nonlinear kolmogorov equations in infinite dimensional spaces the backward stochastic differential equations approach and applications to optimal control solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control

a critical evaluation of the resolution properties of b spline and compact finite difference methods resolution properties of b spline and compact finite difference schemes are compared using fourier analysis in periodic domains and tests based on solution of the wave and heat equations in finite domains with uniform and nonuniform grids results show that compact finite difference schemes have a higher convergence rate and in some cases better resolution however b spline schemes have a more straightforward and robust formulation particularly near boundaries on nonuniform meshes

a stochastic nonlinear schrodinger equation with multiplicative noise we study a conservative stochastic nonlinear schrodinger equation with a multiplicative noise we show the global existence and uniqueness of square integrable solutions for subcritical nonlinearities the critical exponent being the same in dimension 1 or 2 as the critical exponent of the deterministic equation

interface conditions of finite difference compact schemes for computational aeroacoustics the interface condition provides flexibility to handle complex geometries on a set of structured grids however when used with a compact scheme to apply in computational aeroacoustics its accuracy and stability may be degraded seriously by the boundary treatment that arises at a grid interface to minimize this effect a modified characteristic condition is incorporated in the finite difference compact scheme by using an explicit difference form at the boundary the extension of this approach is naturally attained through the coordinate transformation of convective terms across an interface in multidimensions the validity of current development is demonstrated through a simulation of trailing edge noise generation from an airfoil at a low mach number

development of noncentered wavenumber based optimized interpolation schemes with amplification control for overlapping grids a noncentered interpolation technique has been constructed to perform simulations using overlapping grids for complex geometries high order centered lagrange polynomial interpolations and interpolations optimized in the fourier space are first generalized to the noncentered case these noncentered interpolations either generate significant dispersion errors or strongly amplify high wavenumber components accordingly a noncentered high order wavenumber based optimized interpolation method is developed with the addition of a nonlinear constraint for the control of the amplitude amplification induced by decentering high order piecewise polynomial regressions of the obtained interpolation coefficients are performed the time stability of the method is investigated in the 1 d case when the interpolation method is used in conjunction with explicit high order differencing filtering schemes as well as a 6 step runge kutta time integration algorithm a criterion is formulated to predict its stability as a function of the filtering strength and the courant friedrichs lewy constant finally 1 d convection simulations are presented to illustrate the stability and the accuracy of the developed noncentered interpolations

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

scattering of sound from axisymetric sources by multiple circular cylinders a general analytic method for calculating the scattering of sound by multiple rigid circular cylinders arranged in an arbitrary parallel configuration is presented the sound scattered by this collection of cylinders is generated by a time periodic spatially distributed axisymmetric source located within the domain of interest a hankel transform method is used to calculate the incident field while separation of variables is used to obtain the scattered fields from each cylinder in the collection the unknown scattering coefficients are determined through the use of general addition theorems that allows the various fields to be readily transformed between coordinate systems the method is validated using various two three and four cylinder configurations and the number of coefficients that must be retained in the truncated series is examined benchmark configurations consisting of two and three cylinder systems with cylinders of varying radii are also presented these solutions have been used to validate computational aeroacoustic solvers developed for complex geometries

compact finite difference method for american option pricing a compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems the problem of pricing an american option can be cast as a partial differential equation using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem the complicating factor for american options is the existence of an optimal exercise boundary which is jointly determined with the value of the option in this article we develop three ways of combining compact finite difference methods for american option price on a single asset with methods for dealing with this optimal exercise boundary compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value this method is very fast and accurate even when the spatial step size h is large h 0 1 compact difference method two must solve an algebraic nonlinear equation obtained by pantazopoulos 1998 at every time step this method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the crank nicolson projected successive over relaxation method compact finite difference method three refines the free boundary value by a method developed by barone adesi and lugano the saga of the american put 2003 and this method can obtain high accuracy for space x the last two of these three methods are convergent moreover all the three methods work for both short term and long term options through comparison with existing popular methods by numerical experiments our work shows that compact finite difference methods provide an exciting new tool for american option pricing

direct numerical simulations of flow past an array of distributed roughness elements direct numerical simulation was used to describe the subsonic flow past an array of distributed cylindrical roughness elements mounted on a flat plate solutions were obtained for element heights corresponding to a roughness based reynolds number re k of both 202 and 334 the numerical method used a sixth order accurate centered compact finite difference scheme to represent spatial derivatives which was used in conjunction with a tenth order low pass pade type nondispersive filter operator to maintain stability an implicit approximately factored time marching algorithm was employed and newton like subiterations were applied to achieve second order temporal accuracy calculations were carried out on a massively parallel computing platform using domain decomposition to distribute subzones on individual processors a high order overset grid approach preserved spatial accuracy on the mesh system used to represent the roughness elements features of the flowfields are described and results of the computations are compared with experimentally measured velocity components of the time mean flowfield which are available only for re k 202 flow about the elements is characterized by a system of two weak corotating horseshoe vortices for re k 334 an unstable shear layer emanating from the top of the cylindrical element generated nonlinear unsteady disturbances of sufficient amplitude to produce explosive bypass transition downstream of the array the re k 202 case displayed exponential growth of turbulence energy in the streamwise direction which may eventually result in transition

coupling of dirichlet to neumann boundary condition and finite difference methods in curvilinear coordinates for multiple scattering the applicability of the dirichlet to neumann technique coupled with finite difference methods is enhanced by extending it to multiple scattering from obstacles of arbitrary shape the original boundary value problem bvp for the multiple scattering problem is reformulated as an interface bvp a heterogenous medium with variable physical properties in the vicinity of the obstacles is considered a rigorous proof of the equivalence between these two problems for smooth interfaces in two and three dimensions for any finite number of obstacles is given the problem is written in terms of generalized curvilinear coordinates inside the computational region then novel elliptic grids conforming to complex geometrical configurations of several two dimensional obstacles are constructed and approximations of the scattered field supported by them are obtained the numerical method developed is validated by comparing the approximate and exact far field patterns for the scattering from two circular obstacles in this case for a second order finite difference scheme a second order convergence of the numerical solution to the exact solution is easily verified

on the effect of a noise on the solutions of the focusing supercritical nonlinear schrodinger equation we investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear schrodinger equation we prove that contrary to the deterministic case any initial data gives birth to a solution which develops singularities moreover the singularities appear immediately we use a stochastic generalization of the variance identity and a control argument

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 1 in this paper we prove global well posedness and scattering for the defocusing one dimensional mass critical nonlinear schr o dinger equation we make use of a long time strichartz estimate and a frequency localized morawetz estimate this continues work begun in earlier papers for dimensions d geq 3 and d 2

quasi disjoint pentadiagonal matrix systems for the parallelization of compact finite difference schemes and filters this paper proposes a novel systematic approach for the parallelization of pentadiagonal compact finite difference schemes and filters based on domain decomposition the proposed approach allows a pentadiagonal banded matrix system to be split into quasi disjoint subsystems by using a linear algebraic transformation technique as a result the inversion of pentadiagonal matrices can be implemented within each subdomain in an independent manner subject to a conventional halo exchange process the proposed matrix transformation leads to new subdomain boundary sb compact schemes and filters that require three halo terms to exchange with neighboring subdomains the internode communication overhead in the present approach is equivalent to that of standard explicit schemes and filters based on seven point discretization stencils the new sb compact schemes and filters demand additional arithmetic operations compared to the original serial ones however it is shown that the additional cost becomes sufficiently low by choosing optimal sizes of their discretization stencils compared to earlier published results the proposed sb compact schemes and filters successfully reduce parallelization artifacts arising from subdomain boundaries to a level sufficiently negligible for sophisticated aeroacoustic simulations without degrading parallel efficiency the overall performance and parallel efficiency of the proposed approach are demonstrated by stringent benchmark tests

global well posedness and scattering for the energy critical nonlinear schrodinger equation in r 3 we obtain global well posedness scattering and global l 10 spacetime bounds for energy class solutions to the quintic defocusing schrodinger equa tion in r 1 3 which is energy critical in particular this establishes global existence of classical solutions our work extends the results of bourgain 4 and grillakis 20 which handled the radial case the method is similar in spirit to the induction on energy strategy of bourgain 4 but we perform the induction analysis in both frequency space and physical space simultaneously and replace the morawetz inequality by an interaction variant first used in 12 13 the principal advantage of the interaction morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions in particular this interaction estimate together with an almost conservation argument controlling the movement of l 2 mass in fre quency space rules out the possibility of energy concentration

a comparison of techniques for overcoming non uniqueness of boundary integral equations for the collocation partition of unity method in two dimensional acoustic scattering the partition of unity method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements the non uniqueness of solution obtained from the conventional boundary integral equation cbie is well known the cbie derived through green s identities suffers from a problem of non uniqueness at certain characteristic frequencies two of the standard methods of overcoming this problem are the so called combined helmholtz integral equation formulation chief method and that of burton and miller the latter method introduces a hypersingular integral which may be treated in various ways in this paper we present the collocation partition of unity boundary element method pubem for the helmholtz problem and compare the performance of chief against a burton miller formulation regularised using the approach of li and huang

acoustic scattering by two spheres multiple scattering and symmetry considerations abstract acoustic scattering by two identical spheres is theoretically numerically and experimentally studied by highlighting the role of the symmetries of the scatterer incident and scattered fields are expanded over the different irreducible representations of d h the continuous symmetry group of the scatterer then from the boundary conditions one obtains for each irreducible representation an infinite system of linear complex algebraic equations where the unknown scattering coefficients are uncoupled this feature greatly simplifies the treatment of the problem and speeds up calculations farfield form functions are computed in the cases of neumann boundary conditions rigid spheres and elastic boundary conditions elastic spheres immersed in water a series of experiments based on ultrasonic spectroscopy is performed in the case of two stainless steel spheres immersed in water the comparison between the theoretical and the experimental results provides quite a good agreement

martingale solutions for the stochastic nonlinear schrodinger equation in the energy space we consider a stochastic nonlinear schrodinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic nlse in h 1 on compact manifolds and bounded domains we construct a martingale solution using a modified faedo galerkin method based on the littlewood paley decomposition for the 2d manifolds with bounded geometry we use the strichartz estimates to show the pathwise uniqueness of solutions

a new time space domain high order finite difference method for the acoustic wave equation a new unified methodology was proposed in finkelstein and kastner 2007 39 to derive spatial finite difference fd coefficients in the joint time space domain to reduce numerical dispersion the key idea of this method is that the dispersion relation is completely satisfied at several designated frequencies we develop this new time space domain fd method further for 1d 2d and 3d acoustic wave modeling using a plane wave theory and the taylor series expansion new spatial fd coefficients are frequency independent though they lead to a frequency dependent numerical solution we prove that the modeling accuracy is 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation however under the same discretization the new 1d method can reach 2m th order accuracy and is always stable the 2d method can reach 2m th order accuracy along eight directions and has better stability similarly the 3d method can reach 2m th order accuracy along 48 directions and also has better stability than the conventional fd method the advantages of the new method are also demonstrated by the results of dispersion analysis and numerical modeling of acoustic wave equation for homogeneous and inhomogeneous acoustic models in addition we study the influence of the fd stencil length on numerical modeling for 1d inhomogeneous media and derive an optimal fd stencil length required to balance the accuracy and efficiency of modeling a new time space domain high order staggered grid fd method for the 1d acoustic wave equation with variable densities is also developed which has similar advantages demonstrated by dispersion analysis stability analysis and modeling experiments the methodology presented in this paper can be easily extended to solve similar partial difference equations arising in other fields of science and engineering

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

dispersive estimates for principally normal pseudodifferential operators the aim of these notes is to describe some recent re sults concerning dispersive estimates for principally normal pseu dodifferential operators the main motivation for this comes from unique continuation problems such estimates can be used to prove l q carleman inequalities which in turn yield unique continuation results for various partial differential operators with rough poten tials dispersive estimates are l q estimates for nonelliptic partial differ ential operators which are a consequence of the decay properties of their fundamental solutions these decay properties follow from spa tial spreading of the singularities of the solutions since solutions prop agate in directions conormal to the characteristic set of the operator this spreading can be related to nonzero curvatures of the characteristic set dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis various types of dispersive estimates are known to be true for op erators such as the wave operator the schrodinger operator and the linear kdv see ginibre velo 4 keel tao 11 they have proved to be useful in the study of nonlinear problems as well as of problems with unbounded potentials more recently similar estimates have been obtained for wave op erators with variable coefficients beginning with the smooth case in kapitanskii 10 mockenhaupt seeger and sogge 14 up to operators with c 2 coefficients in smith 15 and tataru 21 23 similar results were obtained for the schrodinger equation in staffilani tataru 19 c 2 coefficients and in burq gerard tzvetkov 1 smooth coeffic ients in the variable coefficient elliptic case one should also mention sogge s l q

simple intuitive relations concerning the effect of boundaries on the stability of the fdtd method simple intuitive relations are derived to study the effect of some boundaries on the courant friedrich levy number cfln in the finite difference time domain method a novel von neumann stability analysis is proposed to study the effect of perfect electric conductor perfect magnetic conductor and perfect electromagnetic conductor boundaries on the cfln also gershgorin s theorem is applied to determine the cfln at the interface between two dielectrics studies show that the aforesaid boundaries do not restrict the cfln of the computational space

a robust higher order compact scheme for solving general second order partial differential equation with derivative source terms on nonuniform curvilinear meshes abstract a fourth order compact finite difference scheme is proposed for solving general second order steady partial differential equation pde in two dimension 2d on geometries having nonuniform curvilinear grids in this work the main efforts are focused not only on nonorthogonal curvilinear grids but also on the presence of mixed derivative term and nonhomogeneous derivative source terms in the governing equation this is in turn suitable for solving fluid flow and heat transfer problems governed by navier stokes n s equations on geometries having nonuniform and nonorthogonal curvilinear grids the newly proposed scheme has been applied to solve general second order partial differential equation having analytical solution and some pertinent fluid flow problems namely viscous flows in a lid driven cavity such as trapezoidal cavity using nonorthogonal grid square cavity using distorted grid complicated enclosures using curvilinear grid and mixed convection flow in a bottom wavy wall cavity it is seen to efficiently capture steady state solutions of the n s equations with dirichlet as well as neumann boundary conditions detailed comparison data produced by the proposed scheme for all the test cases are provided and compared with existing analytical as well as established numerical results available in the literature excellent comparison is obtained in all the cases

a new family of fourth order locally one dimensional schemes for the three dimensional wave equation in this paper we present a new family of locally one dimensional lod schemes with fourth order accuracy in both space and time for the three dimensional 3d acoustic wave equation it is well known that high order explicit schemes offer efficient time steppings but with restrictive cfl conditions implicit discretization gives unconditional stable schemes but with inefficient time steppings our lod schemes can be seen as a compromise between explicit and implicit schemes in the way that our schemes have more relax cfl conditions compared with traditional explicit schemes and only require solutions of tridiagonal systems of linear algebraic equations which is more efficient than implicit schemes moreover the new scheme is four layer in time and three layer in space so that boundary conditions can be imposed in the classical way the computations of the initial conditions for the three intermediate time layers are explicitly constructed furthermore the stability condition is derived and the restriction for the time step is given explicitly finally numerical examples are completed to show the performance of our new method

energy stable numerical methods for hyperbolic partial differential equations using overlapping domain decomposition overlapping domain decomposition methods otherwise known as overset grid or chimera methods are useful for simplifying the discretization of partial differential equations in or around complex geometries though in wide use such methods are prone to numerical instability unless numerical diffusion or some other form of regularization is used especially for higher order methods to address this shortcoming high order provably energy stable overlapping domain decomposition methods are derived for hyperbolic initial boundary value problems the overlap is treated by splitting the domain into pieces and using generalized summation by parts derivative operators and polynomial interpolation new implicit and explicit operators are derived that do not require regularization for stability in the linear limit applications to linear and nonlinear problems in one and two dimensions are presented where it is found the explicit operators are preferred to the implicit ones

high order interpolation method for overset grid based on finite volume method to enhance accuracy in a high order flow solver with an overset grid method a high order interpolation method was developed based on finite volume method in the euler equations to improve stability when calculating a nonlinear discontinuity with the high order interpolation the interpolation was combined with a multidimensional limiting process to remove oscillatory phenomena thus the proposed method can be robustly applied to a discontinuous region as well as a continuous one it was compared with conventional methods using test cases thereby verifying its accuracy and efficiency it can be extendable to the navier stokes equations without any modification even though it was tested in the euler equations in the present paper

on the effective accuracy of spectral like optimized finite difference schemes for computational aeroacoustics the present article aims at highlighting the strengths and weaknesses of the so called spectral like optimized explicit central finite difference schemes when the latter are used for numerically approximating spatial derivatives in aeroacoustics evolution problems with that view we first remind how differential operators can be approximated using explicit central finite difference schemes the possible spectral like optimization of the latter is then discussed the advantages and drawbacks of such an optimization being theoretically studied before they are numerically quantified for doing so two popular spectral like optimized schemes are assessed via a direct comparison against their standard counterparts such a comparative exercise being conducted for several academic test cases at the end general conclusions are drawn which allows us discussing the way spectral like optimized schemes shall be preferred or not to standard ones when it comes to simulate real life aeroacoustics problems

computational and experimental investigation of a nonslender delta wing computational simulations have been performed for a 50 deg sweep delta wing with a sharp leading edge at a 15 deg angle of attack and moderate reynolds numbers of re 2 x 10 5 re 6 2 x 10 5 and re 2 x 10 6 a sixth order compact difference scheme with an eighth order low pass filter is used to solve the navier stokes equations turbulence modeling has been accomplished using an implicit large eddy simulation method that exploits the high order accuracy of the compact difference scheme and uses the discriminating higher order filter to regularize the solution computations have been performed on a baseline mesh of 11 3 x 10 6 grid points and a refined mesh of 35 x 10 6 grid points an assessment of grid resolution showed that significantly finer scale features of the flow could be captured on the refined mesh providing a more accurate representation of the complex unsteady separated flow comparisons are also made with high resolution particle image velocimetry images obtained for the two lower reynolds numbers the numerical results are examined to provide a description of the mean and instantaneous flow structure over the delta wing including the separated vortical flow vortex breakdown surface flow features and surface boundary layer transition near the symmetry plane the effect of reynolds number on each of these features is assessed

stochastic maximum principle for optimal control of a class of nonlinear spdes with dissipative drift we prove a version of the stochastic maximum principle in the sense of pontryagin for the finite horizon optimal control of a stochastic partial differential equation driven by an infinite dimensional additive noise in particular we treat the case in which the nonlinear term is of nemytskii type dissipative and with polynomial growth the performance functional to be optimized is fairly general and may depend on point evaluation of the controlled equation the results can be applied to a large class of nonlinear parabolic equations such as reaction diffusion equations

algebraic aspects of multiple scattering by two parallel cylinders classification and physical interpretation of scattering resonances abstract acoustic scattering by a pair of identical parallel cylinders is studied by emphasizing the role of the symmetries of the scatterer incident and scattered fields are expanded over the different irreducible representations ofc2v the symmetry group of the scatterer then from the boundary conditions one obtains an infinite set of four linear complex algebraic equations each one associated with a representation where the unknown coefficients of the scattered fields are uncoupled this method significantly simplifies the numerical treatment of the problem as a consequence positions of the scatterer resonances are determined in the complex plane of the reduced frequency and a partial algebraic classification of the resonances is obtained for various boundary conditions soft cylinders hard cylinders and elastic cylinders immersed in water a physical interpretation of certain resonances in terms of trapped geometrical paths is provided

artificial dissipation models for the euler equations various artificial dissipation models that are used with central difference algorithms for the euler equations are analyzed for their effect on accuracy stability and convergence rates in particular linear and nonlinear models are investigated using an implicit approximate factorization code arc2d for transonic airfoils fully implicit application of the dissipation models is shown to improve robustness and convergence rates the treatment of dissipation models at boundaries will be examined it will be shown that accurate error free solutions with sharp shocks can be obtained using a central difference algorithm coupled with an appropriate nonlinear artificial dissipation model i introduction t he solution of the euler equations using numerical techniques requires the use of either a differencing method with inherent dissipation or the addition of dissipation terms to a nondissipative scheme this is because the euler equations do not provide any natural dissipation mechanism such as viscosity in the navier stokes equations that would eliminate high frequencies which are caused by nonlinearitie s and especially shocks a variety of numerical algorithms and computer codes for the euler equations have been developed methods such as maccormack s1 explicit

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

provably stable overset grid methods for computational aeroacoustics the simulation of sound generating flows in complex geometries requires accurate numerical methods that are non dissipative and stable and well posed boundary conditions a structured mesh approach is often desired for a higher order discretization that better uses the provided grids but at the expense of complex geometry capabilities relative to techniques for unstructured grids one solution is to use an overset mesh based discretization where locally structured meshes are globally assembled in an unstructured manner this article discusses recent advancements in overset methods also called chimera methods concerning boundary conditions parallel methods for overset grid management and stable and accurate interpolation between the grids several examples are given some of which include moving grids

strichartz estimates and local smoothing estimates for asymptotically flat schrodinger equations abstract in this article we study global in time strichartz estimates for the schrodinger evolution corresponding to long range perturbations of the euclidean laplacian this is a natural continuation of a recent article d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 of the third author where it is proved that local smoothing estimates imply strichartz estimates by d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 the local smoothing estimates are known to hold for small perturbations of the laplacian here we consider the case of large perturbations in three increasingly favorable scenarios i without non trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term ii with non trapping assumptions we prove global estimates modulo a lower order spatially localized error term and iii for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum

optimal staggered grid finite difference schemes by combining taylor series expansion and sampling approximation for wave equation modeling high order staggered grid finite difference sfd schemes have been universally used to improve the accuracy of wave equation modeling however the high order sfd coefficients on spatial derivatives are usually determined by the taylor series expansion te method which just leads to great accuracy at small wavenumbers for wave equation modeling some conventional optimization methods can achieve high accuracy at large wavenumbers but they hardly guarantee the small numerical dispersion error at small wavenumbers in this paper we develop new optimal explicit sfd esfd and implicit sfd isfd schemes for wave equation modeling we first derive the optimal esfd and isfd coefficients for the first order spatial derivatives by applying the combination of the te and the sampling approximation to the dispersion relation and then analyze their numerical accuracy finally we perform elastic wave modeling with the esfd and isfd schemes based on the te method and the optimal method respectively when the appropriate number and interval for the sampling points are chosen these optimal schemes have extremely high accuracy at small wavenumbers and can also guarantee small numerical dispersion error at large wavenumbers numerical accuracy analyses and modeling results demonstrate the optimal esfd and isfd schemes can efficiently suppress the numerical dispersion and significantly improve the modeling accuracy compared to the te based esfd and isfd schemes

a transformation free hoc scheme for incompressible viscous flows past an impulsively started circular cylinder in this paper we present a higher order compact scheme for the unsteady two dimensional 2d navier stokes equations on nonuniform polar grids specifically designed for the incompressible viscous flows past a circular cylinder the scheme is second order accurate in time and at least third order accurate in space the scheme very efficiently computes both unsteady and time marching steady state flow for a wide range of reynolds numbers re ranging from 10 to 9500 for the impulsively started cylinder the robustness of the scheme is highlighted when it accurately captures the vortex shedding for moderate re represented by the von karman street and the so called and s phenomena for higher re comparisons are made with established numerical and experimental results and excellent agreement is found in all the cases both qualitatively and quantitatively

global well posedness of the energy critical nonlinear schrodinger equation with small initial data in h1 t3 a refined trilinear strichartz estimate for solutions to the schrodinger equation on the flat rational torus t3 is derived by a suitable modification of critical function space theory this is applied to prove a small data global well posedness result for the quintic nonlinear schrodinger equation in hs t3 for all s 1 this is the first energy critical global well posedness result in the setting of compact manifolds

the cauchy problem for quasi linear schrodinger equations with x rn t r with x1 xn and summation convention one may think of the equation in 1 1 as a non linear schrodinger equation where the operator modeling the dispersion relation is non isotropic and depends also on the unknown function its conjugate and their gradients in the space variables under appropriate assumptions on the function f the method presented here extends to fully non linear schrodinger equations of the form

high order compact finite difference methods on general overset grids this work investigates the coupling of a very high order finite difference algorithm for the solution of conservation laws on general curvilinear meshes with overset grid techniques originally developed to address complex geometric configurations the solver portion of the algorithm is based on pade type compact finite differences of up to sixth order with up to 10th order filters employed to remove spurious waves generated by grid non uniformities boundary conditions and flow non linearities the overset grid approach is utilized as both a domain decomposition paradigm for implementation of the algorithm on massively parallel machines and as a means for handling geometric complexity in the computational domain two key features have been implemented in the current work the ability of the high order algorithm to accommodate holes cut in grids by the overset grid approach and the use of high order interpolation at non coincident grid overlaps several high order high accuracy interpolation methods were considered and a high order explicit non optimized lagrangian method was found to be the most accurate and robust for this application several two dimensional benchmark problems were examined to validate the interpolation methods and the overall algorithm these included grid to grid interpolation of analytic test functions the inviscid convection of a vortex laminar flow over single and double cylinder configurations and the scattering of acoustic waves from one and three cylinder configurations the employment of the overset grid techniques coupled with high order interpolation at overset boundaries was found to be an effective way of employing the high order algorithm for more complex geometries than was previously possible

a high order immersed boundary method for acoustic wave scattering and low mach number flow induced sound in complex geometries a new sharp interface immersed boundary method based approach for the computation of low mach number flow induced sound around complex geometries is described the underlying approach is based on a hydrodynamic acoustic splitting technique where the incompressible flow is first computed using a second order accurate immersed boundary solver this is followed by the computation of sound using the linearized perturbed compressible equations lpce the primary contribution of the current work is the development of a versatile high order accurate immersed boundary method for solving the lpce in complex domains this new method applies the boundary condition on the immersed boundary to a high order by combining the ghost cell approach with a weighted least squares error method based on a high order approximating polynomial the method is validated for canonical acoustic wave scattering and flow induced noise problems applications of this technique to relatively complex cases of practical interest are also presented

grid adaptation using chimera composite overlapping meshes the objective of this paper is to perform grid adaptation using composite over lapping meshes in regions of large gradient to capture the salient features accurately during computation the chimera grid scheme a multiple overset mesh technique is used in combination with a navier stokes solver the numerical solution is first converged to a steady state based on an initial coarse mesh solution adaptive enhancement is then performed by using a secondary fine grid system which oversets on top of the base grid in the high gradient region but without requiring the mesh boundaries to join in any special way communications through boundary interfaces between those separated grids are carried out using tri linear interpolation applications to the euler equations for shock reflections and to a shock wave boundary layer interaction problem are tested with the present method the salient features are well resolved

optimal bilinear control of gross pitaevskii equations a mathematical framework for optimal bilinear control of nonlinear schrodinger equations of gross pitaevskii type arising in the description of bose einstein condensates is presented the obtained results generalize earlier efforts found in the literature in several aspects in particular the cost induced by the physical workload over the control process is taken into account rather than the often used l 2 or h 1 norms for the cost of the control action well posedness of the problem and existence of an optimal control are proved in addition the first order optimality system is rigorously derived also a numerical solution method is proposed which is based on a newton type iteration and used to solve several coherent quantum control problems

the nonlinear schrodinger equation driven by jump processes abstract the main result of the paper is the existence of a martingale solution of the nonlinear schrodinger equation with a levy noise with infinite activity to be more precise let a be the laplace operator with d a u l 2 r d u l 2 r d let z l 2 r d be a function space and be a poisson random measure on z with intensity measure let g r c and h r c be some given functions satisfying certain conditions specified later let 1 and 0 we are interested in the solution of the following equation 0 1 i d u t x u t x d t u t x 1 u t x d t u 0 z u t x g z x d z d t z u t x h z x d z d t u 0 u 0 where d z d t denotes the compensator of the random measure first we consider the case where the levy process is a compound poisson process then we use this result to show that 0 1 has a solution in the general case

implicit lyapunov control of finite dimensional schrodinger equations an implicit lyapunov based approach is proposed for generating trajectories of a finite dimensional controlled quantum system the main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable the controlled lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem the convergence analysis is done using lasalle invariance principle closed loop simulations illustrate the performance of such feedback laws for the open loop control of a test case considered by chemists

a wavenumber based extrapolation and interpolation method for use in conjunction with high order finite difference schemes the errors incurred in using extrapolation and interpolation in large scale computations are analyzed and quantified in the wavenumber space if a large extrapolation stencil is used the errors in the low wavenumbers can be significantly reduced however the errors in the high wavenumbers are at the same time greatly increased the opposite is true if the stencil size is reduced based on the wavenumber analysis an optimized extrapolation and interpolation method is proposed the optimization is carried out over a selected band of wavenumbers it is known that extrapolation often leads to numerical instability the instability is the result of large error amplification in the high wavenumber range to reduce the tendency to trigger numerical instability it is proposed that an extra constraint be imposed on the optimized extrapolation method the added constraint aims to reduce error amplification over the high wavenumbers numerical examples are provided to illustrate that accurate and stable numerical results can be obtained in large scale simulation using a high order finite difference scheme and the proposed optimized extrapolation method when the same problems are recomputed using the familiar high order polynomials extrapolation method in the lagrange form in one case the numerical results are plagued by large errors and ultimately instability in another problem it is found that the use of the lagrange polynomials extrapolation method would lead immediately to numerical instability

subcritical approximations to stochastic defocusing mass critical nonlinear schrodinger equation on r abstract we show robustness of various truncated and subcritical approximations to the stochastic defocusing mass critical nonlinear schrodinger equation nls in dimension d 1 whose solution was constructed in 8 with one particular such approximation the key ingredient in the proof is a uniform bound of the solutions to the family of deterministic mass subcritical defocusing nls

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

solving navier stokes equation for flow past cylinders using single block structured and overset grids many complex fluid motions are driven by physical processes of instability transition and turbulence dependent upon nonlinear mechanisms here we solve the flow past cylinder s using single block structured and overset grids by computing navier stokes equation in two dimensions the suitability of a compact scheme in discretizing convection and diffusion terms are investigated first by looking at relevant numerical properties also for the overset grid method one of the methods is identified that shows the best results in minimizing interpolation error at sub domain boundaries for an analytical test function we provide extensive comparisons with experimental and other computational results for flow past a single cylinder utilizing both single block structured and chimera or overset grids apart from showing instability of this flow calculated by these methods we also compare the computed vorticity and velocity data using these two grids by employing the proper orthogonal decomposition pod we have analyzed and developed an overset grid method with compact scheme that does not need any filtering to control error this has been ascertained by performing pod analysis to show that the developed method is capable of handling complex geometries we have computed flow past two cylinders in side by side arrangement results obtained capture the known flow characteristics for this arrangement well using relatively fewer number of grid points

stochastic nonlinear schrodinger equations abstract this paper is devoted to the well posedness of stochastic nonlinear schrodinger equations in the energy space h 1 r d and is a continuation of our recent work barbu et al 2014 we allow general complex coefficients in the noise thus covering both the conservative and non conservative case the latter being important to include quantum measurement effects in the underlying physical model we consider both focusing and defocusing nonlinearities and prove global well posedness in h 1 r d including also the pathwise continuous dependence on initial conditions with exponents of the nonlinearity in exactly the same sharp range as in the deterministic case in particular in the special conservative case this work improves earlier results in de bouard and debussche 2003 where the critical exponents could not be reached for d 6 in the defocusing and for d 4 in the focusing case respectively moreover the local existence uniqueness and blowup alternative are also established for the energy critical case the approach presented here is mainly based on the rescaling approach already used in barbu et al 2014 to study the l 2 case and also on the strichartz estimates established in marzuola et al 2008 for large perturbations of the laplacian

b spline method and zonal grids for simulations of complex turbulent flows a numerical technique for computations of turbulent flows is described the technique is based on b splines and allows grid embedding in physically significant flow regions numerical tests which include solutions of nonlinear advection diffusion equations and computations of flow over a circular cylinder at reynolds numbers up to 300 indicate that the method is accurate and efficient in computations of flow over a cylinder the lift drag and base suction coefficients agree well with existing experimental data and previous numerical simulations

optimal bilinear control of stochastic nonlinear schrodinger equations mass sub critical case we study optimal control problems for stochastic nonlinear schrodinger equations in both the mass subcritical and critical case for general initial data of the minimal l 2 regularity we prove the existence and first order lagrange condition of an open loop control in particular these results apply to the stochastic nonlinear schrodinger equations with the critical quintic and cubic nonlinearities in dimensions one and two respectively furthermore we obtain uniform estimates of backward stochastic solutions in new spaces of type u 2 and v 2 adapted to evolution operators related to linear schrodinger equations with lower order perturbations these estimates yield a new temporal regularity of backward stochastic solutions which is crucial for the tightness of approximating controls induced by ekeland s variational principle

global well posedness and scattering for the defocusing energy critical nonlinear schrodinger equation in r1 4 we obtain global well posedness scattering uniform regularity and global l6t x spacetime bounds for energy space solutions to the defocusing energy critical nonlinear schrodinger equation in a 4 our arguments closely follow those of colliander keel et al though our derivation of the frequency localized interaction morawetz estimate is somewhat simpler as a consequence our method yields a better bound on the l6t x norm

an efficient implementation of fourth order compact finite difference scheme for poisson equation with dirichlet boundary conditions fourth order compact finite difference scheme has been proposed for solving the poisson equation with dirichlet boundary conditions for some time an efficient implementation of such numerical scheme is often desired for practical usage in this paper based on fast discrete sine transform we design an efficient algorithm to implement this scheme to do this poisson equation is first discretized by fourth order compact finite difference method the subsequent discretized system is not solved by the usual method matrix inversion instead it is solved with the fast discrete sine transform by doing this way the computational cost of proposed algorithm for such scheme with large grid numbers can be greatly reduced detailed numerical algorithm of this fast solver for one dimensional two dimensional and three dimensional poisson equation has been presented numerical results in one dimension two dimensions three dimensions and four dimensions have shown that the applied compact finite difference scheme has fourth order accuracy and can be efficiently implemented

blow up for the stochastic nonlinear schrodinger equation with multiplicative noise we study the influence of a multiplicative gaussian noise white in time and correlated in space on the blow up phenomenon in the supercritical nonlinear schrodinger equation we prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability

construction of invariant compact finite difference schemes in this paper we propose a method which is based on equivariant moving frames for development of high order accurate invariant compact finite difference schemes that preserve lie symmetries of underlying partial differential equations in this method variable transformations that are obtained from the extended symmetry groups of partial differential equations pdes are used to transform independent and dependent variables and derivative terms of compact finite difference schemes constructed for these pdes such that the resulting schemes are invariant under the chosen symmetry groups the unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames in some cases owing to selection of convenient moving frames numerical representation of invariant or symmetry preserving compact numerical schemes is found to be notably simpler than that of standard noninvariant compact numerical schemes further the accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite difference algorithms application of the proposed method is demonstrated through construction of invariant compact finite difference schemes for some common linear and nonlinear pdes including the linear advection diffusion equation in one or two dimensions the inviscid burgers equation in one dimension viscous burgers equation in one or two dimensions spherical burgers equation in one dimension and shallow water equations in two dimensions results obtained from our numerical simulations indicate that invariant compact finite difference schemes not only inherit selected symmetry properties of underlying pdes but are also comparably more accurate than the standard noninvariant base numerical schemes considered here

time space domain dispersion relation based finite difference method with arbitrary even order accuracy for the 2d acoustic wave equation high order finite difference fd methods have been widely used for numerical solution of acoustic wave equations it has been reported that the modeling accuracy is of 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation under the same discretization the present version of time space domain dispersion relation based fd method can improve the accuracy from 2nd order to 2m th order along eight directions for the 2d acoustic wave equation to increase the accuracy further we propose a new fd stencil for 2d acoustic wave equation modeling this new time space domain dispersion relation based fd stencil can reach the same arbitrary even order accuracy along all directions and is more accurate and more stable than the conventional one for the same m dispersion analysis and modeling examples demonstrate its advantages

analysis of exponential time differencing for fdtd in lossy dielectrics we determine the stability condition and analyze the accuracy of the exponential and centered time differencing schemes for finite difference time domain fdtd in an isotropic homogeneous lossy dielectric with electric and magnetic conductivities spl sigma and spl sigma respectively we show that these schemes are equivalent and determine that for accuracy both schemes must be used with a time step that finely resolves the electric and magnetic conduction current relaxation time scales the implications of these results for perfectly matched layer pml type absorbing boundary conditions are discussed

generalized stability criterion of 3 d fdtd schemes for doubly lossy media this paper presents the generalized stability criterion of 3 d finite difference time domain fdtd schemes for doubly lossy media where both electric and magnetic conductivities coexist the generalized stability criterion is applicable for all 3 d fdtd schemes such as time average ta time forward tf time backward tb and exponential time differencing etd it is reducible to either electrically lossy magnetically lossy or lossless media the stability criterion for perfectly matched layer pml matching condition can also be obtained as a special case to the doubly lossy media it is shown that for doubly lossy media the stability criterion for etd and tf becomes even more relaxed and for tb even more stringent compared to either electrically lossy magnetically lossy or lossless media on the other hand the stability criterion for ta remains unchanged even in doubly lossy media as numerical demonstration the tunneling of electromagnetic wave through a very thin doubly lossy conductor is simulated numerical experiments further show the maximum allowed time step as dictated by the derived stability criterion for different schemes

on the use of higher order finite difference schemes on curvilinear and deforming meshes this study enables the use of very high order finite difference schemes for the solution of conservation laws on stretched curvilinear and deforming meshes to illustrate these procedures we focus on up to 6th order pade type spatial discretizations coupled with up to 10th order low pass filters these are combined with explicit and implicit time integration methods to examine wave propagation and wall bounded flows described by the navier stokes equations it is shown that without the incorporation of the filter application of the high order compact scheme to nonsmooth meshes results in spurious oscillations which inhibit their applicability inclusion of the discriminating low pass high order filter restores the advantages of high order approach even in the presence of large grid discontinuities when three dimensional curvilinear meshes are employed the use of standard metric evaluation procedures significantly degrades accuracy since freestream preservation is violated to overcome this problem a simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes for dynamically deforming grids an effective numerical treatment is described to evaluate expressions containing the time varying transformation metrics with these techniques metric cancellation is guaranteed regardless of the manner in which grid speeds are defined the efficacy of the new procedures is demonstrated by solving several model problems as well as by application to flow past a rapidly pitching airfoil and past a flexible panel

optimal bilinear control of an abstract schro dinger equation well posedness of abstract quantum mechanical systems is considered and the existence of optimal control of such systems is proved first order optimality systems are derived convergence of the monotone scheme for the solution of the optimality system is proved

elastic wave scattering and dynamic stress concentrations in cylindrical shells with a circular cutout in this paper based on the theory of elastic wave motion for open cylindrical shell wave scattering and dynamic stress concentrations in open cylindrical shells with a hole are studied by making use of small parameter perturbation methods and boundary integral equation techniques the boundary integral equations and iterative imminent series of scattered waves around the cavity of the cylindrical shell are derived by employing this method the approximately analytical solutions of scattered waves on the edge of cutout are gained the computational formula for getting the dynamic stress concentration factors on the contour of cavity is developed as an example the numerical results of these dynamic stress concentration factors are graphically presented and discussed the analytical methods put forward in the present work have practical significances for solving the problem of elastic wave scattering and dynamic stress concentrations in cylindrical shells with a circular cutout

an efficient time splitting compact finite difference method for gross pitaevskii equation abstract we propose an efficient time splitting compact finite difference method for gross pitaevskii equation gpe in our method we solve the gpe in time with time splitting technique and in space by the compact finite difference method to find the numerical solution of the resulting discretized system in one dimension 1d two dimensions 2d and three dimensions 3d we apply the fast discrete sine transform in 1d 2d and 3d respectively and get an efficient solver for the discretized system in 1d 2d and 3d respectively our numerical algorithm at every time step does not need linear algebraic equations solver whose computation cost will be much higher when the spatial dimension increases the method also has the merit that it is unconditionally stable and conservative moreover the method can achieve spectral like accuracy in space when high order compact finite difference method is applied extensive numerical tests for the gpe in 1d 2d and 3d are presented to demonstrate the power and accuracy of the proposed numerical method

stochastic nonlinear schrodinger equations no blow up in the non conservative case abstract this paper is devoted to the study of noise effects on blow up solutions to stochastic nonlinear schrodinger equations it is a continuation of our recent work 2 where the local well posedness is established in h 1 also in the non conservative critical case here we prove that in the non conservative focusing mass super critical case by adding a large multiplicative gaussian noise with high probability one can prevent the blow up on any given bounded time interval 0 t 0 t moreover in the case of spatially independent noise the explosion even can be prevented with high probability on the whole time interval 0 the noise effects obtained here are completely different from those in the conservative case studied in 5

lyapunov control of bilinear schrodinger equations a lyapunov based approach for trajectory tracking of the schrodinger equation is proposed in the finite dimensional case convergence is precisely analyzed connection between the controllability of the linearized system around the reference trajectory and asymptotic tracking is studied when the linearized system is controllable such a feedback ensures almost global asymptotic convergence when the linearized system is not controllable the stability of the closed loop system is not asymptotic to overcome such lack of convergence we propose when the reference trajectory is an eigenstate a modification based on adiabatic invariance simulations illustrate the simplicity and also the interest for trajectory generation

a block interface flux reconstruction method for numerical simulation with high order finite difference scheme overlap grid is usually used in numerical simulation of flow with complex geometry by high order finite difference scheme it is difficult to generate overlap grid and the connectivity information between adjacent blocks especially when interpolation is required for non coincident overlap grids in this study an interface flux reconstruction ifr method is proposed for numerical simulation using high order finite difference scheme with multi block structured grids in this method the neighboring blocks share a common face and the fluxes on each block are matched to set the boundary conditions for each interior block therefore this method has the promise of allowing discontinuous grids on either side of an interior block interface the proposed method is proven to be stable for 7 point central drp scheme coupled with 4 point and 5 point boundary closure schemes as well as the 4th order compact scheme coupled with 3rd order boundary closure scheme four problems are numerically solved with the developed code to validate the interface flux reconstruction method in this study the ifr method coupled with the 4th order drp scheme or compact scheme is validated to be 4th order accuracy with one and two dimensional waves propagation problems two dimensional pulse propagation in mean flow is computed with wavy mesh to demonstrate the ability of the proposed method for non uniform grid to demonstrate the ability of the proposed method for complex geometry sound scattering by two cylinders is simulated and the numerical results are compared with the analytical data it is shown that the numerical results agree well with the analytical data finally the ifr method is applied to simulate viscous flow pass a cylinder at reynolds number 150 to show its capability for viscous problem the computed pressure coefficient on the cylinder surface the frequency of vortex shedding the lift and drag coefficients are presented the numerical results are compared with the data of other researchers and a good agreement is obtained the validations imply that the proposed ifr method is accurate and effective for inviscid and viscous problems with complex geometry