an unconditionally energy stable scheme based on an implicit auxiliary energy variable for incompressible two phase flows with different densities involving only precomputable coefficient matrices abstract we present an energy stable scheme for numerically approximating the governing equations for incompressible two phase flows with different densities and dynamic viscosities for the two fluids the proposed scheme employs a scalar valued auxiliary energy variable in its formulation and it satisfies a discrete energy stability property more importantly the scheme is computationally efficient within each time step it computes two copies of the flow variables velocity pressure phase field function by solving individually a linear algebraic system involving a constant and time independent coefficient matrix for each of these field variables the coefficient matrices involved in these linear systems only need to be computed once and can be pre computed additionally within each time step the scheme requires the solution of a nonlinear algebraic equation about a scalar valued number using the newton s method the cost for this nonlinear solver is very low accounting for only a few percent of the total computation time per time step because this nonlinear equation is about a scalar number not a field function extensive numerical experiments have been presented for several two phase flow problems involving large density ratios and large viscosity ratios comparisons with theory show that the proposed method produces physically accurate results simulations with large time step sizes demonstrate the stability of computations and verify the robustness of the proposed method an implication of this work is that energy stable schemes for two phase problems can also become computationally efficient and competitive eliminating the need for expensive re computations of coefficient matrices even at large density ratios and viscosity ratios

a roadmap for discretely energy stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity abstract we present a framework for devising discretely energy stable schemes for general dissipative systems based on a generalized auxiliary variable the auxiliary variable a scalar number can be defined in terms of the energy functional by a general class of functions not limited to the square root function adopted in previous approaches the current method has another remarkable property the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level regardless of the time step sizes or the external forces this property of guaranteed positivity is not available in previous approaches a unified procedure for treating the dissipative governing equations and the generalized auxiliary variable on the discrete level has been presented the discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems the current method termed gpav generalized positive auxiliary variable requires only the solution of linear algebraic equations within a time step with appropriate choice of the operator in the algorithm the resultant linear algebraic systems upon discretization involve only constant and time independent coefficient matrices which only need to be computed once and can be pre computed several specific dissipative systems are studied in relative detail using the gpav framework ample numerical experiments are presented to demonstrate the performance of the method and the robustness of the scheme at large time step sizes

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

practical stabilization of a quantum particle in a one dimensional infinite square potential well we consider a nonrelativistic charged particle in a one dimensional infinite square potential well this quantum system is subjected to a control which is a uniform in space time depending electric field it is represented by a complex probability amplitude solution of a schrodinger equation on a one dimensional bounded domain with dirichlet boundary conditions we prove the almost global practical stabilization of the eigenstates by explicit feedback laws

multiple scalar auxiliary variable msav approach and its application to the phase field vesicle membrane model we consider in this paper gradient flows with disparate terms in the free energy that cannot be efficiently handled with the scalar auxiliary variable sav approach and we develop the multiple sc

the scalar auxiliary variable sav approach for gradient flows abstract we propose a new approach which we term as scalar auxiliary variable sav approach to construct efficient and accurate time discretization schemes for a large class of gradient flows the sav approach is built upon the recently introduced ieq approach it enjoys all advantages of the ieq approach but overcomes most of its shortcomings in particular the sav approach leads to numerical schemes that are unconditionally energy stable and extremely efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step the scheme is not restricted to specific forms of the nonlinear part of the free energy so it applies to a large class of gradient flows numerical results are presented to show that the accuracy and effectiveness of the sav approach over the existing methods

manifolds of g2 holonomy from the n 4 sigma model using the two dimensional 2d n 4 sigma model with u 1 r gauge symmetry and introducing the ade cartan matrices as gauge matrix charges we build toric hyper kahler eight real dimensional manifolds x8 dividing by one toric geometry circle action of x8 manifolds we present examples describing quotients x7 x8 u 1 of g2 holonomy in particular for the ar cartan matrix the quotient space is a cone on an s2 bundle over r intersecting wcp2 1 2 1 weighted projective spaces according to the ar dynkin diagram

optimal control of pdes in a complex space setting application to the schrodinger equation in this paper we discuss optimality conditions for abstract optimization problems over complex spaces we then apply these results to optimal control problems with a semigroup structure as an application we detail the case when the state equation is the schrodinger one with pointwise constraints on the bilinear control we derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly

an energy stable scheme for incompressible navier stokes equations with periodically updated coefficient matrix abstract we present an energy stable scheme for simulating the incompressible navier stokes equations based on the generalized positive auxiliary variable gpav framework in the gpav reformulated system the original nonlinear term is replaced by a linear term plus a correction term where the correction term is put under control by an auxiliary variable the proposed scheme incorporates a pressure correction type strategy into the gpav procedure and it satisfies a discrete energy stability property the scheme entails the computation of two copies of the velocity and pressure within a time step by solving an individual de coupled linear equation for each of these field variables upon discretization the pressure linear system involves a constant coefficient matrix that can be pre computed while the velocity linear system involves a coefficient matrix that is updated periodically once every k 0 time steps in the current work where k 0 is a user specified integer the auxiliary variable being a scalar valued number is computed by a well defined explicit formula which guarantees the positivity of its computed values it is observed that the current method can produce accurate simulation results at large or fairly large time step sizes for the incompressible navier stokes equations the impact of the periodic coefficient matrix update on the overall cost of the method is observed to be small in typical numerical simulations several flow problems have been simulated to demonstrate the accuracy and performance of the method developed herein

analysis and comparison of numerical methods for the klein gordon equation in the nonrelativistic limit regime we analyze rigourously error estimates and compare numerically temporal spatial resolution of various numerical methods for solving the klein gordon kg equation in the nonrelativistic limit regime involving a small parameter 0 varepsilon ll 1 which is inversely proportional to the speed of light in this regime the solution is highly oscillating in time i e there are propagating waves with wavelength of o varepsilon 2 and o 1 in time and space respectively we begin with four frequently used finite difference time domain fdtd methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step as well as the small parameter varepsilon based on the error bounds in order to compute correct solutions when 0 varepsilon ll1 the four fdtd methods share the same varepsilon scalability tau o varepsilon 3 then we propose new numerical methods by using either fourier pseudospectral or finite difference approximation for spatial derivatives combined with the gautschi type exponential integrator for temporal derivatives to discretize the kg equation the new methods are unconditionally stable and their varepsilon scalability is improved to o 1 and tau o varepsilon 2 for linear and nonlinear kg equations respectively when 0 varepsilon ll1 numerical results are reported to support our error estimates

fully discrete second order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities we develop spatial temporally second order energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities one is quasi incompressible while the other is incompressible we introduce a novel energy quadratization technique to arrive at fully discrete linear schemes where in each time step only a linear system needs to be solved these schemes are then shown to be unconditionally energy stable rigorously subject to periodic boundary conditions so that a large time step is plausible both spatial and temporal mesh refinements are conducted to illustrate the second order accuracy of the schemes the linearization technique developed in this paper is so general that it can be applied to any thermodynamically consistent hydrodynamic theories so long as their energies are bounded below numerical examples on coarsening dynamics of two immiscible fluids and a heavy fluid drop settling in a lighter fluid matrix are presented to show the

smoothing property for schrodinger equations with potential superquadratic at infinity we prove a smoothing property for one dimensional time dependent schrodinger equations with potentials which satisfy at infinity k 2 as an application we show that the initial value problem for certain nonlinear schrodinger equations with such potentials is l2 well posed we also prove a sharp asymptotic estimate of the lp norm of the normalized eigenfunctions of h v for large energy

a time stepping scheme involving constant coefficient matrices for phase field simulations of two phase incompressible flows with large density ratios we present an efficient time stepping scheme for simulations of the coupled navier stokes cahn hilliard equations for the phase field approach the scheme has several attractive characteristics i it is suitable for large density ratios and numerical experiments with density ratios up to 1000 have been presented ii it involves only constant time independent coefficient matrices for all flow variables which can be pre computed during pre processing so it effectively overcomes the performance bottleneck induced by variable coefficient matrices associated with the variable density and variable viscosity iii it completely de couples the computations of the velocity pressure and the phase field function strategy for spectral element type spatial discretizations to overcome the difficulty associated with the large spatial order of the cahn hilliard equation is also discussed ample numerical simulations demonstrate that the current algorithm together with the navier stokes cahn hilliard phase field approach is an efficient and effective method for studying two phase flows involving large density ratios moving contact lines and interfacial topology changes

constructive solution of a bilinear optimal control problem for a schrodinger equation abstract often considered in numerical simulations related to the control of quantum systems the so called monotonic schemes have not been so far much studied from the functional analysis point of view yet these procedures provide an efficient constructive method for solving a certain class of optimal control problems this paper aims both at extending the results already available about these algorithms in the finite dimensional case i e the time discretized case and at completing those of the continuous case this paper starts with some results about the regularity of a functional related to a wide class of models in quantum chemistry these enable us to extend an inequality due to ojasiewicz to the infinite dimensional case finally some inequalities proving the cauchy character of the monotonic sequence are obtained followed by an estimation of the rate of convergence

highly efficient invariant conserving explicit runge kutta schemes for nonlinear hamiltonian differential equations abstract a unified framework of invariant conserving explicit runge kutta schemes for the nonlinear hamiltonian odes and pdes are proposed by utilizing the invariant energy quadratization technique first the nonlinear hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant for the nonlinear ode the reformulation is then discretized using a class of relaxation runge kutta schemes for the nonlinear pde the reformulation is first discretized in the space direction by adopting the fourier pseudo spectral discretization which preserves the semi discrete quadratic conservation laws then the semi discrete system is integrated in the time direction using the explicit relaxation runge kutta method the obtained method can conserve different quadratic conservation laws to machine accuracy which improves the numerical stability during long time computations besides the proposed method keeps the same convergence rate of the standard runge kutta scheme when computed with time step size rescaling and gives a rate of convergence reduced by at most one when computed without step size rescaling numerical experiments for several odes and pdes are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis

m theory on g2 manifolds and the method of p q brane webs using a reformulation of the method of p q webs we study the four dimensional n 1 quiver theories from m theory on seven dimensional manifolds with g2 holonomy we first construct such manifolds as u 1 quotients of eight dimensional toric hyper kahler manifolds using n 4 supersymmetric sigma models we show that these geometries in general are given by real cones on s2 bundles over complex two dimensional toric varieties v2 cr 2 c r then we discuss the connection between the physics content of m theory on such g2 manifolds and the method of p q webs motivated by a result of acharya and witten preprint hep th 0109152 we reformulate the method of p q webs and reconsider the derivation of the gauge theories using toric geometry mori vectors of v2 and brane charge constraints for we find that the gauge group is given by g u w1n u w2n u w3n this is required by the anomaly cancellation condition

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

gpav based unconditionally energy stable schemes for the cahn hilliard equation stability and error analysis abstract we present several first order and second order numerical schemes for the cahn hilliard equation with unconditional energy stability in terms of a discrete energy these schemes stem from the generalized positive auxiliary variable gpav idea and require only the solution of linear algebraic systems with a constant coefficient matrix more importantly the computational complexity operation count per time step of these schemes is approximately a half of those of the gpav and the scalar auxiliary variable sav methods in previous works we investigate the stability properties of the proposed schemes to establish stability bounds for the field function and the auxiliary variable and also provide their error analyses numerical experiments are presented to verify the theoretical analyses and also demonstrate the stability of the schemes at large time step sizes

local exact controllability of a one dimensional nonlinear schrodinger equation we consider a one dimensional nonlinear schrodinger equation modeling a bose einstein condensate in an infinite square well potential box this is a nonlinear control system in which the state is the wave function of the bose einstein condensate and the control is the length of the box we prove that local exact controllability around the ground state associated with a fixed length of the box holds generically with respect to the chemical potential mu i e up to an at most countable set of mu values the proof relies on the linearization principle and the inverse mapping theorem as well as ideas from analytic perturbation theory

on imposing dynamic contact angle boundary conditions for wall bounded liquid gas flows we present an efficient scheme within the phase field framework for imposing dynamic contact angle boundary conditions for wall bounded flows of two immiscible incompressible fluids with large density ratios first we develop an algorithm for imposing the dynamic contact angle boundary conditions to the cahn hilliard equation our algorithm consists of two components i we ignore the boundary conditions and transform the cahn hilliard equation into two nominally de coupled helmholtz type equations ii we treat the dynamic contact angle boundary conditions in such a manner that the two helmholtz type equations are truly de coupled then we combine this algorithm together with a scheme for variable density navier stokes equations we developed recently to form an efficient method for the coupled system of navier stokes and cahn hilliard equations for contact line problems involving large density ratios the overall method can deal with moving contact lines under dynamic and also static contact angle boundary conditions it is endowed with several attractive features that make the method very efficient in particular computations for all flow variables are completely decoupled the resultant linear algebraic systems after discretization for all flow variables involve only constant and time independent coefficient matrices which can be pre computed during pre processing even though the coupled navier stokes cahn hilliard system involves variable density and variable viscosity ample numerical simulations of wall bounded air water two phase flows have been presented to demonstrate the capability of the method for dealing with contact line problems under dynamic and static contact angle conditions involving large density ratios

diffuse interface model for incompressible two phase flows with large density ratios we investigate the applicability of an incompressible diffuse interface model for two phase incompressible fluid flows with large viscosity and density contrasts diffuse interface models have been used previously primarily for density matched fluids and it remains unclear to what extent such models can be used for fluids of different density thereby potentially limiting the application of these models in this paper the convective cahn hilliard equation and the condition that the velocity field is divergence free are derived from the conservation law of mass of binary mixtures in a straightforward way for fluids with large density and viscosity ratios differences in the equations of motion with a previously derived quasi incompressible model are shown to result from the respective assumptions made regarding the relationship between the diffuse fluxes of two species the convergence properties of the model are investigated for cases with large density ratio quantitative comparisons are made with results from previous studies to validate the model and its numerical implementation tests show that the variation in volume during the computation is of the order of machine accuracy which is consistent with our use of a conservative discretization scheme finite volume methods for the cahn hilliard equation results of the method are compared with previous work for the change in topology of rising bubbles and rayleigh taylor instability additional results are presented for head on droplet collision and the onset of droplet entrainment in stratified flows

numerical approximation of incompressible navier stokes equations based on an auxiliary energy variable abstract we present a numerical scheme for approximating the incompressible navier stokes equations based on an auxiliary variable associated with the total system energy by introducing a dynamic equation for the auxiliary variable and reformulating the navier stokes equations into an equivalent system the scheme satisfies a discrete energy stability property in terms of a modified energy and it allows for an efficient solution algorithm and implementation within each time step the algorithm involves the computations of two pressure fields and two velocity fields by solving several de coupled individual linear algebraic systems with constant coefficient matrices together with the solution of a nonlinear algebraic equation about a scalar number involving a negligible cost a number of numerical experiments are presented to demonstrate the accuracy and the performance of the presented algorithm

on local calabi yau supermanifolds and their mirrors we use local mirror symmetry to study a class of local calabi yau supermanifolds with bosonic sub variety vb having a vanishing first chern class solving the usual super cy condition requiring the equality of the total u 1 gauge charges of bosons b and the ghost like fields f one bqb fqf as bqb 0 and fqf 0 several examples are studied and explicit results are given for local ar supergeometries a comment on purely fermionic super cy manifolds corresponding to the special case where qb 0 b and fqf 0 is also made

regularity for a schrodinger equation with singular potentials and application to bilinear optimal control abstract we study the schrodinger equation i t u u v 0 u v 1 u 0 on r 3 0 t where v 0 x t x a t 1 with a w 2 1 0 t r 3 is a coulombian potential singular at finite distance and v 1 is an electric potential possibly unbounded the initial condition u 0 h 2 r 3 is such that r 3 1 x 2 2 u 0 x 2 dx the potential v 1 is also real valued and may depend on space and time variables we prove that if v 1 is regular enough and at most quadratic at infinity this problem is well posed and the regularity of the initial data is conserved for the solution we also give an application to the bilinear optimal control of the solution through the electric potential

optimal bilinear control of nonlinear schrodinger equations with singular potentials abstract we consider an optimal bilinear control problem for the nonlinear schrodinger equations with singular potentials we show the well posedness of the problem and the existence of an optimal control in addition the first order optimality system is rigorously derived our results give an improvement for some recent results

an efficient algorithm for incompressible n phase flows abstract we present an efficient algorithm within the phase field framework for simulating the motion of a mixture of n n 2 immiscible incompressible fluids with possibly very different physical properties such as densities viscosities and pairwise surface tensions the algorithm employs a physical formulation for the n phase system that honors the conservations of mass and momentum and the second law of thermodynamics we present a method for uniquely determining the mixing energy density coefficients involved in the n phase model based on the pairwise surface tensions among the n fluids our numerical algorithm has several attractive properties that make it computationally very efficient i it has completely de coupled the computations for different flow variables and has also completely de coupled the computations for the n 1 phase field functions ii the algorithm only requires the solution of linear algebraic systems after discretization and no nonlinear algebraic solve is needed iii for each flow variable the linear algebraic system involves only constant and time independent coefficient matrices which can be pre computed during pre processing despite the variable density and variable viscosity of the n phase mixture iv within a time step the semi discretized system involves only individual de coupled helmholtz type including poisson equations despite the strongly coupled phase field system of fourth spatial order at the continuum level v the algorithm is suitable for large density contrasts and large viscosity contrasts among the n fluids extensive numerical experiments have been presented for several problems involving multiple fluid phases large density contrasts and large viscosity contrasts in particular we compare our simulations with the de gennes theory and demonstrate that our method produces physically accurate results for multiple fluid phases we also demonstrate the significant and sometimes dramatic effects of the gravity density ratios pairwise surface tensions and drop sizes on the n phase configurations and dynamics the numerical results show that the method developed herein is capable of dealing with n phase systems with large density ratios large viscosity ratios and pairwise surface tensions and that it can be a powerful tool for studying the interactions among multiple types of fluid interfaces

optimal bilinear control of gross pitaevskii equations with coulombian potentials in this paper we consider an optimal bilinear control problem for the gross pitaevskii equations with coulombian potentials we show the well posedness of the problem and the existence of an optimal control in addition the first order optimality system is rigorously derived in particular we prove the frechet differentiability of the unconstrained functional we extend the study of hintermuller et al 2013 15 to more general power nonlinearities and unbounded potentials c 2015 elsevier inc all rights reserved

multiphase allen cahn and cahn hilliard models and their discretizations with the effect of pairwise surface tensions in this paper the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an n component mixture are studied for the general choice of phase variables the unisolvent property of the coefficient matrix involved in the n phase models based on the pairwise surface tensions is established moreover the symmetric positive definite property of the coefficient matrix on an n 1 n 1 dimensional hyperplane which is of fundamental importance to the well posedness of the models can be proved equivalent to some physical condition for pairwise surface tensions the n phase allen cahn and n phase cahn hilliard equations can then be derived from the free energy functional a natural property is that the resulting dynamics of concentrations are independent of phase variables chosen finite element discretizations for n phase models can be obtained as a natural extension of the existing discretizations for the two phase model the discrete energy law of the numerical schemes can be proved and numerically observed under some restrictions pertaining to time step size numerical experiments including the spinodal decomposition and the evolution of triple junctions are described in order to investigate the effect of pairwise surface tensions

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

on geometric transitions in string compactifications we reconsider the study of the geometric transitions and brane flux dualities in various dimensions first we give toric interpretations of the topology changing transitions in the calabi yau conifold and the spin 7 manifold the latter for instance can be viewed as three intersecting calabi yau conifolds according to the toric graph orbifolds of such geometries are given in terms of del pezzo complex surfaces second we propose a four dimensional f theory interpretation of type iib geometric transitions on the calabi yau conifold this gives a dual description of the m theory flop in terms of toric mirror symmetry in two dimensions we study the geometric transition in a singular spin 7 manifold constructed as a cone on su 3 u 1 in particular we discuss brane flux duality in such a compactification in both type iia and type iib superstrings these examples preserve one supercharge and so have supersymmetry in two dimensions then an interpretation in terms of f theory is given

linear first and second order unconditionally energy stable numerical schemes for the phase field model of homopolymer blends abstract in this paper we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends the governing system is derived from the energetic variational approach of a total free energy that consists of a nonlinear logarithmic flory huggins potential and a gradient entropy with a concentration dependent de gennes type coefficient the main challenging issue to solve this kind of models numerically is about the time marching problem i e how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level we solve this issue in this paper by developing the first and second order temporal approximation schemes based on the invariant energy quadratization method where all nonlinear terms are treated semi explicitly consequently the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step the unconditional energy stabilities are further proved various numerical simulations of 2d and 3d are presented to demonstrate the stability and the accuracy of the proposed schemes

linear schrodinger evolution equations with moving coulomb singularities abstract the cauchy problem for schrodinger evolution equations with a finite number of moving coulomb singularities is investigated the case of a single singularity has been studied by several authors see e g baudouin et al 2005 1 and okazawa et al 2010 19 however it seems to be no previous work on plural singularities we shall show that the problem has a unique classical solution by using a time dependent linear transformation of the unknown function which locally freezes the motion of the whole singularities under the simplest collisionless condition in fact a new existence and uniqueness theorem is available for the transformed problem such an abstract framework is established from the viewpoint of linear evolution equations of hyperbolic type in a hilbert space as an innovative modification of those in okazawa 1998 17 and okazawa and yoshii 2011 20

convergence and error analysis for the scalar auxiliary variable sav schemes to gradient flows we carry out convergence and error analysis of the scalar auxiliary variable sav methods for l 2 and h 1 gradient flows with a typical form of free energy we first derive h 2 bounds under certain assumptions suitable for both the gradient flows and the sav schemes which allow us to establish the convergence of the sav schemes under mild conditions we then derive error estimates with further regularity assumptions we also discuss several other gradient flows which cannot be cast in the general framework used in this paper but for which convergence and error analysis can still be established using a similar procedure

a general framework for deriving integral preserving numerical methods for pdes a general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented in particular linearly implicit methods preserving a time discretized version of the invariant are developed for systems of partial differential equations with polynomial nonlinearities the framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order it is proved formally that second order convergence is obtained the procedure is applied to a test case and numerical experiments are provided

f theory duals of m theory on g2 manifolds from mirror symmetry using mirror pairs m3 w3 in type ii superstring compactifications on calabi yau threefolds we study geometrically f theory duals of m theory on seven manifolds with g2 holonomy we first develop a way of obtaining landau ginzburg lg calabi yau threefolds w3 embedded in four complex dimensional toric varieties mirror to the sigma model on toric calabi yau threefolds m3 this method gives directly the right dimension without introducing non dynamical variables then using toric geometry tools we discuss the duality between m theory on s1 m3 z2 with g2 holonomy and f theory on elliptically fibred calabi yau fourfolds with su 4 holonomy containing w3 mirror manifolds illustrative examples are presented

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

energy and entropy preserving numerical approximations of thermodynamically consistent crystal growth models abstract we present a numerical scheme that preserves the total energy and the entropy production rate termed the energy and entropy production rate preserving scheme for a general class of thermodynamically consistent phase field models for dendritic crystal growth derived from the first and second law of thermodynamics the scheme is second order in time linear and energy and entropy production rate preserving for any time steps the scheme is first discretized in time aided by the energy quadratization eq method and then in space using compact finite difference methods the linear system resulting from the scheme is shown to be uniquely solvable at both the semi discrete and the fully discrete level mesh refinement tests are performed to show the second order time convergence rate in the scheme several numerical examples of dendritic crystal growth are provided to demonstrate the accuracy and efficiency of the scheme the effects of various model parameters on growth patterns of the crystal are further investigated in details with the numerical solver the approach to developing the energy and entropy production rate preserving numerical scheme proposed in this study is so general that it can be applied to a wide range of thermodynamically consistent models not limited to phase field models

a family of second order energy stable schemes for cahn hilliard type equations abstract we focus on the numerical approximation of the cahn hilliard type equations and present a family of second order unconditionally energy stable schemes by reformulating the equation into an equivalent system employing a scalar auxiliary variable we approximate the system at the time step n n denoting the time step index and is a real valued parameter and devise a family of corresponding approximations that are second order accurate and unconditionally energy stable this family of approximations contains the often used crank nicolson scheme and the second order backward differentiation formula as particular cases we further develop two efficient solution algorithms for the resultant discrete system of equations to overcome the difficulty caused by the unknown scalar auxiliary variable within each time step our method requires only the solution of either four de coupled individual helmholtz type equations or two separate individual systems with each system consisting of two coupled helmholtz type equations all the resultant linear algebraic systems involve only constant and time independent coefficient matrices that can be pre computed a number of numerical examples are presented to demonstrate the performance of the family of schemes developed herein we note that this family of second order approximations can be readily applied to devise energy stable schemes for other types of gradient flows when combined with the auxiliary variable approaches

linearly decoupled energy stable numerical methods for multicomponent two phase compressible flow in this paper for the first time we propose two linear decoupled energy stable numerical schemes for multicomponent two phase compressible flow with a realistic equation of state e g peng ro

mirror symmetry and landau ginzburg calabi yau superpotentials in f theory compactifications we study the landau ginzburg lg theory mirror to two dimensional 2d n 2 gauged linear sigma models on toric calabi yau manifolds we derive and solve new constraint equations for lg elliptic calabi yau superpotentials depending on the physical data of dual linear sigma models in the calabi yau threefold case we consider two examples first we give the mirror symmetry of the canonical line bundle over the hirzebruch surfaces fn second we find a special geometry with the affine so 8 lie algebra toric data extending the geometry of elliptically fibred k3 this geometry leads to a pure n 1 six dimensional so 8 gauge model from the f theory compactification for calabi yau fourfolds we give a new algebraic realization for ade hypersurfaces

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

nc calabi yau orbifolds in toric varieties with discrete torsion using the algebraic geometric approach of berenstein et al hep th 005087 and hep th 009209 and methods of toric geometry we study non commutative nc orbifolds of calabi yau hypersurfaces in toric varieties with discrete torsion we first develop a new way of getting complex d mirror calabi yau hypersurfaces h d in toric manifolds m d 1 with a c r action and analyse the general group of the discrete isometries of h d then we build a general class of d complex dimensional nc mirror calabi yau orbifolds where the non commutativity parameters are solved in terms of discrete torsion and toric geometry data of m d 1 in which the original calabi yau hypersurfaces are embedded next we work out a generalization of the nc algebra for generic d dimensional nc calabi yau manifolds and give various representations depending on different choices of the calabi yau toric geometry data we also study fractional d branes at orbifold points we refine and extend the result for nc t 2 x t 2 x t 2 z 2 x z 2 to higher dimensional torii orbifolds in terms of clifford algebra

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

local controllability and non controllability for a 1d wave equation with bilinear control abstract we consider a linear wave equation on the interval 0 1 with bilinear control and neumann boundary conditions we study the controllability of this nonlinear control system locally around a constant reference trajectory we prove that the following results hold generically for every t 2 this system is locally controllable in h 3 h 2 in time t with controls in l 2 0 t r for t 2 this system is locally controllable up to codimension one in h 3 h 2 in time t with controls in l 2 0 t r the reachable set is locally a non flat submanifold of h 3 h 2 with codimension one for every t 2 this system is not locally controllable more precisely the reachable set with controls in l 2 0 t r is contained in a non flat submanifold of h 3 h 2 with infinite codimension the proof of these results relies on the inverse mapping theorem and second order expansions