energy conserving explicit local time stepping for second order wave equations locally refined meshes impose severe stability constraints on explicit time stepping methods for the numerical simulation of time dependent wave phenomena to overcome that stability restriction local time stepping methods are developed which allow arbitrarily small time steps precisely where small elements in the mesh are located when combined with a symmetric finite element discretization in space with an essentially diagonal mass matrix the resulting discrete numerical scheme is explicit is inherently parallel and exactly conserves a discrete energy starting from the standard second order leap frog scheme time stepping methods of arbitrary order of accuracy are derived numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory

marker method for the solution of nonlinear diffusion equations the marker method for the solution of nonlinear diffusion equations is described the method relies on the definition of a convective field associated with the underlying partial differential equation the information about the approximate solution is associated with the response of an ensemble of markers to this convective field some key aspects of the method such as the selection of the shape function and the initial loading are discussed in some detail numerical experiments show that the method is accurate in determining the long time behavior of nonlinear diffusion equations the marker method can be applied to an ensemble of nonlinear dispersive partial differential equations

on conservation at grid interfaces this paper considers the solution of hyperbolic systems of conservation laws on discontinuous grids in particular we consider what happens to conservation at grid interfaces a procedure is presented to derive conservative difference approximations at the grid interfaces for two dimensional grids which overlap in an arbitrary configuration the same procedures are applied to compute interface formulas for grids which are refined in space and or time and for continuous grids where a switch in the scheme causes the discontinuity

a note on the numerical solution of the wave equation with piecewise smooth coefficients the numerical solution of the initial value problem for the wave equation is considered for the case when the equation coefficients are piecewise smooth this problem models linear wave propagation in a medium in which the properties of the medium change discontinuously at interfaces convergent difference approximations can be found that do not require the explicit specification of the boundary conditions at interfaces in the medium and hence are simple to program although such difference approximations typically can only be expected to be first order accurate the numerical phase velocity has the same accuracy as the difference approximation would if the coefficients in the differential equation were smoooth this is proved for the one dimensional case and demonstrated numerically for an example in two space dimensions in which the interface is not aligned with the computational mesh

time domain numerical modeling of wave propagation in 2d heterogeneous porous media this paper deals with the numerical modeling of wave propagation in porous media described by biot s theory the viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity which is valid in the low frequency range the coexistence of propagating fast compressional wave and shear wave and of a diffusive slow compressional wave makes numerical modeling tricky to avoid restrictions on the time step the biot s system is splitted into two parts the propagative part is discretized by a fourth order ader scheme while the diffusive part is solved analytically near the material interfaces a space time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave the jump conditions along the interfaces are discretized by an immersed interface method numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling the efficiency of the approach is illustrated by simulations of multiple scattering

a conservative space time mesh refinement method for the 1 d wave equation part i construction we propose a new method for space time refinement for the 1 d wave equation this method is based on the conservation of a discrete energy through two different discretization grids which guarantees the stability of the scheme our approach results in a non interpolatory scheme whose stability condition is not affected by the transition between the two grids

fdtd local grid with material traverse often a finite difference time domain fdtd calculation requires a relatively higher mesh resolution in only small subvolumes of the total mesh space by locally applying finer grids local grids to these volumes the necessary resolution can be obtained computation time and memory requirements may be far less than for an fdtd space with the smaller mesh resolution throughout in many situations it is important that these local grids function when materials traverse the main grid local grid mg lg boundary surfaces since the volumes that require local grids may not be isolated in a homogeneous medium a local grid method which allows dielectric and or conducting materials to traverse the boundaries is developed three different fdtd problems that utilize the local grid method are used as validation tests results are compared with uniform mesh fdtd solutions

a simple derivation of the class operator of the su 2 group an elementary derivation of the class operator of the su 2 group is presented considerably simplifying the recent calculations of fan and ren

instability of difference models for hyperbolic initial boundary value problems a th00 eory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary according to this theory instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity c 0 to make this point of view precise we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries from these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in l2 norms which extend earlier results due to osher and to gustafsson kreiss and sundstrom in particular we investigate l2 instability with respect to both initial and boundary data and show how they are affected by a finite versus infinite reflection coefficients and b wave radiation with c 0 versus c 0

interface conditions for wave propagation through mesh refinement boundaries we study the propagation of waves across fixed mesh refinement boundaries in linear and nonlinear model equations in 1 d and 2 d and in the 3 d einstein equations of general relativity we demonstrate that using linear interpolation to set the data in guard cells leads to the production of reflected waves at the refinement boundaries implementing quadratic interpolation to fill the guard cells suppresses these spurious signals

computational framework for coupling compressible and low mach number codes a hybrid multicode computational method is developed that allows combining compressible and low mach number codes into a single computational solver the proposed methodology can be used for integrated simulations of multicomponent engineering problems the unsteady coupling between the two codes is performed via exchanging time dependent state information through the interfaces using overlapping meshes the proposed coupling procedure is validated on laminar steady and unsteady test problems both constant density and variable density regimes of the low mach number code are investigated results of the test cases performed in frameworks of one way coupling and two way coupling are documented several methods of implementation of interface conditions for the compressible code are compared by looking at the levels of dilatation as well as numerical errors the injection method when all the variables are simply transformed from one code to another through an interpolation procedure is chosen for its superior stability accuracy and simplicity of implementation numerical results for the coupled calculations obtained with the injection method show good agreement with both standalone calculations and analytical solutions

introduction and study of fourth order theta schemes for linear wave equations a new class of high order implicit three time step schemes for semi discretized wave equations is introduced and studied these schemes are constructed using the modified equation approach generalizing the q scheme their stability properties are investigated via an energy analysis which enables us to design super convergent schemes and also optimal stable schemes in terms of consistency errors specific numerical algorithms for the fully discrete problem are tested and discussed showing the efficiency of our approach compared to second order q schemes

normally ordering some multimode exponential operators by virtue of the iwop technique the author shows that the technique of integration within an ordered product iwop provides one with a very simple approach to deriving the normal product form of some multimode exponential operators which greatly simplifies the calculations of normalising some state vectors in hilbert space

adaptive grid refinement for numerical weather prediction abstract an adaptive atmospheric flow model is described and results of integrations with this model are presented the adaptive technique employed is that of berger and oliger the technique uses a finite difference method to integrate the dynamical equations first on a coarse grid and then on finer grids which have been placed based on a richardson type estimate of the truncation error in the coarse grid solution by correctly coupling the integrations on the various grids periodically re estimating the error and recreating the finer grids uniformily accurate solutions are economically produced the primitive hydrostatic equations of meteorology are solved for the advection of a barotropic cyclone and for the development of a baroclinic disturbance which results from the perturbation of an unstable jet these integrations demonstrate the feasibility of using multiple rotated overlapping fine grids direct computations of the truncation error are used to confirm the accuracy of the richardson type truncation error estimates

numerical wave propagation on non uniform one dimensional staggered grids the wave propagation behaviour of centered difference schemes on one dimensional non uniform staggered grids is investigated previous results for the linear advection equation are extended to the case of the shallow water equations on staggered grids for waves of a given frequency the wave field is decomposed into right and left propagating components and a wave energy conservation law is derived in terms of these components for slowly varying grids separate evolution equations for the right and left propagating components are derived leading to the result that there is asymptotically no reflection in the limit of a slowly varying grid provided that waves of that frequency are resolvable however there will be reflection from any location at which the wave group velocity goes to zero the possibility for wave energy to tunnel through a narrow region of the grid too coarse for propagation is noted grids with an abrupt jump in resolution are also investigated it is possible to tailor the scheme at the jump to minimize spurious wave reflection over a range of frequencies provided the waves are resolvable on both sides of the jump however it does not appear possible to avoid complete reflection except by introducing extra dissipation terms if the waves are not resolvable on one side of the jump an example is presented of a second order accurate scheme that spontaneously radiates waves from the resolution jump

stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations abstract implicit noniterative finite difference schemes have recently been developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations when applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable a stable as applied in practice the algorithms often face a severe time step restriction a major source of the difficulty is the treatment of the numerical boundary conditions one conjecture has been that unconditional stability requires implicit numerical boundary conditions an apparent counterexample was the space time extrapolation considered by gustafsson kreiss and sundstrom in this paper we examine space implicit and space time explicit extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals the results indicate that for unconditional stability with a finite number of spatial mesh intervals the numerical boundary conditions must be implicit

a class of energy stable high order finite difference interface schemes suitable for adaptive mesh refinement of hyperbolic problems we present a class of energy stable high order finite difference interface closures for grids with step resolution changes these grids are commonly used in adaptive mesh refinement of hyperbolic problems the interface closures are such that the global accuracy of the numerical method is that of the interior stencil the summation by parts property is built into the stencil construction and implies asymptotic stability by the energy method while being non dissipative we present one dimensional closures for fourth order explicit and compact pade type finite differences tests on the scalar one and two dimensional wave equations the one dimensional navier stokes solution of a shock and two dimensional inviscid compressible vortex verify the accuracy and stability of this class of methods

local time space mesh refinement for simulation of elastic wave propagation in multi scale media this paper presents an original approach to local time space grid refinement for the numerical simulation of wave propagation in models with localized clusters of micro heterogeneities the main features of the algorithm are the application of temporal and spatial refinement on two different surfaces the use of the embedded stencil technique for the refinement of grid step with respect to time the use of the fast fourier transform fft based interpolation to couple variables for spatial mesh refinement the latter makes it possible to perform filtration of high spatial frequencies which provides stability in the proposed finite difference schemes in the present work the technique is implemented for the finite difference simulation of seismic wave propagation and the interaction of such waves with fluid filled fractures and cavities of carbonate reservoirs however this approach is easy to adapt and or combine with other numerical techniques such as finite elements discontinuous galerkin method or finite volumes used for approximation of various types of linear and nonlinear hyperbolic equations

a generalization of the construction of the class operator it is shown that the construction of the class operator for su 2 is a partial case of a much more general problem that of decomposing an operator into components transforming under conjugation according to a given irreducible representation the problem is solved generally for arbitrary compact groups and some possibilities for extensions of this procedure to the case of non compact groups are indicated

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

bypassing the groenewold van hove obstruction on mathbb r 2n a new argument in favor of born jordan quantization there are known obstructions to a full quantization of in the spirit of dirac s approach the most known being the groenewold and van hove no go result we show following a suggestion of s k kauffmann that it is possible to construct a well defined quantization procedure by weakening the usual requirement that commutators should correspond to poisson brackets the weaker requirement consists in demanding that this correspondence should only hold for hamiltonian functions of the type this reformulation leads to a non injective quantization of all observables which when restricted to polynomials is the rule proposed by born and jordan in the early days of quantum mechanics

improved fdtd subgridding algorithms via digital filtering and domain overriding in numerical simulations of maxwell s equations for problems with disparate geometric scales it is often advantageous to use grids of varying densities over different portions of the computational domain in simulations involving structured finite difference time domain fdtd grids this strategy is often referred as subgridding sg although sg can lead to major computational savings it is known to cause instabilities spurious reflections and other accuracy problems in this paper we introduce two strategies to combat these problems first we present an overlapped sg osg approach combined with digital filters in space osg can recover standard sg ssg schemes but it is based upon a more general explicit separation between interpolation decimation operations and the fdtd field update itself this allows for a better classification of errors associated with the subgrid interface more importantly digital filters and phase matching techniques can be then employed to combat those errors second we introduce sg with a domain overriding sg do strategy consisting of overlapped sub grid regions that contain auxiliary buffer subdomains with perfectly matched layers pml to allow explicit control on the reflection and transmission properties at sg interfaces we provide two dimensional 2 d numerical examples showing that residual errors from 2 d sg do fdtd simulations can be significantly reduced when compared to ssg schemes

permutation operators in hilbert space gained via iwop technique a kind of permutation operators in hilbert space are derived in terms of the technique of integration within ordered product iwop these operators are shown to be quantum maps imaged by certain permutation transformations in classical space some operator identities about the relationship between permutation operators and transposition operators are also derived

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

additive and iterative operator splitting methods and their numerical investigation as an alternative to the classical splitting methods two new splitting schemes have been developed recently the additive and the iterative splitting in this paper we discuss the most important properties the advantages and disadvantages of these schemes and investigate their performance on simple examples as well as on more complex physical models

conservative space time mesh refinement methods for the fdtd solution of maxwell s equations a new variational space time mesh refinement method is proposed for the fdtd solution of maxwell s equations the main advantage of this method is to guarantee the conservation of a discrete energy that implies that the scheme remains l 2 stable under the usual cfl condition the only additional cost induced by the mesh refinement is the inversion at each time step of a sparse symmetric positive definite linear system restricted to the unknowns located on the interface between coarse and fine grid the method is presented in a rather general way and its stability is analyzed an implementation is proposed for the yee scheme in this case various numerical results in 3 d are presented in order to validate the approach and illustrate the practical interest of space time mesh refinement methods

strict stability of high order compact implicit finite difference schemes temporal or strict stability of approximation to pdes is much more difficult to achieve than the classical lax stability in this paper we present a class of finite difference schemes for hyperbolic initial boundary value problems in one and two space dimensions that possess the property of strict stability the approximations are constructed so that all eigenvalues of corresponding differentiation matrix have a nonpositive real part boundary conditions are imposed by using penalty like terms fourth and sixth order compact implicit finite difference schemes are constructed and analyzed computational efficacy of the approach is corroborated by a series of numerical tests in 1 d and 2 d scalar problems

high order explicit local time stepping methods for damped wave equations locally refined meshes impose severe stability constraints on explicit time stepping methods for the numerical simulation of time dependent wave phenomena local time stepping methods overcome that bottleneck by using smaller time steps precisely where the smallest elements in the mesh are located starting from classical adams bashforth multi step methods local time stepping methods of arbitrarily high order of accuracy are derived for damped wave equations when combined with a finite element discretization in space with an essentially diagonal mass matrix the resulting time marching schemes are fully explicit and thus inherently parallel numerical experiments with continuous and discontinuous galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time stepping methods

a cartesian based embedded geometry technique with adaptive high order finite differences for compressible flow around complex geometries an immersed boundary methodology to solve the compressible navier stokes equations around complex geometries in cartesian fluid dynamics solvers is described the objective of the new approach is to enable smooth reconstruction of pressure and viscous stresses around the embedded objects without spurious numerical artifacts a standard level set represents the boundary of the object and defines a fictitious domain into which the flow fields are smoothly extended boundary conditions on the surface are enforced by an approach inspired by analytic continuation each fluid field is extended independently constrained only by the boundary condition associated with that field unlike most existing methods no jump conditions or explicit derivation of them from the boundary conditions are required in this approach numerical stiffness that arises when the fluid solid interface is close to grid points of the mesh is addressed by preconditioning in addition the embedded geometry technique is coupled with a stable high order adaptive discretization that is enabled around the object boundary to enhance resolution the stencils used to transition the order of accuracy of the discretization are derived using the summation by parts technique that ensures stability applications to shock reflections shock ramp interactions and supersonic and low mach number flows over two and three dimensional geometries are presented

coupling discontinuous galerkin methods and retarded potentials for transient wave propagation on unbounded domains this work deals with the numerical simulation of wave propagation on unbounded domains with localized heterogeneities to do so we propose to combine a discretization based on a discontinuous galerkin method in space and explicit finite differences in time on the regions containing heterogeneities with the retarded potential method to account the unbounded nature of the computational domain the coupling formula enforces a discrete energy identity ensuring the stability under the usual cfl condition in the interior moreover the scheme allows to use a smaller time step in the interior domain yielding to quasi optimal discretization parameters for both methods the aliasing phenomena introduced by the local time stepping are reduced by a post processing by averaging in time obtaining a stable and second order consistent in time coupling algorithm we compute the numerical rate of convergence of the method for an academic problem the numerical results show the feasibility of the whole discretization procedure

weyl ordered polynomials in fractional dimensional quantum mechanics we develop algebraic properties of weyl ordered polynomials in the momentum and position operators p q which satisfy the r deformed heisenberg algebra representations of which describe quantum mechanics in fractional dimensions by viewing weyl ordered polynomials as tensor operators with respect to the lie algebra we derive a specific form for these polynomials including an expression in terms of hypergeometric functions and determine various algebraic properties such as recurrence relations symmetries and also a general product formula from which all commutators and anti commutators may be calculated we briefly discuss several applications to quantum mechanics in fractional dimensions

the choice of numerical boundary conditions for hyperbolic systems abstract the aim of this paper is the discussion of two fundamental problems for mixed initial boundary value problems with applications in fluid mechanics first different stability properties are discussed which are of importance for long time integrations and steady state calculations second a new numerical technique for problems with an artificial boundary is introduced

weighted class operators a new approach to tensor operators for compact groups weighted class operators defined by averaging conjugation operators within fixed representation of a compact group against a weight function living on a conjugacy class are discussed and their connection to the tensor operators is studied making extensive use of the theory of induced representations we prove several results for weighted class operators and in particular identify them for weight functions being suitable matrix elements of the given group with irreducible tensor operators the results complement and extend those obtained earlier by the authors a orlowski and a strasburger j phys a 27 1971 1976 1994 and others

nondissipative and energy stable high order finite difference interface schemes for 2 d patch refined grids a class of finite difference interface schemes suitable for two dimensional cell centered grids with patch refinement and step changes in resolution is presented grids of this type are generated by adaptive mesh refinement methods according to resolution needs dictated by the physics of the problem being modeled for these grids coarse and fine nodes are not aligned at the mesh interfaces resulting in hanging nodes three distinct geometries are identified at the interfaces of a domain with interior patch refinement edges concave corners and convex corners asymptotic stability in time of the numerical scheme is achieved by imposing a summation by parts condition on the interface closure which is thus also nondissipative interface stencils corresponding to an explicit fourth order finite difference scheme are presented for each geometry to preserve stability a reduction in local accuracy is required at the corner geometries it is also found that no second order accurate solution exists that satisfies the summation by parts condition tests using the 2 d scalar advection equation and an inviscid compressible vortex support the stability and accuracy of these stencils for both linear and nonlinear problems

stability of interfaces with mesh refinement a study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite difference approximations first a root condition which implies stability for the initial boundary value problem for this type of interface is derived from the root condition the stability of several interface equations is proved using the maximum principle in some cases the final verification steps can be done analytically in other cases a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle using this method stability for lax wendroff with all the interface conditions considered and for leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved

a finite volume method for the approximation of maxwell s equations in two space dimensions on arbitrary meshes a new finite volume method is presented for discretizing the two dimensional maxwell equations this method may be seen as an extension of the covolume type methods to arbitrary possibly non conforming or even non convex n sided polygonal meshes thanks to an appropriate choice of degrees of freedom an equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles the main properties of the scheme are its energy conservation its stability under a cfl like condition and the fact that it preserves gauss law and divergence free magnetic fields second order convergence is demonstrated numerically on non conforming and distorted meshes

anticommutator analogues of certain identities involving repeated commutators the generalization of the baker hausdorff lemma and its anticommutator analogue is formulated additionally the anticommutator analogues of several well known operator identities involving repeated commutators are derived it is pointed out that these are more convenient for application whenever the operators in question satisfy simpler repeated anticommutator relations and in particular when they anticommute than the repeated commutator relations the diagonalization of two spin 1 hamiltonians in which the anticommutator analogue of the baker hausdorff lemma is used to good advantage is presented

uniqueness of steady state solutions for difference equations on overlapping grids the uniqueness of continuous and discontinuous steady state solutions is studied for difference approximations of hyperbolic problems on a strip with overlapping grids in the continuous case an easily verifiable uniqueness criterion is obtained from a normal mode analysis this criterion is applied to examine the uniqueness of commonly used interface treatments in the discontinuous case a shock is assumed to be contained in the overlap analysis for this case is developed on a linear model equation most of the results are further confirmed by computing smooth and shocked flows in a divergent duct the present theory can also be applied to analyze uniqueness for more general boundary value problems

numerical modeling of 1d transient poroelastic waves in the low frequency range propagation of transient mechanical waves in porous media is numerically investigated in 1d the framework is the linear biot model with frequency independent coefficients the coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky a method combining three numerical tools is proposed a fourth order ader scheme with time splitting to deal with the time marching a space time mesh refinement to account for the small scale evolution of the slow wave and an interface method to enforce the jump conditions at interfaces comparisons with analytical solutions confirm the validity of this approach

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

euler polynomials and identities for non commutative operators three kinds of identities involving non commutating operators and euler and bernoulli polynomials are studied the first identity as given by bender and bettencourt phys rev d 54 12 7710 7723 1996 expresses the nested commutator of the hamiltonian and momentum operators as the commutator of the momentum and the shifted euler polynomial of the hamiltonian the second one by pain j phys a math theor 46 035304 2013 links the commutators and anti commutators of the monomials of the position and momentum operators the third appears in a work by figuieira de morisson and fring j phys a math gen 39 9269 2006 in the context of non hermitian hamiltonian systems in each case we provide several proofs and extensions of these identities that highlight the role of euler and bernoulli polynomials

the perturbed hydrogen atom some new algebraic results the authors have employed algebraic methods to calculate the bound state spectra of a non relativistic hydrogen atom subjected to a wide class of perturbations their procedure exploits the linearity of the complete perturbed hamiltonian in the generators of the so 2 2 lie algebra which follows naturally from the separation of variables in schrodinger s equation in parabolic coordinates appropriate transformations then allow the hamiltonian to be expressed as a linear combination of the compact generators of the two underlying so 2 1 algebras they give some examples for which the bound state spectra can be obtained completely analytically

stability of finite difference models containing two boundaries or interfaces it is known that the stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves here the waves point of view is applied to models containing two boundaries or interfaces where repeated reflection of trapped packets is a potential new source of instability our analysis accounts for various known instability phenomena in a unified way and leads to several new results three of which are as follows 1 dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface 2 algebraic gks instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present 3 gks stability and p stability can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one

ordering of some boson operator functions the problem of normal ordering of the following boson operator functions a a m ar a m a n m a2 n m where n a a is solved explicitly simple algebraic methods are used as an intermediate result the ordering formulae for the symmetrised products of boson operators are presented there is agreement with earlier results of other authors and generalisation of these results a new representation of the generalised stirling numbers used for the ordering of the binomial a2 n m is given and several principal properties of these numbers are derived

some grid refinement schemes for hyperbolic equations with piecewise constant coefficients discontinuities in the coefficients of hyperbolic equations occur both naturally and artificially and must be treated in numerical schemes schemes for handling these discontinuities are derived an interesting stability result is derived and the schemes are shown to be exact under certain restric tions

an efficient implicit spectral element method for time dependent nonlinear diffusion equations by evaluating integrals at one quadrature point we present an implicit spectral element method to approximate the solution of time dependent nonlinear diffusion equations in complex geometries we propose a nodal expansion to approximate derivatives of an unknown function so integrals involving nonlinearities are evaluated at one quadrature point and discrete form of nonlinear diffusion equations is obtained without forming local and global stiffness matrices using the method of lines mol the nonlinear partial differential equation is reduced to a nonlinear system of ordinary differential equations such that the mol formulation is derived in terms of the differentiation matrix instead of the stiffness matrix since the arising system is stiff an implicit method is employed to solve it also with a simple algorithm we obtain a closed analytical form for the jacobian matrix of the system of nonlinear algebraic equations in addition the differentiation matrices for triangular elements and deformed quadrilateral and hexahedral elements are obtained for deformed elements an isoparametric mapping is used where the expansion coefficients are obtained with the aid of the gordon hall mapping several benchmarks including the scalar and system of these equations are carried out as physical applications the cell motion model over long time the tumour angiogenesis model on a mixed quadrangle triangle mesh and two and three dimensional p laplace equations for various choices of the parameter p are considered

grid free simulation of diffusion using random walk methods abstract the simulation of diffusion of a continuum field by the random walk displacement of a set of particles is investigated in detail computational particles are used to transport elements of the gradients of the diffusive concentration one dimensional and quasi one dimensional cases are treated for a generalized diffusion variable different types of boundary conditions of the diffusion equation are considered as well as the extensions to a system of coupled diffusion equations the reaction diffusion equations and the convection diffusion set of equations it is shown that by using concentration gradients in the random walk process statistical errors are reduced and each realization of the numerical solution is a representation of the exact solution moreover transport of higher order derivatives can be utilized to improve the smoothness the algorithm is grid free and the computational elements move to follow the gradients thus it is self adaptive and uniform resolution is attained for all times the method is particularly suitable for the simulation of diffusion in systems which involve more than one transport mechanism when the diffusivity is small and when lagrangian elements are used to model the other mechanisms

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

anticommutator analogue of the baker hausdorff lemma the anticommutator analogue of the baker hausdorff lemma is formulated and proved by the differential equation method it is pointed out that this analogue when suitably transformed is more convenient for application whenever the operators in question satisfy simpler repeated anticommutator than repeated commutator relations

new applications of the coherent state in calculating the class operator of the rotation group the authors show by exploiting the properties of the coherent state and the integration within the ordered product technique that it is possible to derive an explicit expression for the class operator of the su 2 group which has not been obtained before

an adaptive multiresolution scheme with local time stepping for evolutionary pdes we present a fully adaptive numerical scheme for evolutionary pdes in cartesian geometry based on a second order finite volume discretization a multiresolution strategy allows local grid refinement while controlling the approximation error in space for time discretization we use an explicit runge kutta scheme of second order with a scale dependent time step on the finest scale the size of the time step is imposed by the stability condition of the explicit scheme on larger scales the time step can be increased without violating the stability requirement of the explicit scheme the implementation uses a dynamic tree data structure numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid fully adaptive three dimensional computations for reaction diffusion equations illustrate the additional speed up of the local time stepping for a thermo diffusive flame instability

a node centered local refinement algorithm for poisson s equation in complex geometries this paper presents a method for solving poisson s equation with dirichlet boundary conditions on an irregular bounded three dimensional region the method uses a nodal point discretization and adaptive mesh refinement amr on cartesian grids and the amr multigrid solver of almgren the discrete laplacian operator at internal boundaries comes from either linear or quadratic shortley weller extrapolation and the two methods are compared it is shown that either way solution error is second order in the mesh spacing error in the gradient of the solution is first order with linear extrapolation but second order with shortley weller examples are given with comparison with the exact solution the method is also applied to a heavy ion fusion accelerator problem showing the advantage of adaptivity

a two dimensional adaptive mesh generation method abstract an automatic two dimensional adaptive mesh generation method is persented the method is designed so that a small portion of the mesh can be modified without disturbing a large number of adjacent mesh points the method can be used with or without boundary fitted coordinate generation procedures on the generated mesh a differential equation can be discretized by using classical difference formulas designed for uniform meshes as well as the difference formulas developed in this work both cases are illustrated by applying the method to the hiemenz flow for which the exact solution of the navier stokes equation is known 1 and to a two dimensional viscous internal flow model problem

fourth order energy preserving locally implicit time discretization for linear wave equations a family of fourth order coupled implicit explicit time schemes is presented as a special case of fourth order coupled implicit schemes for linear wave equations the domain of interest is decomposed into several regions where different fourth order time discretization are used chosen among a family of implicit or explicit fourth order schemes the coupling is based on a lagrangian formulation on the boundaries between the several non conforming meshes of the regions a global discrete energy is shown to be preserved and leads to global fourth order consistency in time numerical results in 1d and 2d for the acoustic and elastodynamics equations illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries for which using everywhere an explicit scheme can be extremely penalizing accuracy up to fourth order reduces the numerical dispersion inherent to implicit methods used with a large time step and makes this family of schemes attractive compared to second order accurate methods

dynkin s method of computing the terms of the baker campbell hausdorff series the infinite series for log exp x exp y for noncommuting x and y is expressible in terms of iterated commutators of x and y except for the linear term x y dynkin derived an explicit expression for the terms as a sum of iterated commutators over a certain set of sequences this paper presents a practical algorithm for applying dynkin s formula and gives several illustrative examples

preconditioned implicit solution of linear hyperbolic equations with adaptivity this paper describes a method for solving hyperbolic partial differential equations using an adaptive grid the spatial derivatives are discretised with a finite volume method on a grid which is structured and partitioned into blocks which may be refined and derefined as the solution evolves the solution is advanced in time via a backward differentiation formula the discretisation used is second order accurate and stable on cartesian grids the resulting system of linear equations is solved by gmres at every time step with the convergence of the iteration being accelerated by a semi toeplitz preconditioner the efficiency of this preconditioning technique is analysed and numerical experiments are presented which illustrate the behaviour of the method on a parallel computer

on convergence of a domain decomposition method for a scalar conservation law in this paper we prove convergence of a domain decomposition method for one dimensional scalar conservation laws by dealing carefully with nonconservative terms at the interface of subdomains the method consists of an explicit scheme in some subdomains and an implicit scheme in other subdomains with a numerical flux being the same as the one used in the explicit scheme although such a multidomain algorithm is not strictly conservative the conservation error ce 0 n delta t is equal to mathcal o delta t regardless of the smoothness of the solution finally two test examples are given to validate convergence and the computational efficiency of the present method

multigrid elliptic equation solver with adaptive mesh refinement in this paper we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement our code uses truncation error estimates to adaptively refine the grid as part of the solution process the presentation includes a discussion of the orders of accuracy that we use for prolongation and restriction operators to ensure second order accurate results and to minimize computational work code tests are presented that confirm the overall second order accuracy and demonstrate the savings in computational resources provided by adaptive mesh refinement

local time stepping and discontinuous galerkin methods for symmetric first order hyperbolic systems we present a new non conforming space time mesh refinement method for the symmetric first order hyperbolic system this method is based on the one hand on the use of a conservative higher order discontinuous galerkin approximation for space discretization and a finite difference scheme in time on the other hand on appropriate discrete transmission conditions between the grids we use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method

explicit local time stepping methods for maxwell s equations explicit local time stepping methods are derived for time dependent maxwell equations in conducting and non conducting media by using smaller time steps precisely where smaller elements in the mesh are located these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators when combined with a finite element discretisation in space with an essentially diagonal mass matrix the resulting discrete time marching schemes are fully explicit and thus inherently parallel in a non conducting source free medium they also conserve a discrete energy which provides a rigorous criterion for stability starting from the standard leap frog scheme local time stepping methods of arbitrarily high accuracy are derived for non conducting media numerical experiments with a discontinuous galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes

stability analysis of interface temporal discretization in grid overlapping methods we investigate the stability of a temporal discretization of interface terms in grid overlapping methods a matrix stability analysis is performed on a model problem of the one dimensional diffusion equation on overlapping grids the scheme stability is first analyzed theoretically and a proof of the unconditional stability of the first order interface extrapolation scheme with the first and second order time integration for any overlap size is presented for the higher order schemes we obtain explicit estimates of the spectral radius of the corresponding discrete matrix operator and document the values of the stability threshold depending on the number of grid points and the size of overlap the influence of iterations on stability properties is also investigated numerical experiments are then presented relating the obtained stability bounds to the observed numerical values semidiscrete analysis confirms the derived scaling for the stability bounds

on the evaluation of the class operator for the rotation group a simple derivation is given of the explicit value of the class operator for the rotation group so 3

the baker campbell hausdorff formula and the convergence of the magnus expansion the authors show that for a wide class of dynamical systems described by hamiltonians of the form usually considered in time dependent perturbation theory the divergence of the magnus expansion in the schrodinger picture for large time intervals is due to pole singularities inherent to the baker campbell hausdorff formula

commutation relations of operator monomials in this short paper the commutator of monomials of operators obeying constant commutation relations is expressed in terms of anti commutators the formula involves bernoulli numbers or euler polynomials evaluated in zero the role of bernoulli numbers in quantum mechanical identities such as the baker campbell hausdorff formula is emphasized and applications connected to ordering problems as well as to the ehrenfest theorem are proposed

similarity transformations in one and two mode fock space the normally ordered hilbert space image of the general complex linear similarity transformation in phase space is obtained in a coherent state representation although preserving the commutator of a and a these quantum mechanical images of classical transformations are generally not unitary remarkably although the kets and bras produced by the non unitary similarity transformations are not hermitian conjugates squeezed state analogues produced using the similarity transformation still satisfy an overcompleteness relation the results are extended to two mode fock space and simple examples of the utility of the similarity transformation are presented the evaluation of the normally ordered operators is greatly facilitated by the use of integration within ordered products

space time mesh refinement for elastodynamics numerical results we propose here a generalization to the elastodynamic equations of the space time mesh refinement technique intro duced in raffinement de maillage spatio temporel pour les equations de maxwell ph d thesis universitede dau phine paris 2000 for the maxwells equations this method uses a discrete energy conservation to ensure the stability the method is presented in a variational way applicable to other type of hyperbolic systems several numerical experiments are provided to show the efficiency of this approach 2004 elsevier b v all rights reserved

a conservative space time mesh refinement method for the 1 d wave equation part ii analysis we introduced in 2 a new method for space time refinement for the 1 d wave equation this method is based on the conservation of a discrete energy through two different discretization grids which guarantees the stability of the scheme in this second part we analyse the accuracy of this scheme in a detailed way by means of a plane wave analysis and numerical experiments that permit us to point out spurious numerical phenomena and explain how to control them

h box methods for the approximation of hyperbolic conservation laws on irregular grids we study generalizations of the high resolution wave propagation algorithm for the approximation of hyperbolic conservation laws on irregular grids that have a time step restriction based on a reference grid cell length that can be orders of magnitude larger than the smallest grid cell arising in the discretization this godunov type scheme calculates fluxes at cell interfaces by solving riemann problems defined over boxes of a reference grid cell length h we discuss stability and accuracy of the resulting so called h box methods for one dimensional systems of conservation laws an extension of the method for the two dimensional case which is based on the multidimensional wave propagation algorithm is also described

numerical methods for solid mechanics on overlapping grids this paper presents a new computational framework for the simulation of solid mechanics on general overlapping grids with adaptive mesh refinement amr the approach described here for time dependent linear elasticity in two and three space dimensions is motivated by considerations of accuracy efficiency and flexibility we consider two approaches for the numerical solution of the equations of linear elasticity on overlapping grids in the first approach we solve the governing equations numerically as a second order system sos using a conservative finite difference approximation the second approach considers the equations written as a first order system fos and approximates them using a second order characteristic based godunov finite volume method a principal aim of the paper is to present the first careful assessment of the accuracy and stability of these two representative schemes for the equations of linear elasticity on overlapping grids this is done by first performing a stability analysis of analogous schemes for the first order and second order scalar wave equations on an overlapping grid the analysis shows that non dissipative approximations can have unstable modes with growth rates proportional to the inverse of the mesh spacing this new result which is relevant for the numerical solution of any type of wave propagation problem on overlapping grids dictates the form of dissipation that is needed to stabilize the scheme numerical experiments show that the addition of the indicated form of dissipation and or a separate filter step can be used to stabilize the sos scheme they also demonstrate that the upwinding inherent in the godunov scheme which provides dissipation of the appropriate form stabilizes the fos scheme we then verify and compare the accuracy of the two schemes using the method of analytic solutions and using problems with known solutions these latter problems provide useful benchmark solutions for time dependent elasticity we also consider two problems in which exact solutions are not available and use a posterior error estimates to assess the accuracy of the schemes one of these two problems is additionally employed to demonstrate the use of dynamic amr and its effectiveness for resolving elastic shock waves finally results are presented that compare the computational performance of the two schemes these demonstrate the speed and memory efficiency achieved by the use of structured overlapping grids and optimizations for cartesian grids

strict stability of high order compact implicit finite difference schemes the role of boundary conditions for hyperbolic pdes ii abstract this paper deals with the problem of systems of hyperbolic pdes in one and two space dimensions using the theory of part i 7