convergence and superconvergence of staggered discontinuous galerkin methods for the three dimensional maxwell s equations on cartesian grids in this paper a new type of staggered discontinuous galerkin methods for the three dimensional maxwell s equations is developed and analyzed the spatial discretization is based on staggered cartesian grids so that many good properties are obtained first of all our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the gauss law moreover the mass matrices are diagonal thus time marching is explicit and is very efficient our method is high order accurate and the optimal order of convergence is rigorously proved it is also very easy to implement due to its cartesian structure and can be regarded as a generalization of the classical yee s scheme as well as the quadrilateral edge finite elements furthermore a superconvergence result that is the convergence rate is one order higher at interpolation nodes is proved numerical results are shown to confirm our theoretical statements and applications to problems in unbounded domains with the use of pml are presented a comparison of our staggered method and non staggered method is carried out and shows that our method has better accuracy and efficiency

high order exactly well balanced numerical methods for shallow water systems abstract this paper is concerned with the development of high order methods for the shallow water system that preserve every stationary solution of the system and not only the water at rest ones the pursued strategy is based on the generalized hydrostatic reconstruction introduced in 10 and on the technique proposed in 8 to obtain high order well balanced reconstruction operators the more difficult step in the reconstruction procedure is the first one given a cell value a stationary solution of the system has to be found whose average at the cell equals the given value the implementation of this step is discussed in detail as well as the similarities and differences with some previous approaches to obtain well balanced methods for the shallow water system finally the well balanced properties of the developed numerical methods are checked in some test cases

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

analysis of semi implicit dgfem for nonlinear convection diffusion problems on nonconforming meshes the paper deals with the numerical analysis of a scalar nonstationary nonlinear convection diffusion equation the space discretization is carried out by the discontinuous galerkin finite element method dgfem on general nonconforming meshes formed by possibly nonconvex elements with nonsymmetric treatment of stabilization terms and interior and boundary penalty the time discretization is carried out by a semi implicit euler scheme in which the diffusion and stabilization terms are treated implicitly whereas the nonlinear convective terms are treated explicitly we derive a priori asymptotic error estimates in the discrete l l2 l l2 norm l2 h1 l2 h1 seminorm and l h1 l h1 seminorm with respect to the mesh size h and time step numerical examples demonstrate the accuracy of the method and manifest the effect of nonconvexity of elements and nonconformity of the mesh

matrix pencil of closed loop descriptor systems infinite eigenvalue assignment abstract the regular matrix pencil se a bk is considered that characterizes a closed loop descriptor system u t kx t t a carefully contrived and efficient method is given for the determination of a set of matrices k such that the determinant se a bk is a constant value independent of s the problem is formulated as an infinite eigenvalue assignment by finite gain descriptor variable feedback via a singular value decomposition of the matrix e the result is interesting in its own right and finds application in controller and observer design

maximal rank of an element of a tensor product abstract upper bounds are given for the maximal rank of an element of the tensor product of three vector spaces

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

a well balanced flux vector splitting scheme designed for hyperbolic systems of conservation laws with source terms abstract we propose a way to construct robust numerical schemes for the computations of numerical solutions of one and two dimensional hyperbolic systems of balance laws in order to reduce the computational cost we selected the family of flux vector splitting schemes we reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations this is applied to a 1d shallow water system with topography and to a 2d simplified model of two phase flows with damping effects numerical results and comparisons with a classical centered discretizations scheme are supplied

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

stability accuracy and efficiency of a semi implicit method for three dimensional shallow water flow abstract the stability analysis the accuracy and the efficiency of a semi implicit finite difference scheme for the numerical solution of a three dimensional shallow water model are presented and discussed the governing equations are the three dimensional reynolds equations in which pressure is assumed to be hydrostatic the pressure gradient in the momentum equations and the velocities in the vertically integrated continuity equation are discretized with the method with being an implicitness parameter it is shown that the method is stable for 1 2 1 unstable for 1 2 and highest accuracy and efficiency is achieved when 1 2 the resulting algorithm is mass conservative and naturally allows for the simulation of flooding and drying of tidal flats

controllability of descriptor systems different concepts of controllability for descriptor systems ex ax bu have been proposed and investigated in this paper their relationship their interpretation in terms of the kronecker canonical form of e a and the implications in terms of pole assignment regularity and robustness of the closed loop system are discussed

on the design of observers for generalized state space systems using singular value decomposition in this paper a method is developed for the design of luenberger type observers for linear time invariant control systems whose state equation is of the form ex ax bu where e is a singular square matrix the method is based on the singular value decomposition of the matrix e and on the reduction of the equation ex ax bu to a system consisting of a differential equation of the form w1 f1w1 f2 w2 g1u and an algebraic equation of the form h1w1 h2w2 g1u 0 if w2 can be eliminated from the differential equation by the aid of the algebraic equation and the original output equation of the system the method yields a reduced order observer for the generalized state space system

a nested newton type algorithm for finite volume methods solving richards equation in mixed form a finite volume discretization of the mixed form of richards equation leads to a nonlinear numerical model which yields exact local and global mass conservation the resulting nonlinear system requires sophisticated numerical strategies especially in a variable saturated flow regime in this paper a nested newton type algorithm for the discretized richards equation is proposed and analyzed with a judicious choice of the initial guess the quadratic convergence rate is obtained for any time step size and for all flow regimes

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

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

a staggered semi implicit spectral discontinuous galerkin scheme for the shallow water equations a spatially arbitrary high order semi implicit spectral discontinuous galerkin dg scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed the free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis and as test functions each unknown is however defined on a different set of control volumes that are spatially staggered with respect to each other a semi implicit time integration yields a stable and efficient mass conservative algorithm the use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block penta diagonal system must be solved for the new free surface location which is a scalar quantity subsequently the new momentum components are directly obtained the staggered semi implicit approach makes the present dg scheme different from other published dg schemes the sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix free gmres algorithm furthermore for the shallow water equations the proposed scheme can be written as a quadrature free method where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors for the special case n 0 the proposed method reduces to a classical semi implicit finite difference scheme the proposed semi implicit scheme is particularly well suited for low froude number flows the method is validated on some typical academic benchmark problems using polynomial degrees of up to n 20

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

iterative solutions of mildly nonlinear systems the correct numerical modelling of free surface hydrodynamics often requires the solution of diagonally nonlinear systems in doing this one may substantially enhance the model accuracy while fulfilling relevant physical constraints this is the case when a suitable semi implicit discretization is used e g to solve the one dimensional or the multi dimensional shallow water equations to model axially symmetric flows in compliant arterial systems to solve the boussinesq equation in confined unconfined aquifers or to solve the mixed form of the richards equation in this paper two nested iterative methods for solving a mildly nonlinear system of the form v h email protected b are proposed and analysed it is shown that the inner and the outer iterates are monotone and converge to the exact solution for a wide class of mildly nonlinear systems of applied interest a simple and yet non trivial test problem derived from the mathematical modelling of flows in porous media is formulated and solved with the proposed methods

hp adaptive discontinuous galerkin finite element methods for first order hyperbolic problems we consider the a posteriori error analysis of hp discontinuous galerkin finite element approximations to first order hyperbolic problems in particular we discuss the question of error estimation for linear functionals such as the outflow flux and the local average of the solution based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance this involves exploiting both local polynomial degree variation and local mesh subdivision the theoretical results are illustrated by a series of numerical experiments

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

why many theories of shock waves are necessary convergence error in formally path consistent schemes we are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics and for solutions containing shock waves we investigate the convergence of finite difference schemes applied to such systems according to dal maso lefloch and murat s theory a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space in the present paper we consider schemes that are formally consistent with a given family of paths and we investigate their limiting behavior as the mesh is refined we first generalize to systems a property established earlier by hou and lefloch for scalar conservation laws and we prove that nonconservative schemes generate at the level of the limiting hyperbolic system an convergence error source term which provided the total variation of the approximations remains uniformly bounded is a locally bounded measure this convergence error measure is supported on the shock trajectories and as we demonstrate here is usually small in the special case that the scheme converges in the sense of graphs a rather strong convergence property often violated in practice then this measure source term vanishes we also discuss the role of the equivalent equation associated with a difference scheme here the distinction between scalar equations and systems appears most clearly since for systems the equivalent equation of a scheme that is formally path consistent depends upon the prescribed family of paths the core of this paper is devoted to investigate numerically the approximation of several simplified or full hyperbolic models arising in fluid dynamics this leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields the convergence error vanishes for more general models this measure is evaluated very accurately especially by plotting the shock curves associated with each scheme under consideration as we demonstrate plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme

doubly coprime matrix fraction representations using proportional and derivative feedback concepts in generalized state space systems the concepts of proportional and derivative state feedback are used to derive some explicit formulas for doubly coprime matrix fraction representations of generalized state space systems specifically proportional and derivative state feedback is used to determine the doubly coprime matrix fraction representations and to solve the corresponding generalized bezout identity in polynomial matrix form

iterative solution of piecewise linear systems and applications to flows in porous media the correct numerical modeling of free surface hydrodynamic problems often requires to have the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself in doing so one may fulfill relevant physical constraints the existence the uniqueness and two constructive iterative methods to solve a piecewise linear system of the form max boldsymbol l min boldsymbol u mathbf x t mathbf x mathbf b are analyzed the methods are shown to have a finite termination property i e they converge to an exact solution in a finite number of steps and actually they converge very quickly as confirmed by a few numerical tests which are derived from the mathematical modeling of flows in porous media

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

a high order semi implicit discontinuous galerkin method for the two dimensional shallow water equations on staggered unstructured meshes a well balanced spatially arbitrary high order accurate semi implicit discontinuous galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non orthogonal grids the semi implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime for p 0 the resulting scheme becomes a generalization of the classical semi implicit finite volume finite difference scheme of casulli and walters 2000 25 but with less conditions on the grid geometry the method proposed in this paper allows large time steps with respect to the surface wave speed gh and is thus particularly suitable for low froude number flows the approach is validated on some typical academic benchmark problems using polynomial degrees up to p 6

singular systems state feedback problem in this paper the problem of kronecker invariants assignment by state feedback in singular linear systems is studied and resolved this result presents a generalization of the previous results of state feedback action on singular systems

deflation in eigenvalue assignment of descriptor systems using state feedback it is shown how deflation by unitary equivalence transformations can be used to solve the eigenvalue assignment problem for descriptor systems using state feedback the system is first transformed to a form that reveals its controllability the uncontrollable part of the system if any is then discarded and the assignments are continued with the completely controllable part of the system to give insight into the approach a mathematical method for the solution is derived without considering numerical issues using the idea of deflation it is shown how this method can be modified to derive a numerically efficient algorithm the numerical stability of the algorithm is proved and a numerical example is presented

a semi implicit semi lagrangian p adaptive discontinuous galerkin method for the shallow water equations a semi implicit and semi lagrangian discontinuous galerkin method for the shallow water equations is proposed for applications to geophysical scale flows a non conservative formulation of the advection equation is employed in order to achieve a more treatable form of the linear system to be solved at each time step the method is equipped with a simple p adaptivity criterion that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution numerical results show that the method captures well the main features of gravity and inertial gravity waves as well as reproducing correct solutions in nonlinear test cases with analytic solutions the accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high courant numbers and with automatic choice of the local approximation degree

a new algorithm for an eigenvalue assignment problem from singular control theory in this paper a new numerical algorithm for an eigenvalue assignment problem which arises from a singular system control is developed the algorithm is based on orthogonal row column compressions which can be implemented in a numerically reliable way

eigenvalue placement for generalized linear systems this paper deals with some aspects of eigenvalue placement by state feedback for generalized linear systems described by e ax bu where e is a singular map it is shown that controllability of the infinite eigenvalues of the pencil se a is equivalent to the existence of a state feedback map which assigns those eigenvalues to pre specified complex numbers a procedure for the assignment of all eigenvalues to distinct complex numbers is also discussed

an optimal order interior penalty discontinuous galerkin discretization of the compressible navier stokes equations in this article we propose a new symmetric version of the interior penalty discontinuous galerkin finite element method for the numerical approximation of the compressible navier stokes equations here particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest such as the lift and drag coefficients of a body immersed in a viscous fluid with this in mind the key ingredients in the construction of the method include i an adjoint consistent imposition of the boundary conditions ii an adjoint consistent reformulation of the underlying target functional of practical interest iii design of appropriate interior penalty stabilization terms numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the l 2 norm as well as for certain target functionals computational comparisons with other discontinuous galerkin schemes proposed in the literature including the second scheme of bassi and rebay cf f bassi s rebay gmres discontinuous galerkin solution of the compressible navier stokes equations in b cockburn g karniadakis c w shu eds discontinuous galerkin methods lecture notes in comput sci engrg vol 11 springer berlin 2000 pp 197 208 f bassi s rebay numerical evaluation of two discontinuous galerkin methods for the compressible navier stokes equations int j numer methods fluids 40 2002 197 207 the standard sipg method outlined in r hartmann p houston symmetric interior penalty dg methods for the compressible navier stokes equations i method formulation int j numer anal model 3 1 2006 1 20 and an nipg variant of the new scheme will be undertaken

semi implicit finite difference methods for the two dimensional shallow water equation abstract in this paper a semi implicit finite difference method for the 2 dimensional shallow water equations is derived and discussed a characteristic analysis of the governing equations is carried out first in order to determine those terms to be discretized implicitly so that the stability of the method will not depend upon the celerity such terms are the gradient of the water surface elevation in the momentum equations and the velocity divergence in the continuity equation the convective terms are discretized explicitly the simpler explicit discretization for the convective terms is the upwind discretization which is conditionally stable and introduces some artificial viscosity it is shown that the stability restriction is eliminated and the artificial viscosity is reduced when an eulerian lagrangian approach with large time steps is used to discretize the convective terms this method at each time step requires the solution of a linear symmetric 5 diagonal system such a system is diagonally dominant with positive elements on the main diagonal and negative ones elsewhere thus existence and uniqueness of the numerical solution is assured the resulting algorithm is mass conservative and fully vectorizable for an efficient implementation on modern vector computers the performance of this method is further improved when used in combination with an adi technique which results in two sets of simpler linear 3 diagonal systems and maintains all the properties described above

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed

observers for discrete singular systems the author considers the design of observers for the discrete singular system ex k 1 ax k bu k y k cx k placing special emphasis on the problems of state reconstruction and minimal time state reconstruction it is shown that for a singular system finite poles can be moved to infinity by state feedback and the state can be reconstructed by causal observers

parametric eigenstructure assignment for continuous time descriptor systems abstract eigenstructure assignment for continuous time descriptor systems ex t ax t bu t via descriptor variable feedback control u t kx t is treated a new parametric form for the gain matrix k which assigns rank e arbitrary finite eigenvalues to the closed loop system is developed this form embraces two sets of design parameter vectors characterizing respectively the assignable eigenvectors associated with the finite eigenvalues and a necessary and sufficient condition for closed loop system regularity in addition a certain submatrix of k turns out to be totally free whenever nullity e exceeds rank b unlike earlier methods the eigenvectors associated with the infinite eigenvalues are dispensed with in the solution algorithm

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