symmetry preserving difference schemes for some heat transfer equations lie group analysis of differential equations is a generally recognized method which provides invariant solutions integrability conservation laws etc in this paper we present three characteristic examples of the construction of invariant difference equations and meshes where the original continuous symmetries are preserved in discrete models conservation of symmetries in difference modelling helps to retain qualitative properties of the differential equations in their difference counterparts

an implicit fourth order compact finite difference scheme for one dimensional burgers equation abstract a fourth order compact finite difference method is proposed in this paper to solve one dimensional burgers equation the newly proposed method is based on the hopf cole transformation which transforms the original nonlinear burgers equation into a linear heat equation and transforms the dirichlet boundary condition into the robin boundary condition the linear heat equation is then solved by an implicit fourth order compact finite difference scheme a compact fourth order formula is also developed to approximate the robin boundary conditions while the initial condition for the heat equation is approximated using simpson s rule to maintain overall fourth order accuracy numerical experiments have been conducted to demonstrate the efficiency and high order accuracy of this method the numerical results also show that the method is unconditionally stable as there is no constraint on time step size

weak formulation based haar wavelet method for solving differential equations the haar wavelet based discretization method for solving differential equations is developed nonlinear burgers equation is considered as a test problem both strong and weak formulations based approaches are discussed the discretization scheme proposed is based on the weak formulation an attempt is made to combine the advantages of the fem and haar wavelets the obtained numerical results have been validated against a closed form analytical solution as well as fem results good agreement with the exact solution has been observed

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

surface states and spectra let d 1 let h 0 be the discrete laplacian on the hilbert space l 2 d 1 with a dirichlet boundary condition and let v be a potential supported on the boundary d 1 we introduce the notions of surface states and surface spectrum of the operator h h 0 v and explore their properties our main result is that if the potential v is random and if the disorder is either large or small enough then in dimension two h has no surface spectrum on h 0 with probability one to prove this result we combine aizenman molchanov theory with techniques of scattering theory

variational multiscale element free galerkin method for 2d burgers equation a new numerical method which is based on the coupling between variational multiscale method and meshfree methods is developed for 2d burgers equation with various values of re the proposed method takes full advantage of meshfree methods therefore no mesh generation and mesh recreation are involved meanwhile compared with the variational multiscale finite element method a strong assumption that is the fine scale vanishes identically over the element boundaries although non zero within the elements is not needed subsequently two problems which have an available analytical solution of their own are solved to analyze the convergence behaviour of the proposed method finally a 2d burgers equation having large re is solved and the results have also been compared with the ones computed by two other methods the numerical results show that the proposed method can indeed obtain accurate numerical results for 2d burgers equation having large re which does not refer to the choice of a proper stabilization parameter

comparison of some lie symmetry based integrators lie symmetry based integrators are constructed in order to preserve the local invariance properties of the equations the geometrical methods leading to discretized equations for numerical computations involve many different concepts therefore they give rise to numerical schemes that vary in the accuracy in the computational cost and in the implementation in this paper a comparison is made between some alternative lie symmetry based methods illustrated on the example of the burgers equation the importance of the symmetry preservation is numerically highlighted

new characterizations of the region of complete localization for random schrodinger operators we study the region of complete localization in a class of random operators which includes random schrodinger operators with anderson type potentials and classical wave operators in random media as well as the anderson tight binding model we establish new characterizations or criteria for this region of complete localization given either by the decay of eigenfunction correlations or by the decay of fermi projections these are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

localization bounds for multiparticle systems we consider the spectral and dynamical properties of quantum systems of n particles on the lattice mathbb z d of arbitrary dimension with a hamiltonian which in addition to the kinetic term includes a random potential with iid values at the lattice sites and a finite range interaction two basic parameters of the model are the strength of the disorder and the strength of the interparticle interaction it is established here that for all n there are regimes of high disorder and or weak enough interactions for which the system exhibits spectral and dynamical localization the localization is expressed through bounds on the transition amplitudes which are uniform in time and decay exponentially in the hausdorff distance in the configuration space the results are derived through the analysis of fractional moments of the n particle green function and related bounds on the eigenfunction correlators

invariant discretization schemes for the shallow water equations invariant discretization schemes are derived for the one and two dimensional shallow water equations with periodic boundary conditions while originally designed for constructing invariant finite difference schemes we extend the use of difference invariants to allow construction of invariant finite volume methods as well it is found that the classical invariant schemes converge to the lagrangian formulation of the shallow water equations these schemes require redistribution of the grid points according to the physical fluid velocity i e the mesh cannot remain fixed in the course of the numerical integration invariant eulerian discretization schemes are proposed for the shallow water equations in computational coordinates instead of using the fluid velocity as the grid velocity an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level the numerical conservation of energy mass and momentum is evaluated for both the invariant

invariantization of the crank nicolson method for burgers equation abstract in this article a geometric technique to construct numerical schemes for partial differential equations pdes that inherit lie symmetries is proposed the moving frame method enables one to adjust the numerical schemes in a geometric manner and systematically construct proper invariant versions of them to illustrate the method we study invariantization of the crank nicolson scheme for burgers equation with careful choice of normalization equations the invariantized schemes are shown to surpass the standard scheme successfully removing numerical oscillation around sharp transition layers

exponential decay of correlations implies area law we prove that a finite correlation length i e exponential decay of correlations implies an area law for the entanglement entropy of quantum states defined on a line the entropy bound is exponential in the correlation length of the state thus reproducing as a particular case hastings s proof of an area law for groundstates of 1d gapped hamiltonians

a finite element approach for solution of burgers equation in this study for the burgers equation which is used a model problem in turbulence and shock wave theory was given a finite element solution considering the numerical results obtained for different values of viscosity compared with cole s analytical solution it was seen that they were in good agreement

convecting reference frames and invariant numerical models abstract in the recent paper by bernardini et al 1 the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied in a simplified investigation carried out using the viscous burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow here we further discuss this problem and relate it to known results on invariant discretization schemes non invariant and invariant finite difference discretizations of burgers equation are proposed and compared with the discretization using the remedy proposed by bernardini et al

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

recovery of jumps and singularities of an unknown potential from limited data in dimension 1 we prove that in dimension 1 the jumps and singularities of an unknown potential appearing in the schrodinger equation can be recovered using the born approximation the result is based on an accurate determination of the first nonlinear term of the born series we shall also work in wider function space than the previous publications on this topic numerical examples illustrate the feasibility of this technique

numerical solution of burgers equation by quadratic b spline finite elements in this paper a galerkin quadratic b spline finite element method is used to solve numerically the one dimensional burgers equation reduced by the hopf cole transformation the performance of the method is tested on two model problems involving moderately reynolds numbers with known exact solutions the obtained numerical results show that the method is efficient robust and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails computed results are compared with other numerical results in the literature a stability analysis of the method is also investigated

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

a fully implicit finite difference scheme for two dimensional burgers equations the two dimensional burgers equations are discretized in fully implicit finite difference form this scheme leads to a system of nonlinear difference equations to be solved at each time step newton s method is used to solve this nonlinear system the linear system is solved by a direct method at each iteration of newton s method the accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results the present method performs well

numerical integrators that preserve symmetries and reversing symmetries we consider properties of flows the relationships between them and whether numerical integrators can be made to preserve these properties this is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields this new framework unifies several known constructions we also use the concept of covariance of a numerical method with respect to a group of coordinate transformations the main application is to explore the relationship between spatial symmetries reversing symmetries and time symmetry of flows and numerical integrators

invariant parameterization and turbulence modeling on the beta plane abstract invariant parameterization schemes for the eddy vorticity flux in the barotropic vorticity equation on the beta plane are constructed and then applied to turbulence modeling this construction is realized by the exhaustive description of differential invariants for the maximal lie invariance pseudogroup of this equation using the method of moving frames which includes finding functional bases of differential invariants of arbitrary order a minimal generating set of differential invariants and a basis of operators of invariant differentiation in an explicit form special attention is paid to the problem of two dimensional turbulence on the beta plane it is shown that classical hyperdiffusion as used to initiate the energy enstrophy cascades violates the symmetries of the vorticity equation invariant but nonlinear hyperdiffusion like terms of new types are introduced and then used in the course of numerically integrating the vorticity equation and carrying out freely decaying turbulence tests it is found that the invariant hyperdiffusion scheme is closely reproducing the theoretically predicted k 1 shape of enstrophy spectrum in the enstrophy inertial range by presenting conservative invariant hyperdiffusion terms we also demonstrate that the concepts of invariant and conservative parameterizations are consistent

fourth order finite difference method for solving burgers equation abstract in this paper we present fourth order finite difference method for solving nonlinear one dimensional burgers equation this method is unconditionally stable the convergence analysis of the present method is studied and an upper bound for the error is derived numerical comparisons are made with most of the existing numerical methods for solving this equation

an accurate approximation algorithm for burgers equation in the presence of small viscosity abstract most of the existing numerical schemes developed to solve burgers equation cannot exhibit its correct physical behavior for very small values of viscosity this difficulty can be overcome by using splitting methods derived for near integrable system this class of methods has positive real coefficients and can be used for non reversible systems such as burgers equation it also has the advantage of being able to account small viscosity in the accuracy the algorithm is based on the combination of implicit explicit finite difference schemes to solve each simplified problem and filtering technique to treat nonlinear instability the resulting algorithm is accurate efficient and easy to implement the new numerical results are compared with numerical and exact solutions reported in the literature and found that they are very accurate for small values of the viscosity

structure from motion theoretical foundations of a novel approach using custom built invariants we present a novel simple formulation of the problem of 3d object reconstruction from images in this formulation the object is seen as lying at the intersection of the projection of orbits of custom built lie group actions the group parameters correspond to unknown irrelevant quantities such as the camera orientation the depth parameters of the object with respect to the camera and the focal length we then use an algorithmic method based on moving frames a la fels olver to obtain a fundamental set of invariants of these group actions the invariants are used to define a set of equations determining the 3d object thus providing a mathematical formulation of the problem where the irrelevant parameters do not appear

a direct variational methods applied to burgers equation in this study a compact approximate method in limiting form for calculating the solution of burgers equation with appropriate boundary conditions is presented the results obtained by present method are found to be in good agreement with those due to earlier authors and offers appreciable advantages for burgers like nonlinear problems

a remark on the cluster theorem an improved version of the cluster theorem of relativistic quantum field theory with the correct exponential decay is derived by function theoretical methods

the korteweg de vries equation and its symmetry preserving discretization the korteweg de vries equation is one of the most important nonlinear evolution equations in the mathematical sciences in this article invariant discretization schemes are constructed for this equation both in the lagrangian and in the eulerian form we also propose invariant schemes that preserve the momentum numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes we find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving galilean transformations which is demonstrated numerically as well

geometric integrators for multiple time scale simulation in this paper we review and extend recent research on averaging integrators for multiple time scale simulation such as are needed for physical n body problems including molecular dynamics materials modelling and celestial mechanics a number of methods have been proposed for direct numerical integration of multiscale problems with special structure such as the mollified impulse method garcia archilla sanz serna and skeel 1999 siam j sci comput 20 930 63 and the reversible averaging method leimkuhler and reich 2001 j comput phys 171 95 114 features of problems of interest such as thermostatted coarse grained molecular dynamics require extension of the standard framework at the same time in some applications the computation of averages plays a crucial role but the available methods have deficiencies in this regard we demonstrate that a new approach based on the introduction of shadow variables which mirror physical variables has promised for broadening the usefulness of multiscale methods and enhancing accuracy of or simplifying computation of averages the shadow variables must be computed from an auxiliary equation while a geometric integrator in the extended space is possible in practice we observe enhanced long term energy behaviour only through use of a variant of the method which controls drift of the shadow variables using dissipation and sacrifices the formal geometric properties such as time reversibility and volume preservation in the enlarged phase space stabilizing the corresponding properties in the physical variables the method is applied to a gravitational three body problem as well as a partially thermostatted model problem for a dilute gas of diatomic molecules

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

symmetries and first integrals of ordinary difference equations this paper describes a new symmetry based approach to solving a given ordinary difference equation by studying the local structure of the set of solutions we derive a systematic method for determining one parameter lie groups of symmetries in closed form such groups can be used to achieve successive reductions of order if there are enough symmetries the difference equation can be completely solved several examples are used to illustrate the technique for transitive and intransitive symmetry groups it is also shown that every linear second order ordinary difference equation has a lie algebra of symmetry generators that is isomorphic to sl 3 the paper concludes with a systematic method for constructing first integrals directly which can be used even if no symmetries are known

on the class of high order time stepping schemes based on pade approximations for the numerical solution of burgers equation abstract numerical solution of burgers equation is presented using finite difference methods in space and positivity preserving pade approximations in time a class of high order time stepping schemes is introduced for practical purposes first second third and fourth order schemes are constructed efficient parallel versions of these schemes are given using a splitting technique of rational functions accuracy of the schemes is demonstrated by solving a test problem and comparing numerical results with the exact solution time evolution graphs show the physical phenomenon of the problem convergence tables are given to verify the theoretical order of convergence

a finite difference approach for solution of burgers equation abstract in this paper we have applied restrictive pade approximation classical implicit finite difference method to the burgers equation with a set of initial and boundary conditions to obtain its numerical solution the stability region and truncation error of the new method are discussed the accuracy of the proposed method is demonstrated by the two test problems the numerical results obtained by this method for various values of viscosity have been compared with the exact solution to show the efficiency of the method the numerical results are found in good agreement with the exact solutions

many body localization in the droplet spectrum of the random xxz quantum spin chain abstract we study many body localization properties of the disordered xxz spin chain in the ising phase disorder is introduced via a random magnetic field in the z direction we prove a strong form of dynamical exponential clustering for eigenstates in the droplet spectrum for any pair of local observables separated by a distance l the sum of the associated correlators over these states decays exponentially in l in expectation this exponential clustering persists under the time evolution in the droplet spectrum our result applies to the large disorder regime as well as to the strong ising phase at fixed disorder with bounds independent of the support of the observables

numerical solution of one dimesional burgers equation explicit and exact explicit finite difference methods this paper presents finite difference solution and analytical solution of the finite difference approximations based on the standard explicit method to the one dimensional burgers equation which arises frequently in the mathematical modelling used to solve problems in fluid dynamics results obtained by these ways for some modest values of viscosity have been compared with the exact fourier one it is shown that they are in good agreement with each other

an inverse born approximation for the general nonlinear schrodinger operator on the line this work deals with the inverse scattering problems for the one dimensional schrodinger equation of the most general form with general nonlinearity f and with a real parameter k under some assumptions on f we prove that all singularities and jumps of f x u0 where u0 x k eikx can be recovered by the reflection coefficient with arbitrary large k

anderson localization for random schrodinger operators with long range interactions we prove pure point spectrum with exponentially decaying eigenfunctions at all band edges for schrodinger operators with a periodic potential plus a random potential of the form v w x q i w f x i where f decays at infinity like x m for m 4d resp m 3d depending on the regularity of f the random variables q i are supposed to be independent and identically distributed we assume that their distribution has a bounded density of compact support

localization of surface spectra we study spectral properties of the discrete laplacian h on the half space with random boundary condition the v n are independent random variables on a probability space and is the coupling constant it is known that if the v n have densities then on the interval 2 d 1 2 d 1 h 0 the spectrum of the dirichlet laplacian the spectrum of h is p a s absolutely continuous for all jl1 here we show that if the random potential p satisfies the assumption of aizenman molchanov am then there are constants d and d such that for d the spectrum of h outside h 0 is p a s pure point with exponentially decaying eigenfunctions

the new numerical method for solving the system of two dimensional burgers equations in this paper the system of two dimensional burgers equations are solved by local discontinuous galerkin ldg finite element method the new method is based on the two dimensional hopf cole transformations which transform the system of two dimensional burgers equations into a linear heat equation then the linear heat equation is solved by the ldg finite element method the numerical solution of the heat equation is used to derive the numerical solutions of burgers equations directly such a ldg method can also be used to find the numerical solution of the two dimensional burgers equation by rewriting burgers equation as a system of the two dimensional burgers equations three numerical examples are used to demonstrate the efficiency and accuracy of the method

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

numerical solution of burgers equation with high order splitting methods in this work high order splitting methods have been used for calculating the numerical solutions of burgers equation in one space dimension with periodic dirichlet neumann and robin boundary conditions however splitting methods with real coefficients of order higher than two necessarily have negative coefficients and cannot be used for time irreversible systems such as burgers equations due to the time irreversibility of the laplacian operator therefore the splitting methods with complex coefficients and extrapolation methods with real and positive coefficients have been employed if we consider the system as the perturbation of an exactly solvable problem or one that can be easily approximated numerically it is possible to employ highly efficient methods to approximate burgers equation the numerical results show that both the methods with complex time steps having one set of coefficients real and positive say a i r and b i c and high order extrapolation methods derived from a lower order splitting method produce very accurate solutions of burgers equation

invariant meshless discretization schemes a method is introduced for the construction of meshless discretization schemes which preserve lie symmetries of the differential equations that these schemes approximate the method exploits the fact that equivariant moving frames provide a way of associating invariant functions with non invariant functions an invariant meshless approximation of a nonlinear diffusion equation is constructed comparative numerical tests with a non invariant meshless scheme are presented these tests yield that invariant meshless schemes can lead to substantially improved numerical solutions compared to numerical solutions generated by non invariant meshless schemes

numerical solution of the burgers equation by local discontinuous galerkin method abstract in this paper the burgers equation is transformed into the linear diffusion equation by using the hopf cole transformation the obtained linear diffusion equation is discretized in space by the local discontinuous galerkin method the temporal discretization is accomplished by the total variation diminishing runge kutta method numerical solutions are compared with the exact solution and the numerical solutions obtained by adomian s decomposition method finite difference method b spline finite element method and boundary element method the results show that the local discontinuous galerkin method is one of the most efficient methods for solving the burgers equation even with small viscosity coefficient it can get the satisfied solution

partial recovery of potentials in generalized nonlinear schrodinger equations on the line the inverse born approximation method is applied to inverse scattering problems for two one dimensional nonlinear schrodinger equations we prove that all jumps and singularities of the unknown function are recovered from the born function sequence in particular one can locate any bounded interval on the line the result is based on the accurate determination of the first nonlinear term of the born sequence

a sixth order compact finite difference scheme to the numerical solutions of burgers equation abstract a numerical solution of the one dimensional burgers equation is obtained using a sixth order compact finite difference method to achieve this a tridiagonal sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been combined the scheme is implemented to solve two test problems with known exact solutions comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy and efficiency with minimal computational effort the present results are also seen to be more accurate than some available results given in the literature

a numerical solution of the burgers equation using cubic b splines in the present paper numerical solutions of the one dimensional burgers equation are obtained by a method based on collocation of cubic b splines over finite elements the accuracy of the proposed method is demonstrated by three test problems the numerical results are found in good agreement with exact solutions time space integration of the burgers equation yields a system of difference equation which is shown to be unconditionally stable

an exponentially fitted method for solving burgers equation in this paper an exponentially fitted method is used to numerically solve the one dimensional burgers equation the performance of the method is tested on the model involving moderately large reynolds numbers the obtained numerical results show that the method is efficient stable and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails copyright 2009 john wiley sons ltd

numerically invariant signature curves corrected versions of the numerically invariant expressions for the affine and euclidean signature of a planar curve introduced by calabi et al in int j comput vision 26 107 135 1998 are presented the new formulas are valid for fine but otherwise arbitrary partitions of the curve we also give numerically invariant expressions for the four differential invariants parameterizing the three dimensional version of the euclidean signature curve namely the curvature the torsion and their derivatives with respect to arc length

a numerical method based on crank nicolson scheme for burgers equation abstract in this paper we present a solution based on crank nicolson finite difference method for one dimensional burgers equation burgers equation arises frequently in mathematical modeling of problems in fluid dynamics hopf cole transformation e hopf the partial differential equation u t uu x u xx commun pure appl math 3 1950 201 230 j d cole on a quasilinear parabolic equation occurring in aerodynamics quart appl math 9 1951 225 236 is used to linearize burgers equation the resulting heat equation is discretized by using crank nicolson finite difference scheme this method is shown to be unconditionally stable and second order accurate in space and time numerical results obtained by the present method have been compared with exact solution for different values of reynolds number

quantum harmonic oscillator systems with disorder we study many body properties of quantum harmonic oscillator lattices with disorder a sufficient condition for dynamical localization expressed as a zero velocity lieb robinson bound is formulated in terms of the decay of the eigenfunction correlators for an effective one particle hamiltonian we show how state of the art techniques for proving anderson localization can be used to prove that these properties hold in a number of standard models we also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one particle eigenfunction correlators in particular we show that static correlations decay exponentially fast if the corresponding effective one particle hamiltonian exhibits localization at low energies regardless of whether there is a gap in the spectrum above the ground state or not our results apply to finite as well as to infinite oscillator systems the eigenfunction correlators that appear are more general than those previously studied in the literature in particular we must allow for functions of the hamiltonian that have a singularity at the bottom of the spectrum we prove exponential bounds for such correlators for some of the standard models

a linearized numerical scheme for burgers like equations a linearized implicit finite difference method is presented to find numerical solutions of the one dimensional burgers like equations the method has been used successfully to obtain accurate numerical solutions even for small values of viscosity term n results obtained by the present method using a direct technique for some values of n have been compared with the exact values and are found to be in good agreement with each other

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

lieb robinson bounds and the exponential clustering theorem we give a lieb robinson bound for the group velocity of a large class of discrete quantum systems which can be used to prove that a non vanishing spectral gap implies exponential clustering in the ground state of such systems

spectral gap and exponential decay of correlations we study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short range interactions on a wide class of lattices we prove that if two observables anticommute with each other at large distance then the nonvanishing spectral gap implies exponential decay of the corresponding correlation when two observables commute with each other at large distance the connected correlation function decays exponentially under the gap assumption if the observables behave as a vector under the u 1 rotation of a global symmetry of the system we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself rather than just the connected correlation function decays exponentially under the gap assumption on a lattice with a certain self similarity in fractal dimensions d 2 in particular if the system is translationally invariant in one of the spatial directions then this self similarity condition is automatically satisfied we also treat systems with long range power law decaying interactions

lie group formalism for difference equations the methods of lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations several examples are analysed one of them being a nonlinear difference equation for the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

energy conserving arbitrary order numerical solutions of then body problem in this paper an energy conserving modification of the well known extrapolation methods for solving then body problem is presented the method is compared with the most commonly used extrapolation methods it appears that the presented modification yields a better approximation of the exact solution computer examples are given

a numerical scheme based on a solution of nonlinear advection diffusion equations a mathematical formulation of the two dimensional cole hopf transformation is investigated in detail by making use of the cole hopf transformation a nonlinear two dimensional unsteady advection diffusion equation is transformed into a linear equation and the transformed equation is solved by the spectral method previously proposed by one of the authors thus a solution to initial value problems of nonlinear two dimensional unsteady advection diffusion equations is derived on the base of the solution a numerical scheme explicit with respect to time is presented for nonlinear advection diffusion equations numerical experiments show that the present scheme possesses the total variation diminishing properties and gives solutions with good quality

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions

on smoothing of the crank nicolson scheme and higher order schemes for pricing barrier options most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise price discrete barrier options have not only nonsmooth payoffs but also time dependent discontinuities in pricing barrier options certain aspects are triggered if the asset price becomes too high or too low standard smoothing schemes used to solve problems with nonsmooth payoff do not work well for discrete barrier options because of discontinuities introduced in the time domain when each barrier is applied moreover these unwanted oscillations become worse when estimating the hedging parameters e g delta and gamma we have an improved smoothing strategy for the crank nicolson method which is unique in achieving optimal order convergence for barrier option problems numerical experiments are discussed for one asset and two asset problems time evolution graphs are obtained for one asset problems to show how option prices change with respect to time this smoothing strategy is then extended to higher order methods using diagonal m m pade main schemes under a smoothing strategy of using as damping schemes the 0 2m 1 subdiagonal pade schemes

symmetry adapted moving mesh schemes for the nonlinear schrodinger equation in this paper we consider symmetry preserving difference schemes for the nonlinear schrodinger equation where n is the number of space dimensions this equation describes one dimensional waves in n space dimensions in many physical situations including phenomena in plasma physics and nonlinear optics we will consider the nonintegrable case n 2 for which the equation admits solutions that blow up in a finite time and construct discretizations based upon moving mesh schemes that have the same lie group properties and lagrangian structures as the continuous counterpart

lie point symmetries of the discrete liouville equation the liouville equation is well known to be linearizable by a point transformation it has an infinite dimensional lie point symmetry algebra isomorphic to a direct sum of two virasoro algebras we show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries we do however construct a difference system approximating the liouville equation that is invariant under the maximal finite subgroup the invariant scheme is an explicit one and provides a much better approximation of exact solutions than a comparable standard noninvariant scheme and also than a scheme invariant under an infinite dimensional group of generalized symmetries