the influence of plate boundary conditions plate thickness ratios step location ratios and step thickness ratios on the vibration behaviour of stepped circular mindlin plates are examined the exact vibration results can serve as benchmark values for researchers to validate their numerical methods for such circular plate problems

three dimensional vibrations of thick linearly tapered annular plates abstract the ritz method is applied in a three dimensional 3 d analysis to obtain accurate frequencies for thick linearly tapered annular plates the method is formulated for annular plates having any combination of free or fixed boundaries at both inner and outer edges admissible functions for the three displacement components are chosen as trigonometric functions in the circumferential co ordinate and algebraic polynomials in the radial and thickness co ordinates upper bound convergence of the non dimensional frequencies to the exact values within at least four significant figures is demonstrated comparisons of results for annular plates with linearly varying thickness are made with ones obtained by others using 2 d classical thin plate theory extensive and accurate four significant figures frequencies are presented for completely free thick linearly tapered annular plates having ratios of average plate thickness to difference between outer radius a and inner radius b ratios hm l of 0 1 and 0 2 forb l 0 2 and 0 5 all 3 d modes are included in the analyses e g flexural thickness shear in plane stretching and torsional because frequency data given is exact to at least four digits it is benchmark data against which the results from other methods e g 2 d thick plate theory finite element methods and may be compared throughout this work poisson s ratiovis fixed at 0 3 for numerical calculations

comparisons of vibration frequencies for rods and beams from one dimensional and three dimensional analyses a method of three dimensional 3 d analysis is developed for the free vibration frequencies and mode shapes of solid circular cylinders of elastic material the method assumes for the three displacement components a fourier series in the circumferential direction and algebraic polynomials in the radial r and axial z directions all types of boundary conditions and arbitrary length to diameter ratios l d may be accommodated extensive convergence studies show that frequencies which are exact to five or six significant figures may be obtained for cylinders with free free ends and that three or four figure accuracy is achievable for fixed free ends these accurate frequencies are compared with ones determined from known elementary and improved 1 d theories for longitudinal torsional and bending modes for various l d ratios thereby establishing the ranges of accuracy of the 1 d analyses for circular cross sections

fast estimation from above of the maximum wave speed in the riemann problem for the euler equations this paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the riemann solution for the euler system of gas dynamics with the co volume equation of state the novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy the convergence rate of the algorithm is cubic and the bound is guaranteed for gasses with the co volume equation of state and the heat capacity ratio in the range 1 5 3

on a cell entropy inequality for discontinuous galerkin methods we prove a cell entropy inequality for a class of high order dis continuous galerkin finite element methods approximating conservation laws which implies convergence for the one dimensional scalar convex case

entropy viscosity method for nonlinear conservation laws a new class of high order numerical methods for approximating nonlinear conservation laws is described entropy viscosity method the novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand this new approach does not use any flux or slope limiters applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation various benchmark problems are solved with finite elements spectral elements and fourier series to illustrate the capability of 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

a continuum three dimensional vibration analysis of thick rectangular plates abstract this paper presents a continuum three dimensional ritz formulation for the vibration analysis of homogeneous thick rectangular plates with arbitrary combinations of boundary constraints this model is formulated on the basis of the linear three dimensional small deformation elasticity theory to predict the vibratory responses of these thick rectangular plates the displacement fields in the transverse and in plane directions are expressed by sets of orthogonally generated polynomial functions these shape functions are intrinsically a product of a class of orthogonal polynomial functions and a basic function which are chosen to satisfy the essential geometric boundary conditions at the outset sets of frequency data for plates with various aspect ratios and thickness ratios have been presented these data are used to examine the merits and limitations of the classical plate theory and mindlin plate theory by direct comparisons finally using the three dimensional continuum approach sets of first known deformed mode shapes have been generated thus helping to understand the vibratory motion furthermore these results may also serve as the benchmark to further research into the refined plate theories

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

a mixed discontinuous galerkin method with symmetric stress for brinkman problem based on the velocity pseudostress formulation abstract the brinkman equations can be regarded as a combination of the stokes and darcy equations which model transitions between the fast flow in channels governed by stokes equations and the slow flow in porous media governed by darcy s law the numerical challenge for this model is the designing of a numerical scheme which is stable for both the stokes dominated high permeability and the darcy dominated low permeability equations in this paper we solve the brinkman model in n dimensions n 2 3 by using the mixed discontinuous galerkin mdg method which meets this challenge this mdg method is based on the pseudostress velocity formulation and uses a discontinuous piecewise polynomial pair p k 1 s p k k 0 where the stress field is symmetric the main unknowns are the pseudostress and the velocity whereas the pressure is easily recovered through a simple postprocessing a key step in the analysis is to establish the parameter robust inf sup stability through specific parameter dependent norms at both continuous and discrete levels therefore the stability results presented here are uniform with respect to the permeability thanks to the parameter robust stability analysis we obtain optimal error estimates for the stress in broken h div norm and velocity in l 2 norm furthermore the l 2 error estimate for pseudostress is derived under certain conditions finally numerical experiments are provided to support the theoretical results and to show the robustness accuracy and flexibility of the mdg method

entropy stable reduced order modeling of nonlinear conservation laws abstract reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model we introduce projection based hyper reduced models of nonlinear conservation laws which are globally conservative and inherit a semi discrete entropy inequality independently of the choice of basis and choice of parameters

multidimensional summation by parts operators general theory and application to simplex elements summation by parts sbp finite difference discretizations share many attractive properties with galerkin finite element methods fems including time stability and superconvergent functionals however unlike fems sbp operators are not completely determined by a basis so the potential exists to tailor sbp operators to meet different objectives to date application of high order sbp discretizations to multiple dimensions has been limited to tensor product domains this paper presents a definition for multidimensional sbp finite difference operators that is a natural extension of one dimensional sbp operators theoretical implications of the definition are investigated for the special case of a diagonal norm mass matrix in particular a diagonal norm sbp operator exists on a given domain if and only if there is a cubature rule with positive weights on that domain and the polynomial basis matrix has full rank when evaluated at the cubature nodes appropriate simultaneous approximation terms are devel

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

three dimensional elasticity solutions to vibration of cantilevered skewed trapezoids an accurate three dimensional elasticity model for determining the free vibration response of skewed trapezoidal plates is presented the method is formulated on the basis of a linear small strain three dimensional elasticity theory with the ritz minimization procedure the solution method uses sets of uniquely defined one and twodimensional polynomial functions to approximate the trial displacements in the thickness and surface directions respectively these functions are orthogonally generated and have been proven to be highly efficient and numerically stable the proposed technique yields natural frequencies and mode shapes for a wide range of cantilevered skewed trapezoidal plates of arbitrary planform the accuracy of these results whenever possible is validated through comparison with the analytical and experimental data parametric investigations on the vibration behavior of cantilevered trapezoidal plates with respect to different thickness ratios skew angles and chord ratios have been carried out for the first time the presence of surface parallel symmetric thickness modes for these plates is demonstrated in vivid three dimensional mode shape plots

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

three dimensional vibrations of hollow cones and cylinders with linear thickness variations a three dimensional method of analysis is presented for determining the free vibration frequencies and mode shapes of hollow cones and cylinders with variable thickness displacement components us uz and u in the meridional normal and circumferential directions respectively are taken to be sinusoidal in time periodic in and algebraic polynomials in the s and z directions potential strain and kinetic energies of the cones and cylinders are formulated and upper bound values of the frequencies are obtained by minimizing the frequencies as the degree of the polynomials is increased frequencies converge to the exact values novel numerical results are presented for thick linearly tapered hollow cones and cylinders with completely free boundaries convergence to four digit exactitude is demonstrated for the first five frequencies of the cones and cylinders the method is applicable to thin cones and cylinders as well as thick and very thick ones

identification of the age period cohort model and the extended chain ladder model we consider the identification problem that arises in the age period cohort models as well as in the extended chain ladder model we propose a canonical parameterization based on the accelerations of the trends in the three factors this parameterization is exactly identified and eases interpretation estimation and forecasting the canonical parameterization is applied to a class of index sets which have trapezoidal shapes including various lexis diagrams and the insurance reserving triangles

free axisymmetric vibration of laminated transversely isotropic annular plates based on the basic equations of three dimensional theory of elasticity this paper establishes the state space equation of the axisymmetric vibration of laminated annular plates composed of transversely isotropic layers taking advantage of the finite hankel transform four exact solutions are obtained for four different types of boundary conditions the calculating methods of frequencies and mode shapes are presented lastly numerical results are given to validate the present method

on structure preserving discontinuous galerkin methods for hamiltonian partial differential equations energy conservation and multi symplecticity abstract in this paper we present and study discontinuous galerkin dg methods for one dimensional multi symplectic hamiltonian partial differential equations we particularly focus on semi discrete schemes with spatial discretization only and show that the proposed dg methods can simultaneously preserve the multi symplectic structure and energy conservation with a general class of numerical fluxes which includes the well known central and alternating fluxes applications to the wave equation the benjamin bona mahony equation the camassa holm equation the korteweg de vries equation and the nonlinear schrodinger equation are discussed some numerical results are provided to demonstrate the accuracy and long time behavior of the proposed methods numerically we observe that certain choices of numerical fluxes in the discussed class may help achieve better accuracy compared with the commonly used ones including the central fluxes

three dimensional vibrations of cantilevered right triangular plates abstract the first known three dimensional continuum vibration solutions for cantilevered right triangular plates with variable thickness are obtained using the ritz method assumed displacement functions are in the form of algebraic polynomials which satisfy the fixed face conditions exactly and which are mathematically complete reasonably accurate natural frequencies are calculated for low aspect ratio right triangular thin plates having arbitrary values of thickness taper ratios in the spanwise direction detailed numerical studies show that a three dimensional analysis is essential to monitoring coupled mode sensitivities in the variation of non dimensional natural frequencies with increasing thickness taper ratio upper bound results obtained using the present method are compared with those obtained by other investigators using ordinary beam theories two dimensional finite element and finite difference procedures and experimental methods this unified comparison of upper and lower bound solutions is presented here with the aim of bracketing the exact analytical solution of the subject problem

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

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