smoothness increasing accuracy conserving siac postprocessing for discontinuous galerkin solutions over structured triangular meshes theoretically and computationally it is possible to demonstrate that the order of accuracy of a discontinuous galerkin dg solution for linear hyperbolic equations can be improved from order k 1 to 2 k 1 through the use of smoothness increasing accuracy conserving siac filtering however it is a computationally complex task to perform this in an efficient manner which becomes an even greater issue considering nonquadrilateral mesh structures in this paper we present an extension of this siac filter to structured triangular meshes the basic theoretical assumption in the previous implementations of the postprocessor limits the use to numerical solutions solved over a quadrilateral mesh however this assumption is restrictive which in turn complicates the application of this postprocessing technique to general tessellations additionally moving from quadrilateral meshes to triangulated ones introduces more complexity in the calculations as the number of integrations required increases in this paper we extend the current theoretical results to variable coefficient hyperbolic equations over structured triangular meshes and demonstrate the effectiveness of the application of this postprocessor to structured triangular meshes as well as exploring the effect of using inexact quadrature we show that there is a direct theoretical extension to structured triangular meshes for hyperbolic equations with bounded variable coefficients this is a challenging first step toward implementing siac filters for unstructured tessellations we show that by using the usual b spline implementation we are able to improve on the order of accuracy as well as decrease the magnitude of the errors these results are valid regardless of whether exact or inexact integration is used the results here demonstrate that it is still possible both theoretically and computationally to improve to 2 k 1 over the dg solution itself for structured triangular meshes

stability and error estimates of local discontinuous galerkin methods with implicit explicit time marching for advection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with carefully chosen implicit explicit imex runge kutta time discretization up to third order accuracy for solving one dimensional linear advection diffusion equations in the time discretization the advection term is treated explicitly and the diffusion term implicitly there are three highlights of this work the first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the ldg methods the second is that by aid of the aforementioned relationship and the energy method we show that the imex ldg schemes are unconditionally stable for the linear problems in the sense that the time step tau is only required to be upper bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

three best binary linear block codes of minimum distance fifteen three new best codes 37 9 15 42 12 15 and 48 16 15 their weight distributions and a brief description of the used search algorithms are presented

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

stability of uncertain systems using lyapunov functions with non monotonic terms abstract this paper is concerned with the problem of robust stability of uncertain linear time invariant systems in polytopic domains the main contribution is to present a systematic procedure to check the stability of the uncertain systems by using an arbitrary number of quadratic functions within higher order derivatives of the vector field in the continuous time case and higher order differences of the vector field in the discrete time case the matrices of the lyapunov function appear decoupled from the dynamic matrix of the system in the conditions this fact leads to sufficient conditions that are given in terms of linear matrix inequalities defined at the vertices of the polytope the proposed method does not impose sign condition constraints in the quadratic functions that compose the lyapunov function individually moreover some of the quadratic functions do not decrease monotonically along trajectories however if the sufficient conditions are satisfied then a monotonic standard lyapunov function that depends on the dynamics of the uncertain system can be constructed a posteriori numerical examples from the literature are provided to illustrate the proposed approach

analysis of optimal superconvergence of discontinuous galerkin method for linear hyperbolic equations in this paper we study the superconvergence of the error for the discontinuous galerkin dg finite element method for linear conservation laws when upwind fluxes are used we prove that if we apply piecewise k th degree polynomials the error between the dg solution and the exact solution is k 2 th order superconvergent at the downwind biased radau points with suitable initial discretization moreover we also prove the dg solution is k 2 th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution the superconvergence result in this paper leads to a new a posteriori error estimate our analysis is valid for arbitrary regular meshes and for mathcal p k polynomials with arbitrary k geq1 and for both periodic boundary conditions and for initial boundary value problems we perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal

smoothness increasing accuracy conserving siac filters for derivative approximations of discontinuous galerkin dg solutions over nonuniform meshes and near boundaries accurate approximations for the derivatives are usually required in many application areas such as biomechanics chemistry and visualization applications with the help of smoothness increasing accuracy conserving siac filtering one can enhance the derivatives of a discontinuous galerkin solution however current investigations of derivative filtering are limited to uniform meshes and periodic boundary conditions which do not meet practical requirements the purpose of this paper is twofold to extend derivative filtering to nonuniform meshes and propose position dependent derivative filters to handle filtering near the boundaries through analyzing the error estimates for siac filtering we extend derivative filtering to nonuniform meshes by changing the scaling of the filter for filtering near boundaries we discuss the advantages and disadvantages of two existing position dependent filters and then extend them to position dependent derivative filters respectively further we prove that with the position dependent derivative filters the filtered solutions can obtain a better accuracy rate compared to the original discontinuous galerkin approximation with arbitrary derivative orders over nonuniform meshes one and two dimensional numerical results are provided to support the theoretical results and demonstrate that the position dependent derivative filters in general enhance the accuracy of the solution for both uniform and nonuniform meshes

integrator forwarding a new recursive nonlinear robust design we consider the global stabilization of nonlinear systems in strict feedforward form we show that these systems while not feedback linearizable can be globally transformed by feedback and diffeomorphism into lower triangular form the transformation is explicit and recursive we employ this transformation to design a globally stabilizing controller with optimality properties and an input to state stability property with respect to matched uncertainties c 1997 elsevier science ltd

dynamic approximate solutions of the hj inequality and of the hjb equation for input affine nonlinear systems the solution of most nonlinear control problems hinges upon the solvability of partial differential equations or inequalities in particular disturbance attenuation and optimal control problems for nonlinear systems are generally solved exploiting the solution of the so called hamilton jacobi hj inequality and the hamilton jacobi bellman hjb equation respectively an explicit closed form solution of this inequality or equation may however be hard or impossible to find in practical situations herein we introduce a methodology to circumvent this issue for input affine nonlinear systems proposing a dynamic i e time varying approximate solution of the hj inequality and of the hjb equation the construction of which does not require solving any partial differential equation or inequality this is achieved considering the immersion of the underlying nonlinear system into an augmented system defined on an extended state space in which a locally positive definite storage function or value function can be explicitly constructed the result is a methodology to design a dynamic controller to achieve l2 disturbance attenuation or approximate optimality with asymptotic stability

dynamic lyapunov functions lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of linear and nonlinear systems the existence of lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse lyapunov theorems nevertheless the actual computation of the analytic expression of the function may be difficult herein we propose an approach to avoid the computation of an explicit solution of the lyapunov partial differential inequality introducing the concept of dynamic lyapunov function these functions allow to study stability properties of equilibrium points similarly to standard lyapunov functions in the former however a positive definite function is combined with a dynamical system that render dynamic lyapunov functions easier to construct than lyapunov functions moreover families of standard lyapunov functions can be obtained from the knowledge of a dynamic lyapunov function by rendering invariant a desired submanifold of the extended state space the invariance condition is given in terms of a system of partial differential equations similar to the lyapunov pde differently from the latter however in the former no constraint is imposed on the sign of the solution or on the sign of the term on the right hand side of the equation several applications and examples conclude the paper

enhanced accuracy by post processing for finite element methods for hyperbolic equations we consider the enhancement of accuracy by means of a simple post processing technique for finite element approximations to transient hyperbolic equations the post processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes if the mesh is locally translation invariant the support of the kernel is a cube whose edges are of size of the order of x only for example when polynomials of degree k are used in the discontinuous galerkin dg method and the exact solution is globally smooth the dg method is of order k 1 2 in the l2 norm whereas the post processed approximation is of order 2k 1 if the exact solution is in l2 only in which case no order of convergence is available for the dg method the post processed approximation converges with order k 1 2 in l2 0 where 0 is a subdomain over which the exact solution is smooth numerical results displaying the sharpness of the estimates are presented

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

discontinuous galerkin methods for nonlinear scalar hyperbolic conservation laws divided difference estimates and accuracy enhancement in this paper an analysis of the accuracy enhancement for the discontinuous galerkin dg method applied to one dimensional scalar nonlinear hyperbolic conservation laws is carried out this requires analyzing the divided difference of the errors for the dg solution we therefore first prove that the alpha th order 1 le alpha le k 1 1 k 1 divided difference of the dg error in the l 2 l2 norm is of order k frac 3 2 frac alpha 2 k 32 2 when upwind fluxes are used under the condition that f u f u possesses a uniform positive lower bound by the duality argument we then derive superconvergence results of order 2k frac 3 2 frac alpha 2 2k 32 2 in the negative order norm demonstrating that it is possible to extend the smoothness increasing accuracy conserving filter to nonlinear conservation laws to obtain at least frac 3 2 k 1 32k 1 th order superconvergence for post processed solutions as a by product for variable coefficient hyperbolic equations we provide an explicit proof for optimal convergence results of order k 1 k 1 in the l 2 l2 norm for the divided differences of dg errors and thus 2k 1 2k 1 th order superconvergence in negative order norm holds numerical experiments are given that confirm the theoretical results

two chain rules for divided differences and faa di bruno s formula in this paper we derive two formulas for divided differences of a function of a function both formulas lead to other divided difference formulas such as reciprocal and quotient rules the two formulas can also be used to derive faa di bruno s formula and other formulas for higher derivatives of composite functions we also derive a divided difference version of faa di bruno s determinant formula

superconvergence of discontinuous galerkin and local discontinuous galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension in this paper we study the superconvergence property for the discontinuous galerkin dg and the local discontinuous galerkin ldg methods for solving one dimensional time dependent linear conservation laws and convection diffusion equations we prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection diffusion equations the order of superconvergence for both cases is proved to be k frac 3 2 when piecewise p k polynomials with k geq1 are used the proof is valid for arbitrary nonuniform regular meshes and for piecewise p k polynomials with arbitrary k geq1 improving upon the results in y cheng and c w shu j comput phys 227 2008 pp 9612 9627 y cheng and c w shu computers and structures 87 2009 pp 630 641 in which the proof based on fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise p 1 polynomials

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

new 52 18 16 linear binary block code the generator matrix and the weight distribution of a new 52 18 16 linear binary block code are presented the minimum distance of the known 52 18 15 linear binary block code is improved achieving the upper bound on minimum distance for length 52 and dimension 18

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

postprocessing for the discontinuous galerkin method over nonuniform meshes a postprocessing technique based on negative order norm estimates for the discontinuous galerkin methods was previously introduced by cockburn luskin shu and suli proceedings of the international symposium on discontinuous galerkin methods springer new york pp 291 300 math comput 72 2003 pp 577 606 the postprocessor allows improvement in accuracy of the discontinuous galerkin method for time dependent linear hyperbolic equations from order k 1 to order 2 k 1 over a uniform mesh assumptions on the convolution kernel along with uniformity in mesh size give a local translation invariant postprocessor that allows for simple implementation using small matrix vector multiplications in this paper we present two alternatives for extending this postprocessing technique to include smoothly varying meshes the first method uses a simple local l 2 projection of the smoothly varying mesh to a locally uniform mesh and uses this projected solution to compute the postprocessed solution by using this local l 2 projection recalculating the convolution kernel for every element can be avoided and 2 k 1 order accuracy of the postprocessed solution can be achieved the second method uses the idea of characteristic length based upon the largest element size for the scaling of the postprocessing kernel these two methods local projection and characteristic length are also applied to approximations over a mesh with elements that vary in size randomly we discuss the computational issues in using these two techniques and demonstrate numerically that we obtain the 2 k 1 order of accuracy for the smoothly varying meshes and that although the 2 k 1 order of accuracy is not fully realized for random meshes there is significant improvement in the l 2 errors

a note on the augmented hessian when the reduced hessian is semidefinite certain matrix relationships play an important role in optimality conditions and algorithms for nonlinear and semidefinite programming let h be an n n symmetric matrix a an m n matrix and z a basis for the null space of a in a typical optimization context h is the hessian of a smooth function and a is the jacobian of a set of constraints when the reduced hessian zthz is positive definite augmented lagrangian methods rely on the known existence of a finite bar rho ge 0 such that for all rho bar rho the augmented hessian h rho ata is positive definite in this note we analyze the case when zthz is positive semidefinite i e singularity is allowed and show that the situation is more complicated in particular we give a simple necessary and sufficient condition for the existence of a finite bar rho so that h rho ata is positive semidefinite for rho ge bar rho a corollary of our result is that if h is nonsingular and indefinite while zthz is positive semidefinite and singular no such bar rho exists

superconvergence of discontinuous galerkin methods for linear hyperbolic equations in this paper we study superconvergence properties of the discontinuous galerkin dg method for one dimensional linear hyperbolic equations when upwind fluxes are used we prove for any polynomial degree k the 2k 1 th or 2k 1 2 th superconvergence rate of the dg approximation at the downwind points and for the domain average under quasi uniform meshes and some suitable initial discretization moreover we prove that the derivative approximation of the dg solution is superconvergent with a rate k 1 at all interior left radau points all theoretical findings are confirmed by numerical experiments

superconvergence of discontinuous galerkin methods for scalar nonlinear conservation laws in one space dimension in this paper an analysis of the superconvergence property of the semidiscrete discontinuous galerkin dg method applied to one dimensional time dependent nonlinear scalar conservation laws is carried out we prove that the error between the dg solution and a particular projection of the exact solution achieves left k frac32 right th order superconvergence when upwind fluxes are used the results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree k k ge 1 under the condition that f u possesses a uniform positive lower bound numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions indicating that the restriction on f u is artificial

new 47 15 16 linear binary block code a new 47 15 16 linear binary block code and its weight spectrum is presented the code is better than the previously known 47 15 16 code and it reaches the upper bound on code distance for the codeword length 47 and dimension 15

dynamic generalized controllability and observability functions with applications to model reduction and sensor deployment in this paper we introduce the notion of dynamic generalized controllability and observability functions for nonlinear systems these functions are called dynamic and generalized since they make use of additional states dynamic extension and are such that partial differential inequalities are solved in place of equations the presence of the dynamic extension permits the construction of classes of canonical controllability and observability functions without relying on the solution of any partial differential equation or inequality the effectiveness of the proposed concept is validated by means of two applications the model reduction problem via balancing and the sensor deployment problem in a continuous stirred tank reactor cstr

accuracy enhancement of discontinuous galerkin solutions for convection diffusion equations in multiple dimensions discontinuous galerkin dg methods exhibit hidden accuracy that makes superconvergence of this method an increasing popular topic to address previous investigations have focused on the superconvergent properties of ordinary differential equations and linear hyperbolic equations additionally superconvergence of order k 3 2 for the convection diffusion equation that focuses on a special projection using the upwind flux was presented by cheng and shu in this paper we demonstrate that it is possible to extend the smoothness increasing accuracy conserving siac filter for use on the multidimensional linear convection diffusion equation in order to obtain 2k m order of accuracy where m depends upon the flux and takes on the values 0 1 2 or 1 the technique that we use to extract this hidden accuracy was initially introduced by cockburn luskin shu and suli for linear hyperbolic equations and extended by ryan et al as a smoothness increasing accuracy conserving filter we solve this convection diffusion equation using the local discontinuous galerkin ldg method and show theoretically that it is possible to obtain o h2k m in the negative order norm by post processing the ldg solution to a linear convection equation using a specially designed kernel such as the one by cockburn et al we can compute this same order accuracy in the l2norm additionally we present numerical studies that confirm that we can improve the ldg solution from o hk 1 to o h2k 1 using alternating fluxes and that we actually obtain o h2k 2 for diffusion dominated problems

application of discontinuous galerkin method in supersonic and hypersonic gas flows abstract most high order methods often suffer from two and three dimensional unstructured grids particularly for supersonic and hypersonic gas flows we applied a high order mixed modal discontinuous galerkin method with a coupled slope limiter to investigate multidimensional cases of supersonic and hypersonic gas flows furthermore we provide detailed implementations and validations of two and three dimensional problems examples include double mach reflection problems and supersonic and hypersonic gas flows around a cylinder in addition we provide case studies of a 3d hypersonic vehicle to evaluate its computational performance on complex hypersonic gas flows this study can be regarded as an attempt to apply the discontinuous galerkin method to engineering problems