an adaptive direct variational grid generation method abstract variational grid generation techniques are now used to produce grids suitable for solving numerical partial differential equations in irregular geometries variational grid generation methods are very robust but slow the method considered here is a discrete variational method this method preserves the robustness of the variational method also it is very fast so it can be used in time dependent problems here we discuss geometry and solution adaption for the direct discrete grid generation method and some examples are presented to demonstrate its capabilities

an implicit explicit time discretization scheme for second order semilinear wave equations with application to dynamic boundary conditions we construct and analyze a second order implicit explicit imex scheme for the time integration of semilinear second order wave equations the scheme treats the stiff linear part of the problem implicitly and the nonlinear part explicitly this makes the scheme unconditionally stable and at the same time very efficient since it only requires the solution of one linear system of equations per time step for the combination of the imex scheme with a general abstract nonconforming space discretization we prove a full discretization error bound we then apply the method to a nonconforming finite element discretization of an acoustic wave equation with a kinetic boundary condition this yields a fully discrete scheme and a corresponding a priori error estimate

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

a pde approach to fractional diffusion a space fractional wave equation we study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers of order s in 0 1 of symmetric coercive linear elliptic second order operators in bounded domains varomega we realize fractional diffusion as the dirichlet to neumann map for a nonuniformly elliptic problem posed on the semi infinite cylinder mathcal c varomega times 0 infty we thus rewrite our evolution problem as a quasi stationary elliptic problem with a dynamic boundary condition and derive space time and space time regularity estimates for its solution the latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation we propose and analyze two fully discrete schemes the discretization in time is based on finite difference discretization techniques the trapezoidal and leapfrog schemes the discretization in space relies on the tensorization of a first degree fem in varomega with a suitable hp fem in the extended variable for both schemes we derive stability and error estimates we consider a first degree fem in varomega with mesh refinement near corners and the aforementioned hp fem in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open bounded polytopal but not necessarily convex domains varomega subset mathbb r 2 we discuss implementation details and report several numerical examples

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

a new family of exponential based high order dgtd methods for modeling 3 d transient multiscale electromagnetic problems a new family of exponential based time integration methods are proposed for the time domain maxwell s equations discretized by a high order discontinuous galerkin dg scheme formulated on locally refined unstructured meshes these methods which are developed from the lawson method remove the stiffness on the time explicit integration of the semidiscrete operator associated with the fine part of the mesh and allow for the use of high order time explicit scheme for the coarse part operator they combine excellent stability properties with the ability to obtain very accurate solutions even for very large time step sizes here the explicit time integration of the lawson transformed semidiscrete system relies on a low storage runge kutta lsrk scheme leading to a combined lawson lsrk scheme in addition efficient techniques are presented to further improve the efficiency of this exponential based time integration for the efficient calculation of matrix exponential we employ the krylov subspace method numerical experiments are presented to assess the stability verify the accuracy and numerical convergence of the lawson lsrk scheme they also demonstrate that the dg time domain methods based on the proposed time integration scheme can be much faster than those based on classical fully explicit time stepping schemes with the same accuracy and moderate memory usage increase on locally refined unstructured meshes and are thus very promising for modeling 3 d multiscale electromagnetic problems

linear variational methods and adaptive grids abstract the construction of two and three dimensional computational grids using linear variational methods is analyzed the generating equations include aspect ratios to prevent grid folding and stretching transformations to distribute grid points control functions for adaptive grids can also be included in all cases the grid points are determined by solving a diagonally dominant linear system of equations

an adaptive grid solution procedure for convection diffusion problems abstract a computationally efficient and stable adaptive grid solution procedure is developed for convection diffusion problems in this method grid refinement and adaptation is based on an equidistribution law but is only performed in regions with high error estimates that are flagged from a preliminary coarse grid solution the equidistribution law is implicit in the grid generation procedure which requires the solution of two poisson equations with control functions that are obtained directly from the error estimates or weighting functions at the grid points solution on the refined equidistributed mesh in the flagged region is obtained with boundary conditions interpolated from the coarse grid results accurate solutions in both the flagged region and the coarse grid regions of the domain are obtained with a multigrid approach that requires successive solutions on the refined equidistributed mesh in the flagged region and on the coarse mesh in the entire domain the adaptive grid method including the multigrid calculations can be extended to several levels of refinement the acronym lame is coined for this procedure in view of its local adaptation multigridding and equidistribution features the method is shown to be stable computationally efficient and accurate by applying it to three test problems and comparing with conventional calculations on a fixed curvilinear grid

numerical methods for tracking interfaces an overview is given of special numerical methods for tracking discontinuous fronts and interfaces these methods include surface tracking methods based on connected marker points along the interface volume tracking methods that track the volume occupied by the solution regions bounded by the interfaces and moving mesh methods where the underlying mesh is aligned and moved with the interface the pros and cons of the current methods are discussed and a new method is proposed that overcomes some of the difficulties encountered in approximating equations with multiple interacting interfaces

an implicit hybridized discontinuous galerkin method for the 3d time domain maxwell equations we present a time implicit hybridizable discontinuous galerkin hdg method for numerically solving the system of three dimensional 3d time domain maxwell equations this method can be seen as a fully implicit variant of classical so called dgtd discontinuous galerkin time domain methods that have been extensively studied during the last 10 years for the simulation of time domain electromagnetic wave propagation the proposed method has been implemented for dealing with general 3d problems discretized using unstructured tetrahedral meshes we provide numerical results aiming at assessing its numerical convergence properties by considering a model problem on one hand and its performance when applied to more realistic problems we also include some performance comparisons with a centered flux time implicit dgtd method

temporal convergence analysis of a locally implicit discontinuous galerkin time domain method for electromagnetic wave propagation in dispersive media this paper is concerned with the approximation of the time domain maxwell equations in a dispersive propagation medium by a discontinuous galerkin time domain dgtd method the debye model is used to describe the dispersive behavior of the medium we adapt the locally implicit time integration method from verwer 2010 and derive a convergence analysis to prove that the locally implicit dgtd method for maxwell debye equations retains its second order convergence

rational ruijsenaars schneider hierarchy and bispectral difference operators abstract we show that a monic polynomial in a discrete variable n with coefficients depending on time variables t 1 t 2 is a function for the discrete kadomtsev petviashvili hierarchy if and only if the motion of its zeros is governed by a hierarchy of ruijsenaars schneider systems these functions were considered in l haine p iliev commutative rings of difference operators and an adelic flag manifold int math res not 2000 6 2000 281 323 where it was proved that they parametrize rank one solutions to a difference differential version of the bispectral problem

error analysis of implicit euler methods for quasilinear hyperbolic evolution equations in this paper we study the convergence of the semi implicit and the implicit euler methods for the time integration of abstract quasilinear hyperbolic evolution equations the analytical framework considered here includes certain quasilinear maxwell s and wave equations as special cases our analysis shows that the euler approximations are well posed and convergent of order one the techniques will be the basis for the future investigation of higher order time integration methods and full discretizations of certain quasilinear hyperbolic problems

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

generation of computational grids using optimization a method for generation of computational grids using principles of nonlinear programming optimization is described grids are generated so that certain discrete measures of grid smoothness and orthogonality are max imized using a fast iterative optimization procedure the method can also be used to improve an existing grid ir respective of the method used for its generation if the original grid contains regions of overlap nonpositive jacobian this method is capable of unraveling the grid and making it useful for computations the iterative op timization procedure is efficient due to the use of a conjugate direction method with exact line searching examples are given of the application of this technique to two and three dimensional computational grids the extension of the method to generate solution adaptive grids is discussed n the past few years there has been a great deal of interest in the development of computational grid generation techniques1 for discretizing complex regions for the numerical solution of partial differential equations the commonly used grid generation methods have achieved a high degree of sophistication and ease of use however for complex con figurations the existing methods do not always produce ac ceptable grids for computations grid quality deterioration is especially apparent in three dimensional grid generation methods and when two dimensional grids are combined to form a three dimensio nal grid 2 3 in addition to grid quality problems many methods suffer from computational ineffi ciency for these reasons the authors have perceived a need for a grid generatio n method that could be used to improve the quality of a given computational grid in addition the method should be capable of serving as a stand alone grid generation procedure independent of other techniques the basic concept behind this grid generation method is to assume from the outset that the grid to be generated or op timized consists of straight line segments joining the nodal grid points this is in contrast to methods that solve elliptic parabolic or hyperbolic partial differential equations or use conformal mapping these methods are based on con tinuous not discrete coordinate transformations and therefore suffer from discretization errors for any grid con sisting of a finite number of grid points the grid generation technique described herein formulates discrete measures of grid smoothness and orthogonality at each grid point and therefore does not suffer from truncation errors on coarse grids these measures of grid quality are then minimized us ing a conjugate direction procedure with exact line searching the resulting grids are optimal with respect to the particular smoothness and orthogonality measures chosen and there are many possible choices for their exact forms the next section describes the formulation of the method followed by some typical results of using this method to im prove existing grids or generate new grids finally exten sions of the method for generating solution adaptive grids are discussed and other areas of possible future research are pointed out analysis the formulation of the method will be illustrated for two dimensional grids the details of the method for the three dimensional case are given in the appendix we will discuss the grid generation method by first noting similarities with the saltzman brackbill4 variational grid generation method in the variational method two func tionals are introduced that provide measures of grid smoothness

error analysis of a second order locally implicit method for linear maxwell s equations in this paper we consider the full discretization of linear maxwell s equations on spatial grids which are locally refined for such problems explicit time integration schemes become inefficient because the smallest mesh width results in a strict cfl condition recently locally implicit time integration methods have become popular in overcoming the problem of so called grid induced stiffness various such schemes have been proposed in the literature and have been shown to be very efficient however a rigorous analysis of such methods is still lacking in fact the available literature focuses on error bounds which are valid on a fixed spatial mesh only but deteriorate in the limit where the smallest spatial mesh size tends to zero moreover some important questions cannot be answered without such an analysis for example there has been no study of which elements of the spatial mesh enter the cfl condition in this paper we provide such a rigorous analysis for a locally implicit scheme proposed by verw

calogero moser hierarchy and kp hierarchy identification of the hierarchy of calogero moser dynamical systems with the dynamical systems of poles of certain solutions to the kp hierarchy is achieved

vertex operator solutions to the discrete kp hierarchy vertex operators which are disguised darboux maps transform solutions of the kp equation into new ones in this paper we show that the bi infinite sequence obtained by darboux transforming an arbitrary kp solution recursively forward and backwards yields a solution to the discrete kp hierarchy the latter is a kp hierarchy where the continuous space x variable gets replaced by a discrete n variable the fact that these sequences satisfy the discrete kp hierarchy is tantamount to certain bilinear relations connecting the consecutive kp solutions in the sequence at the grassmannian level these relations are equivalent to a very simple fact which is the nesting of the associated infinite dimensional planes flag the discrete kp hierarchy can thus be viewed as a container for an entire ensemble of vertex or darboux generated kp solutions it turns out that many new and old systems lead to such discrete semi infinite solutions like sequences of soliton solutions with more and more solitons sequences of calogero moser systems having more and more particles just to mention a few examples this is developed in 3 in this paper as another example we show that the q kp hierarchy maps via a kind of fourier transform into the discrete kp hierarchy enabling us to write down a very large class of solutions to the q kp hierarchy this was also reported in a brief note 4