comparison theorems in hereditary structures and applications abstract in this paper some new comparison theorems between lower solutions and upper solutions in the caratheodory sense of functional cauchy problems are established criteria on continuous dependence and a functional extension of muller s theorem are derived as applications

high order exponential operator splitting methods for time dependent schrodinger equations in this paper we deduce high order error bounds for exponential operator splitting methods the employed techniques are specific to linear differential equations of the form u t a u t b u t t geq 0 involving an unbounded operator a in particular evolutionary schrodinger equations with sufficiently regular initial values are included in the analysis

algorithms for infinite quadratic programming in lp spaces we study infinite dimensional quadratic programming problems of an integral type the decision variable is taken in the l p space where 1

convergence analysis of high order time splitting pseudospectral methods for nonlinear schrodinger equations in this work the issue of favorable numerical methods for the space and time discretization of low dimensional nonlinear schrodinger equations is addressed the objective is to provide a stability and error analysis of high accuracy discretizations that rely on spectral and splitting methods as a model problem the time dependent gross pitaevskii equation arising in the description of bose einstein condensates is considered for the space discretization pseudospectral methods collocated at the associated quadrature nodes are analyzed for the time integration high order exponential operator splitting methods are studied where the decomposition of the function defining the partial differential equation is chosen in accordance with the underlying spectral method the convergence analysis relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part essential tools in the derivation of a temporal global error estimate are furthe

existence uniqueness and continuous dependence for hereditary differential equations abstract in the setting of functional differential equations an abstract hereditary structure is introduced which includes and unifies many formulations already considered by other authors for the new structure existence uniqueness and continuous dependence theorems are given

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

an explicit exchange algorithm for linear semi infinite programming problems with second order cone constraints in this paper we propose an explicit exchange algorithm for solving the semi infinite programming problem sip with second order cone soc constraints we prove by using the complementarity slackness conditions that the algorithm terminates in a finite number of iterations and the obtained solution sufficiently approximates the original sip solution in existing studies on sips only the nonnegative constraints were considered and hence the complementarity slackness conditions were separable to each scalar component however in our study the existing componentwise analyses are no longer applicable since the complementarity slackness conditions are associated with socs in order to overcome such a difficulty we introduce a certain coordinate system based on the spectral factorization in the numerical experiments we solve some test problems to see the effectiveness of the proposed algorithm first we compare our algorithm with two other algorithms based on the linear sip lsip reformulation and observe that by exploiting the sdpt3 solver for subproblems our algorithm finds the solution much faster than lsip reformulation approaches when the size of the problem is large also we apply the algorithm to some filter design problems and see that those problems can be solved efficiently

a wigner measure analysis of the dufort frankel scheme for the schrodinger equation we apply wigner transform techniques to the analysis of the dufort frankel difference scheme for the schrodinger equation and to the continuous analogue of the scheme in the case of a small scaled planck constant semiclassical regime in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop numerical test examples are presented to help interpret the theory and to compare the dufort frankel scheme to other difference schemes for the schr odinger equation

equivalence of different hereditary structures in ordinary differential equations abstract we prove that different formulations of hereditary settings for ordinary differential systems which appeared not comparable are actually equivalent

a numerical study of adaptive space and time discretisations for gross pitaevskii equations as a basic principle benefits of adaptive discretisations are an improved balance between required accuracy and efficiency as well as an enhancement of the reliability of numerical computations in this work the capacity of locally adaptive space and time discretisations for the numerical solution of low dimensional nonlinear schrodinger equations is investigated the considered model equation is related to the time dependent gross pitaevskii equation arising in the description of bose einstein condensates in dilute gases the performance of the fourier pseudo spectral method constrained to uniform meshes versus the locally adaptive finite element method and of higher order exponential operator splitting methods with variable time stepsizes is studied numerical experiments confirm that a local time stepsize control based on a posteriori local error estimators or embedded splitting pairs respectively is effective in different situations with an enhancement either in efficiency or reliability as expected adaptive time splitting schemes combined with fast fourier transform techniques are favourable regarding accuracy and efficiency when applied to gross pitaevskii equations with a defocusing nonlinearity and a mildly varying regular solution however the numerical solution of nonlinear schrodinger equations in the semi classical regime becomes a demanding task due to the highly oscillatory and nonlinear nature of the problem the spatial mesh size and the time increments need to be of the size of the decisive parameter 0 e 1 especially when it is desired to capture correctly the quantitative behaviour of the wave function itself the required high resolution in space constricts the feasibility of numerical computations for both the fourier pseudo spectral and the finite element method nevertheless for smaller parameter values locally adaptive time discretisations facilitate to determine the time stepsizes sufficiently small in order that the numerical approximation captures correctly the behaviour of the analytical solution further illustrations for gross pitaevskii equations with a focusing nonlinearity or a sharp gaussian as initial condition respectively complement the numerical study

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics this paper is concerned with transparent boundary conditions tbcs for wide angle parabolic equations wapes in the application to underwater acoustics assuming cylindrical symmetry existing discretizations of these tbcs introduce slight numerical reflections at this artificial boundary and also render the overall crank nicolson finite difference method only conditionally stable here a novel discrete tbc is derived from the fully discretized whole space problem that is reflection free and yields an unconditionally stable scheme while we shall assume a uniform discretization in range the interior depth discretization i e in the water column may be nonuniform and we shall discuss strategies for the best exterior discretization i e in the sea bottom the superiority of the new discrete tbc over existing discretizations is illustrated on several benchmark problems in the literature different wapes or wape and the standard parabolic equation have been coupled in the water and the sea bottom we analyze under which conditions this yields a hybrid model that is conservative for the acoustic field

defect based local error estimators for splitting methods with application to schrodinger equations part i the linear case we introduce a defect correction principle for exponential operator splitting methods applied to time dependent linear schrodinger equations and construct a posteriori local error estimators for the lie trotter and strang splitting methods under natural commutator bounds on the involved operators we prove asymptotical correctness of the local error estimators and along the way recover the known a priori convergence bounds numerical examples illustrate the theoretical local and global error estimates

extremal points and optimal solutions for general capacity problems this paper studies the infinite dimensional linear programming problems in the integration type the variable is taken in the space of bounded regular borel measures on compact hausdorff spaces it will find an optimal measure for a constrained optimization problem namely a capacity problem relations between extremal points of the feasible region and optimal solutions of the optimization problem are investigated the necessary sufficient conditions for a measure to be optimal are established the algorithm for optimal solution of the general capacity problem onx y 0 1 is formulated

numerical approximation of quadratic observables of schrodinger type equations in the semi classical limit we apply wigner transform techniques to the analysis of difference methods for schrodinger type equations in the case of a small planck constant in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not numerical test examples are presented to help interpret the theory

a numerical scheme for the simulation of blow up in the nonlinear schrodinger equation we present a self adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear schrodinger equations nlses as an example we study the character of the focusing singularity of the two dimensional critical nlse

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star