convergence analysis for operator splitting methods applied to conservation laws with stiff source terms we analyze the order of convergence for operator splitting methods applied to conservation laws with stiff source terms we suppose that the source term q u is dissipative it is proved that the l 1 error introduced by the time splitting can be bounded by o d t vert q u 0 vert l 1 which is an improvement of the o q d t upper bound where d t is the splitting time step q is the lipschitz constant of q or q max u vert q u vert in case q is smooth a generic model with a special form of stiff source is also investigated we propose a nonuniform temporal mesh to eliminate the effect of the initial layer introduced by the stiff source term our results are derived by using parabolic regularizations rather than using kuznetsov s approximation theory which has been employed as a standard approach for error analysis to the dimensional or time splitting methods numerical examples are presented to illustrate the theoretical results

l 1 error estimates for difference approximations of degenerate convection diffusion equations we analyze monotone finite difference schemes for strongly degenerate convection diffusion equations in one spatial dimension these nonlinear equations are well posed within a class of discontinuous entropy solutions we prove that the l1 error between the approximate and exact solutions is o x1 3 where x is the spatial grid parameter this result should be compared with the classical o x1 2 error estimate for conservation laws kuznecov 1976 and a recent estimate of o x1 11 for degenerate convection diffusion equations karlsen koley risebro 2012

simulating mesoscopic reaction diffusion systems using the gillespie algorithm we examine an application of the gillespie algorithm to simulating spatially inhomogeneous reaction diffusion systems in mesoscopic volumes such as cells and microchambers the method involves discretizing the chamber into elements and modeling the diffusion of chemical species by the movement of molecules between neighboring elements these transitions are expressed in the form of a set of reactions which are added to the chemical system the derivation of the rates of these diffusion reactions is by comparison with a finite volume discretization of the heat equation on an unevenly spaced grid the diffusion coefficient of each species is allowed to be inhomogeneous in space including discontinuities the resulting system is solved by the gillespie algorithm using the fast direct method we show that in an appropriate limit the method reproduces exact solutions of the heat equation for a purely diffusive system and the nonlinear reaction rate equation describing the cubic autocatalytic reaction

a mixed variational formulation based on exact intrinsic equations for dynamics of moving beams abstract a nonlinear intrinsic formulation for the dynamics of initially curved and twisted beams in a moving frame is presented the equations are written in a compact matrix form without any approximations to the geometry of the deformed beam reference line or to the orientation of the intrinsic cross section frame in accordance with previously published work when effects of warping on the dynamics and local constraints on the cross sectional deformation are ignorable the in and out of plane st venant warping displacements which are fully coupled for nonhomogeneous anisotropic beams need only be included explicitly in the determination of a suitable elastic law a two dimensional problem and need not be considered explicitly in the one dimensional equations governing global deformation in this paper it is presupposed that such an elastic law is given as a one dimensional strain energy function thus the present equations which are based on only six generalized strain variables are valid for beams with closed cross sections and for which warping is unrestrained when simplified for various special cases they agree with similar intrinsic equations in the literature although the resulting equations are newtonian in structure closely resembling euler s dynamical equations for a rigid body the formulation adheres to a variational approach throughout thus providing a link between newtonian and energy based methods in particular the present development provides substantial insight into the relationships among variational formulations in which different displacement and rotational variables are used as well as between these formulations and newtonian ones for computational purposes a compact and complete mixed variational formulation is presented that is ideally suited for development of finite element analyses finally a specialized version of the intrinsic equations is developed in which shear deformations is suppressed

two a posteriori error estimates for one dimensional scalar conservation laws in this paper we propose a posteriori local error estimates for numerical schemes in the context of one dimensional scalar conservation laws we first consider the schemes for which a consistent in cell entropy inequality can be derived then we extend this result to second order schemes written in viscous form satisfying weak entropy inequalities as an illustration we show several numerical tests on the burgers equation and we propose an adaptive algorithm for the selection of the mesh

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

error estimates of a first order lagrange finite element technique for nonlinear scalar conservation equations this paper establishes a mathcal o h frac14 error estimate in the l t infty l x 1 norm for the approximation of scalar conservation equations using an explicit continuous finite element technique a general a priori error estimate based on entropy inequalities is also given in the appendix

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

accuracy analysis of mimetic finite volume operators on geodesic grids and a consistent alternative many newly developed climate weather and ocean global models are based on quasi uniform spherical polygonal grids aiming for high resolution and better scalability thuburn et al 2009 and ringler et al 2010 developed a c staggered finite volume difference method for arbitrary polygonal spherical grids suitable for these next generation dynamical cores this method has many desirable mimetic properties and became popular being adopted in some recent models in spite of being known to possess low order of accuracy in this work we show that for the nonlinear shallow water equations on non uniform grids the method has potentially 3 main sources of inconsistencies local truncation errors not converging to zero as the grid is refined i the divergence term of the continuity equation ii the perpendicular velocity and iii the kinetic energy terms of the vector invariant form of the momentum equations although some of these inconsistencies have not impacted the convergence on some standard shallow water test cases up until now they may constitute a potential problem for high resolution 3d models based on our analysis we propose modifications for the method that will make it first order accurate in the maximum norm it preserves many of the mimetic properties albeit having non steady geostrophic modes on the f sphere experimental results show that the resulting model is a more accurate alternative to the existing formulations and should provide means of having a consistent computationally cheap and scalable atmospheric or ocean model on c staggered voronoi grids

a supraconvergent scheme for nonlinear hyperbolic systems abstract supraconvergent difference schemes are methods for which the local truncation error degrades when applied to an irregular mesh yet the observed error in the solution is as good as if the mesh were uniform for a version of the lax wendroff scheme requiring the computation of the jacobian we show that superconvergence holds on an arbitrary grid one dimensional for the standard version we present a computation which shows that second order accuracy is not always maintained but the convergence is better than first order

on asymptotically correct timoshenko like anisotropic beam theory abstract this paper presents a finite element cross sectional beam analysis capable of capturing transverse shear effects the approach uses the variational asymptotic method and can handle beams of general cross sectional shape and arbitrary anisotropic material the work builds on previous works which deal with development of the classical beam theory which includes only extension torsion and bending a timoshenko like formulation is sought to achieve a refined theory with simple boundary conditions apart from some simple special cases it is shown that this problem is overdetermined however it is possible to obtain the best possible solution using the least squares minimization technique the geometrical meaning of the shear variable for this formulation is also derived results are found to be in good agreement with published results for the shear stiffness coefficients for both isotropic and anisotropic beams

convergence of a non monotone scheme for hamilton jacobi bellman equations with discontinous initial data we prove the convergence of a non monotonous scheme for a one dimensional first order hamilton jacobi bellman equation of the form v t max f x vx 0 v 0 x v0 x the scheme is related to the hjb ultrabee scheme suggested in bokanowski and zidani j sci comput 30 1 1 33 2007 we show for general discontinuous initial data a first order convergence of the scheme in l1 norm towards the viscosity solution we also illustrate the non diffusive behavior of the scheme on several numerical examples

convergence of the discontinuous galerkin method for discontinuous solutions we consider linear first order scalar equations of the form rho t rm div rho v a rho f with appropriate initial and boundary conditions it is shown that approximate solutions computed using the discontinuous galerkin method will converge in ltwoltwo when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions in particular v need not be lipschitz so characteristics of the equation may not be defined and the solutions being approximated my not have bounded variation

a priori mesh quality metric error analysis applied to a high order finite element method characterization of computational mesh s quality prior to performing a numerical simulation is an important step in insuring that the result is valid a highly distorted mesh can result in significant errors it is therefore desirable to predict solution accuracy on a given mesh the hifi sel high order finite element code is used to study the effects of various mesh distortions on solution quality of known analytic problems for spatial discretizations with different order of finite elements the measured global error norms are compared to several mesh quality metrics by independently varying both the degree of the distortions and the order of the finite elements it is found that the spatial spectral convergence rates are preserved for all considered distortion types while the total error increases with the degree of distortion for each distortion type correlations between the measured solution error and the different mesh metrics are quantified identifying the most appropriate overall mesh metric the results show promise for future a priori computational mesh quality determination and improvement

multidimensional model for the stress analysis of reinforced shell structures the present paper proposes an approach that can be used to mix one two and three dimensional refined models derived using the carrera unified formulation to build a variable kinematic model t

on nonlocal monotone difference schemes for scalar conservation laws analyse d erreur d une classe de methodes aux differences finies monotones pour la solution approchee des lois de conservation hyperboliques

analysis of a beam cross section under coupled actions including transversal shear abstract an efficient finite elements fe based method of cross sectional analysis of a longitudinally homogeneous beam with an arbitrary cross section and material model assumed submitted to tension compression bending torsion and shear is developed a new formulation of this classical problem is done as a special case of the homogenization theory it takes into account 3d strain and stress state fulfilling equilibrium compatibility and constitutive equations at any point stress free conditions on lateral surfaces but preserves control by cross sectional stress resultant forces and corresponding generalized strains a source of problems caused by the presence of a shear force is explained and a way of overcoming it is shown in the paper both variational formulation of the problem and its 2d fe solution are given benchmark cases including shear of a square a composite l section an orthotropic square and an elasto plastic analysis of shearing of a square section show correctness of the method finally two practical cases of application of the method to an analysis of reinforced concrete beam sections are shown

stability of markov regenerative switched linear systems in this paper we give a necessary and sufficient condition for mean stability of switched linear systems having a markov regenerative process as its switching signal this class of switched linear systems which we call markov regenerative switched linear systems contains markov jump linear systems and semi markov jump linear systems as special cases we show that a markov regenerative switched linear system is m th mean stable if and only if a particular matrix is schur stable under the assumption that either m is even or the system is positive

a supraconvergent scheme for the korteweg de vries equation a finite difference method for the integration of the korteweg de vries equation on irregular grids is analyzed under periodic boundary conditions the method is shown to be supraconvergent in the sense that though being inconsistent it is second order convergent however such a convergence only takes place on grids with an odd number of points per period when a grid with an even number of points is used the inconsistency of the method leads to divergence numerical results backing the analysis are presented

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

on timoshenko like modeling of initially curved and twisted composite beams abstract a generalized finite element based cross sectional analysis for nonhomogenous initially curved and twisted anistropic beams is formulated from geometrically nonlinear three dimensional 3 d elasticity the 3 d strain field is formulated based on the concept of decomposition of the rotation tensor and is given in terms of one dimensional 1 d generalized strains and a 3 d warping displacement that is obtained from the formulation not assumed the warping is found in terms of the 1 d strains via the variational asymptotic method vam in this paper a timoshenko like model is presupposed for a beam with cross sectional characteristic length h wavelength of deformation given by l and the magnitude of the radius of initial curvature and or twist is taken to be of the order r first a solution for the asymptotically correct refinement of classical anisotropic beam theory for initially curved and twisted beams through o h2 r2 is obtained next the o h2 l2 correction is computed it is known that timoshenko like theory is not capable of capturing all the o h2 l2 corrections for generally anisotropic beams however if all the o h2 l2 terms are known then the corresponding timoshenko like theory is uniquely defined numerical results are presented to illustrate the trends of the various classical extension twist bending twist and extension bending and nonclassical couplings extension shear bending shear and shear torsion as the initial twist and curvatures are varied

validation of the variational asymptotic beam sectional analysis the computer program vabs variational asymptotic beam section analysis uses the variational asymptotic method to split a three dimensional nonlinear elasticity problem into a twodimensional linear cross sectional analysis and a one dimensional nonlinear beam problem this is accomplished by taking advantage of certain small parameters inherent to beam like structures vabs is able to calculate the one dimensional cross sectional stiffness constants with transverse shear and vlasov refinements for initially twisted and curved beams with arbitrary geometry and material properties several validation cases are presented first an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method and the solution is shown to be identical to that obtained from the theory of elasticity the shear center locations calculated by vabs for various cross sections agree well with those obtained from common engineering analyses comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model for numerical validation values of the one dimensional variables are extracted from an abaqus model and compared with results from a one dimensional beam analysis using cross sectional constants from vabs furthermore point wise three dimensional stress and strain fields are recovered using vabs and the correlation with the three dimensional results from abaqus is excellent finally classical theory is shown to be insufficient for general purpose beam modeling appropriate refined theories are recommended for some classes of problems

a note on entropy inequalities and error estimates for higher order accurate finite volume schemes on irregular families of grids recently cockburn coquel and lefloch proved convergence and error estimates for higher order finite volume schemes their result is based on entropy inequalities which are derived under restrictive assumptions on either the flux function or the numerical fluxes moreover they assume that the spatial grid satisfies a standard regularity assumption using instead entropy inequalities derived in previous work by kroner noelle and rokyta and a weaker condition on the grid we can generalize and simplify the error estimates

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation this paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions we introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in mathbf r nd and every nonvanishing vector mathbf a of mathbf r nd first we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the finite volume scheme occurs afterwards we prove that this norm is indeed bounded by the mesh size in several cases including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases such as the one with independent refined meshes

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

grid adaptive algorithms for the solution of the euler equations on irregular grids abstract the effect of grid irregularity on the accuracy of algorithms for one dimensional unsteady flow is investigated it is shown how standard second order accurate algorithms for equally spaced grids may be extended to act as first order accurate algorithms on an irregular grid the loss of accuracy on irregular grids is found to cause a significant reduction in the quality of the results to obtain solutions comparable in quality with second order accurate algorithms on equally spaced grids a class of second order accurate algorithms is derived which are conservative and compatible on randomly spaced grids they are shown to give solutions to the equations similar in quality to those obtainable on equally spaced grids

a priori error estimates for numerical methods for scalar conservation laws part iii multidimensional flux splitting monotone schemes on non cartesian grids this paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we consider multidimensional flux splitting monotone schemes defined on non cartesian grids we identify those schemes which are consistent and prove that the l infty 0 t l 1 bbb r d norm of the error goes to zero as delta x 1 2 when the discretization parameter delta x goes to zero moreover we show that nonconsistent schemes can converge at optimal rates of delta x 1 2 because i the conservation form of the schemes and ii the so called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency

analysis of space time discontinuous galerkin method for nonlinear convection diffusion problems the paper presents the theory of the discontinuous galerkin finite element method for the space time discretization of a nonstationary convection diffusion initial boundary value problem with nonlinear convection and linear diffusion the problem is not singularly perturbed with dominating convection the discontinuous galerkin method is applied separately in space and time using in general different space grids on different time levels and different polynomial degrees p and q in space and time dicretization in the space discretization the nonsymmetric symmetric and incomplete interior and boundary penalty nipg sipg iipg approximation of diffusion terms is used the paper is concerned with the proof of error estimates in l 2 l 2 and dg norm formed by the l 2 h 1 seminorm and penalty terms a special technique based on the use of the gauss radau interpolation and numerical integration has been used for the derivation of an abstract error estimate in the dg norm the error estimates are optimal with respect to the size of the space grid they are optimal with respect to the time step if the dirichlet boundary condition has behaviour in time as a polynomial of degree q

local a posteriori error estimates for time dependent hamilton jacobi equations university of minnesota ph d dissertation may 2010 major mathematics advisor prof bernardo cockburn 1 computer file pdf viii 67 pages ill some col

stability analysis of positive semi markovian jump linear systems with state resets this paper studies the mean stability of positive semi markovian jump linear systems we show that their mean stability is characterized by the spectral radius of a matrix that is easy to compute in deriving the condition we use a certain discretization of a semi markovian jump linear system that preserves stability also we show a characterization for the exponential mean stability of continuous time positive markovian jump linear systems numerical examples are given to illustrate the results

finite volume relaxation schemes for multidimensional conservation laws we consider finite volume relaxation schemes for multidimensional scalar conservation laws these schemes are constructed by appropriate discretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a rate of h 1 4 in l 0 t l loc 1 r d

efficient method for computing lower bounds on the p radius of switched linear systems abstract this paper proposes lower bounds on a quantity called l p norm joint spectral radius or in short p radius of a finite set of matrices despite its wide range of applications to for example stability analysis of switched linear systems and the equilibrium analysis of switched linear economical models algorithms for computing the p radius are only available in a very limited number of particular cases the proposed lower bounds are given as the spectral radius of an average of the given matrices weighted via kronecker products and do not place any requirements on the set of matrices we show that the proposed lower bounds theoretically extend and also can practically improve the existing lower bounds a markovian extension of the proposed lower bounds is also presented

error estimate for finite volume scheme we study the convergence of a finite volume scheme for the linear advection equation with a lipschitz divergence free speed in r d we prove a h1 2 error estimate in the l 0 t l1 norm for bv data this result was expected from numerical experiments and is optimal

refined one dimensional formulations for laminated structure analysis this paper proposes one dimensional formulations based on hierarchical expansions of the unknown displacement variables for the analysis of multilayered structures made of anisotropic composite layers the hierarchical technique shows variable kinematic properties and it is based on the carrera unified formulation two different classes of refined theories are proposed the first expands the unknown variables in terms of power polynomials of the cross sectional coordinates it consists of a taylor like expansion the second class of onedimensional theories uses lagrange polynomials lagrange expansion and subdomain discretizations of the cross section and it leads to only pure displacements as the unknown variables taylor like expansion is used to develop equivalent single layer formulations and lagrange expansion is used to construct both equivalent single layer and layerwise descriptions the finite element method is employed to develop numerical applications using the carrera unified formulation finite element matrices are obtained in terms of a few fundamental nuclei that are formally independent of all the considered one dimensional formulations a number of numerical examples are given concerning on beams plates and more complex structures comparisons with results from plate and solid models are provided the following has been concluded 1 the proposed formulation represents a reliable compact and accurate method to develop refined one dimensional models 2 the present one dimensional models are very effective at detecting both global and local responses of composite structures 3 shell and solidlike results are obtained with a significant reduction in the computational costs

a priori error estimates for approximate solutions to convex conservation laws summary we introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite difference approximation calculated with the scheme of engquistosher lax friedrichs or godunov this technique is a discrete counterpart of the duality technique introduced by tadmor siam j numer anal 1991 the error is related to the consistency error of cell averages of the entropy weak solution this consistency error can be estimated by exploiting a regularity structure of the entropy weak solution one ends up with optimal error estimates

guidelines and recommendations to construct theories for metallic and composite plates this work has evaluated the refinement of some classical theories such as the kirchhoff and reissner mindlin theories adding generalized displacement variables up to fourth order to the taylor type expansion in the thickness plate direction isotropic orthotropic and laminated plates have been analyzed varying the thickness ratio orthotropic ratio and stacking sequence of the layout higher order theories have been implemented according to the compact scheme known as the carrera unified formulation the results have been restricted to simply supported orthotropic plates subjected to harmonic distributions of transverse pressure for which closed form solutions are available for a given plate problem isotropic orthotropic or laminated the effectiveness of each employed generalized displacement variable has been established comparing the error obtained accounting for and removing the variable in the plate governing equations a number of theories have therefore been constructed imposing a given error with respect to the available best results guidelines and recommendations that are focused on the proper selection of the displacement variables that have to be retained in refined plate theories are then furnished it has been found that the terms that have to be used according to a given error vary from problem to problem but they also vary when the variable that has to be evaluated displacement stress components is changed diagrams errors in terms of geometrical and orthotopic ratios and graphical schemes have been built to establish the appropriate theories with respect to the data of the problem under consideration

l infty and l 2 error estimates for a finite volume approximation of linear advection we study the convergence of the upwind finite volume scheme applied to the linear advection equation with a lipschitz divergence free speed in r d we prove an h 1 2 varepsilon error estimate in the l infty r d times 0 t norm for lipschitz initial data the expected optimal result is an h 1 2 error estimate in a second part we also prove an h 1 2 error estimate in the l infty 0 t l 2 r d norm for initial data in h 1 r d

a finite difference formula for the discretization of d 3 dx 3 on nonuniform grids we analyze the use of a five point difference formula for the dis cretization of the third derivative operator on nonuniform grids the formula was derived so as to coincide with the standard five point formula on regular grids and to lead to skew symmetric schemes it is shown that under periodic boundary conditions the formula is supraconvergent in the sense that in spite of being inconsistent it gives rise to schemes with second order convergence however such a supraconvergence only takes place when the number of points in the grid is odd for grids with an even number of points the inconsistency of the formula results in lack of convergence both stationary and evolutionary problems are considered and the analysis is backed by numerical experiments

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

a stable adaptive numerical scheme for hyperbolic conservation laws a new adaptive finite difference scheme for scalar hyperbolic conservation laws is introduced a key aspect of the method is a new automatic mesh selection algorithm for problems with shocks we show that the scheme is l 1 stable in the sense of kuznetsov and that it generates convergent approximations for linear problems numerical evidence is presented that indicates that if an error of size varepsilon is required our scheme takes at most o varepsilon 3 operations standard monotone difference schemes can take up to o varepsilon 4 calculations for the same problems

general behavior and shear center location of prismatic anisotropic beams via power series abstract the behavior of a tip loaded anisotropic cantilever beam with an arbitrary cross section is studied using saint venant s semi inverse method along with a power series solution for the local in plane and out of plane deformation warping functions the power series coefficients are determined by solving a set of variationally derived linear algebraic equations using the resulting three dimensional displacement solutions the shear deformation associated with applied tip loads is investigated as well as the shear center location two different definitions of the shear center are presented for anisotropic beams by extending existing approaches developed for isotropic sections both of the extended definitions reveal the linear dependency of the shear center location on beam length numerical results are presented for three different cross sections ellipse triangle naca 0012 airfoil and two different materials al 6061 t6 off angle high strength graphite epoxy fibers

asymptotic theory for static behavior of elastic anisotropic i beams abstract end effects for prismatic anisotropic beams with thin walled open cross sections are analyzed by the variational asymptotic method the decay rates for disturbances at the ends of prismatic beams are evaluated and the most influential end disturbances are incorporated into a refined beam theory thus the foundations of vlasovs theory as well as restrictions on its applicability are obtained from the variational asymptotic point of view vlasovs theory is proved to be asymptotically correct for isotropic i beams the asymptotically correct generalization of vlasovs theory for static behavior of anisotropic beams is presented in light of this development various published generalizations of vlasovs theory for thin walled anisotropic beams are discussed comparisons with a numerical 3 d analysis are provided showing that the present approach gives the closest agreement of all published theories the procedure can be applied to any thin walled beam with open cross sections

sign stability of positive markov jump linear systems abstract this paper investigates stability of positive markov jump linear systems pmjlss in the absence of a numerical realization it considers the situation when only signs and not magnitudes of the entries of the subsystem matrices and the markov transition matrices are known the result is an analysis of a qualitative notion of stability known as sign stability although the notions of sign stability of pmjlss are natural extensions of standard stochastic stability concepts such as exponential almost sure mean and mean square stability the sign stability notions are proven equivalent which is not the case for their corresponding standard concepts moreover for irreducible markov chains the particular structure of the markov chain is shown to have no bearing on sign stability

stability analysis of linear systems subject to regenerative switchings this paper investigates the stability of switched linear systems whose switching signal is modeled as a stochastic process called a regenerative process we show that the mean stability of such a switched system is characterized by the spectral radius of a matrix the matrix is obtained by taking the expectation of the transition matrix of the system on one cycle of the underlying regenerative process the characterization generalizes floquet s theorem for the stability analysis of linear time periodic systems we illustrate the result with the stability analysis of a linear system with a failure prone controller under periodic maintenance

a priori error estimates for numerical methods for scalar conservation laws part i the general approach we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original kuznetsov approximation theory as a first application of this general technique we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution thus we obtain optimal error estimates for the engquist osher scheme without using the fact i that the solution is uniformly bounded ii that the scheme is total variation diminishing and iii that the discrete semigroup associated with the scheme has the l 1 contraction property which guarantees an upper bound for the modulus of continuity in time of the approximate solution

stability and stabilization for positive systems with semi markov switching abstract in this article the problems of mean stability analysis and control synthesis are studied for stochastic switching systems subject to positive constraint such a switching is governed by a semi markov process subject to a special non exponential distribution considering a linear lyapunov krasovskii function lkf necessary and sufficient conditions are proposed to realize mean stability for the open loop system based on this the solvability conditions for the desired stabilizing controller can be determined under a linear programming lp framework finally the theoretical findings are illustrated by the virus mutation treatment model

generalized beam theory applied to shear stiffness there is no consensus of opinion on the correct expressions for shear stiffness even in the apparently simple case of rectangular sections made from homogeneous isotropic material a general beam theory has been proposed which is applicable to all regular prismatic systems this has been used to find the appropriate beam like flexibilities for trusses the same approach can be used for normal beams giving values for the shear stiffnesses of various cross sections as particular results of a general theory embracing torsion binding extension and shear of regular prismatic systems

the convergence rate of godunov type schemes godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws the authors study the convergence rate of such sche