an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

spatial finite difference approximations for wave type equations the simplest finite difference approximations for spatial derivatives are centered explicit and applied to regular equispaced grids well established generalizations include the use of implicit compact approximations and staggered grids we find here that the combination of these two concepts together with high formal order of accuracy is very effective for approximating the first derivatives in space that occur in many wave type pdes

on asymptotic global error estimation and control of finite difference solutions for semilinear parabolic equations abstract the aim of this paper is to extend the global error estimation and control addressed in lang and verwer siam j sci comput 29 2007 for initial value problems to finite difference solutions of semilinear parabolic partial differential equations the approach presented there is combined with an estimation of the pde spatial truncation error by richardson extrapolation to estimate the overall error in the computed solution approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time asymptotic control in a discrete l 2 norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement numerical examples are used to illustrate the reliability of the estimation and control strategies

indecomposable representations of the nonlinear lie algebras on the two state heisenberg weyl basis the indecomposable representations of the nonlinear lie algebras of vector type r 0 1 one of which is generated by two angular momentum operators j0 j2 and any component of an irreducible tensor operator of rank 1 t 1 0 1 of an so 3 group are studied in detail we give the explicit expressions for the infinite dimensional indecomposable representations of r which further subduce the infinite dimensional indecomposable representations on the invariant subspaces and induce the finite dimensional or infinite dimensional indecomposable irreducible representations on the quotient spaces as special cases the standard angular momentum representations of r are given also

exact evolution equations for su 2 quasidistribution functions we derive an exact differential evolution equation for a class of su 2 quasiprobability distribution functions linear and quadratic cases are considered as well as the quasiclassical limit of the large dimension of representation s 1

canonical structure in so 4 q and relations to e 3 q and so 3 1 q starting with the su 2 q su 2 q basis and using q cg coefficients the closest possible q deformation of the classical canonical gelfand zetlin formalism is constructed explicitly for so 4 q simple introduction of q brackets for each factor of the classical matrix elements valid for su n q is shown to need significant and unexpected modifications for so n q even when n 4 contraction to e 3 q and continuation to so 3 1 q are studied making explicit the possibilities and the problems that arise periodic representations for q a root of unity are commented upon briefly

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

an algorithm for solving singular perturbation problems with mechanization abstract in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of finite difference method for solving singular perturbation problems was established a new matlab procedure ehssp was implemented too several linear and non linear problems are presented to illustrate the implementation of the program

the three dimensional euclidean quantum group e 3 q and its r matrix a contraction procedure starting from so 4 q is used to determine the quantum analog e 3 q of the three dimensional euclidean group and the structure of its representations a detailed analysis of the contraction of the r matrix is then performed and its explicit expression has been found the classical limit of r is shown to produce an integrable dynamical system by means of the r matrix the pseudogroup of the noncommutative representative functions is considered it will finally be shown that a further contraction made on e 3 q produces the two dimensional galilei quantum group and this in turn can be used to give a new realization of e 3 q and e 2 1 q

linear complementarity problems and multiple objective programming an equivalence is demonstrated between solving a linear complementarity problem with general data and finding a certain subset of the efficient points of a multiple objective programming problem a new multiple objective programming based approach to solving linear complementarity problems is presented results on existence uniqueness and computational complexity are included

nonlinear lie algebra and ladder operators for orbital angular momentum based on the nonlinear lie algebra a new framework is put forward to treat the problem of angular momentum from this framework the ladder operators for the orbital angular momentum can be solved as simultaneous eigenvectors of the lie derivations ad l2 and ad l2 in an operator space the ladder operators for solvable central potentials can also be found in a united way

a spatially second order alternating direction implicit adi method for solving three dimensional parabolic interface problems abstract a new matched alternating direction implicit adi method is proposed in this paper for solving three dimensional 3d parabolic interface problems with discontinuous jumps and complex interfaces this scheme inherits the merits of its ancestor for two dimensional problems while possesses several novel features such as a non orthogonal local coordinate system for decoupling the jump conditions two side estimation of tangential derivatives at an interface point and a new douglas rachford adi formulation that minimizes the number of perturbation terms to attack more challenging 3d problems in time discretization this new adi method is found to be of first order and stable in various experiments in space discretization the matched adi method achieves the second order accuracy based on simple cartesian grids for various irregularly shaped surfaces and spatial temporal dependent jumps computationally the matched adi method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3d parabolic equations in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom n i e o n furthermore unlike iterative methods the adi method is an exact or non iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed n therefore the proposed matched adi method provides a very promising tool for solving 3d parabolic interface problems

implicit finite difference simulations of seismic wave propagation we propose a new finite difference modeling method implicit both in space and in time for the scalar wave equation we use a three level implicit splitting time integration method for the temporal derivative and implicit finite difference operators of arbitrary order for the spatial derivatives both the implicit splitting time integration method and the implicit spatial finite difference operators require solving systems of linear equations we show that it is possible to merge these two sets of linear systems one from implicit temporal discretizations and the other from implicit spatial discretizations to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm we give the complete derivations of the implicit splitting time integration method and the implicit spatial finite difference operators and present the resulting discretized formulas for the scalar wave equation we conduct a thorough numerical analysis on grid dispersions of this new implicit

on the numerical solution of nonlinear burgers type equations using meshless method of lines abstract in this paper a meshless method of lines mmol is proposed for the numerical solution of nonlinear burgers type equations this technique does not require a mesh in the problem domain and only a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions rbfs the scheme is tested for various examples the results obtained by this method are compared with the exact solutions and some earlier work

highly accurate compact mixed methods for two point boundary value problems finite difference method and finite element method are widely used partly for their simplicity though these methods can obtain first order or second order of accuracy in this paper we give two highly accurate fourth order and sixth order of accuracy respectively while still quite simple schemes for two point boundary value problems we call them compact mixed methods since they can obtain numerical solutions for unknown function and its first derivative simultaneously to yield best numerical solutions we combine our compact mixed schemes with a direct solver and an iterative solver then we employ fourier method to analyze differencing errors shows that our compact mixed formulae are closer to the true wavenumber numerical experiments show compact mixed schemes are very efficient and highly accurate techniques

nonlinear symmetry algebra of the mic kepler problem on the sphere s3 the quantum mechanical problem of motion in the dual charged coulomb field modified by a centrifugal term mic kepler problem is considered in the three dimensional space of constant positive curvature s3 conserved operators in this problem form a cubic algebra similar to that of the kepler problem on s3 the explicit form of invariants of this algebra shows that its representation associated with the mic kepler problem on s3 is a nondegenerate one the cubic symmetry algebra is used to obtain the energy spectrum of the problem

global 2d digital image correlation for motion estimation in a finite element framework a variational formulation and a regularized pyramidal multi grid implementation in this study the inverse problem of reconstructing the in plane 2d displacements of a monitored surface through a sequence of two dimensional digital images severely ill posed in hadamard s sense is deeply investigated a novel variational formulation is presented for the continuum 2d digital image correlation problem and critical issues such as semi coercivity and solution multiplicity are discussed by functional analysis tools in the framework of a galerkin finite element discretization of the displacement field a robust implementation for 2d digital image correlation is outlined aiming to attenuate the spurious oscillations which corrupt the deformation scenario especially when very fine meshes are adopted recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid multi grid cycles are performed ascending and descending along the pyramid with only one newton iteration per level irrespective of the tolerance satisfaction as if the problem were linear at each level the conventional least square matching functional is herein enriched by a tychonoff regularization provision preserving the solution against an unstable response the algorithm is assessed on the basis of both synthetic and truly experimental image pairs copyright 2013 john wiley sons ltd

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

second order method for solving 3d elasticity equations with complex interfaces elastic materials are ubiquitous in nature and indispensable components in man made devices and equipments when a device or equipment involves composite or multiple elastic materials elasticity interface problems come into play the solution of three dimensional 3d elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equations this work introduces the matched interface and boundary mib method for solving 3d elasticity interface problems the proposed mib elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface the interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines such an enforcement determines the fictitious values a number of new techniques have been developed to construct efficient mib elasticity interface schemes for dealing with cross derivative in coupled governing equations the proposed method is extensively validated over both weak and strong discontinuities of the solution both piecewise constant and position dependent material parameters both smooth and nonsmooth interface geometries and both small and large contrasts in the poisson s ratio and shear modulus across the interface numerical experiments indicate that the present mib method is of second order convergence in both l and l 2 error norms for handling arbitrarily complex interfaces including biomolecular surfaces to our best knowledge this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

on some nonlinear extensions of the angular momentum algebra deformations of the lie algebras so 4 so 3 1 and e 3 that leave their so 3 subalgebra undeformed and preserve their coset structure are considered it is shown that such deformed algebras are associative for any choice of the deformation parameters their casimir operators are obtained and some of their unitary irreducible representations are constructed for vanishing deformation the unitary irreducible representations of the deformed algebras transform into those of the corresponding lie algebras that contain each of the so 3 unitary irreducible representations at most once it is also proved that similar deformations of the lie algebras su 3 sl 3 r and of the semidirect sum of an abelian algebra t 5 and so 3 do not lead to associative algebras

approach of spherical harmonics to the representation of the deformed su 1 1 algebra the m shifting generators of su 2 algebra together with a pair of l shifting ladder symmetry operators have been used in the space of all spherical harmonics ylm in order to introduce a new set of operators expressing the transitions between them it is shown that the space of spherical harmonics whose l 2m or l 2m is given presents negative and positive irreducible representations of a deformed su 1 1 algebra respectively these internal symmetries also suggest new algebraic methods to construct the spherical harmonics in the framework of the spectrum generating algebras

method of lines solutions of the extended boussinesq equations a numerical solution procedure based on the method of lines for solving the nwogu one dimensional extended boussinesq equations is presented the numerical scheme is accurate up to fifth order in time and fourth order accurate in space thus reducing all truncation errors to a level smaller than the dispersive terms retained by most extended boussinesq models exact solitary wave solutions and invariants of motions recently derived by the authors are used to specify initial data for the incident solitary waves in the numerical model of nwogu and for the verification of the associated computed solutions the invariants of motions and several error measures are monitored in order to assess the conservative properties and the accuracy of the numerical scheme the proposed method of lines solution procedure is general and can be easily modified to solve a wide range of boussinesq like equations in coastal engineering

a matlab implementation of upwind finite differences and adaptive grids in the method of lines in this paper we report on the development of a matlab library for the solution of partial differential equation systems following the method of lines in particular we focus attention on upwind finite difference schemes and grid adaptivity i e grid movement or grid refinement several algorithms are presented and their performance is demonstrated with illustrative examples including a fixed bed reactor with periodic flow reversal a model of flame propagation and the korteweg de vries equation

boson and differential realizations of polynomial angular momentum algebra the inhomogeneous multiboson and multivariable differential realizations of the polynomial angular momentum algebra of quadratic type l2 which may be seen as the nonlinear extension of an ordinary angular momentum algebra su 2 or its noncompact form su 1 1 are obtained from fock representations of l2 which are related to master representation of l2 on the space of its universal enveloping algebra u l2 or to induced representations on their respective quotient spaces u l2 ii with different left ideals ii

compact finite difference methods for high order integro differential equations high order integro differential equations ides especially nonlinear are usually difficult to solve even numerically in this chapter we give a highly accurate compact finite difference method to efficiently solve integro differential equations including high order and nonlinear problems by numerical experiments we show that compact finite difference method for integro differential equations is fast and can obtain a high accuracy

on a particular quadratic network problem abstract let c x d x and h x be linear functions defined over a convex polyhedron x x ax r and 0 x u where a is the node edge incidence matrix of a given network let f x c x d x h x be a particular quadratic function which we intend to minimize over x in this paper we use the theory of multiobjective programming do develop algorithms for the semidefinite positive case f x d x 2 h x and is a strictly postive real constant and the semidefinite negative case f x d x 2 h x and is a strictly negative real constant a special indefinite case h x is constant over x is also studied in the proposed algorithms basic solutions are used in all the interactions with possible exception for the last one moreover only the concepts of the simplex method are used consequently these algorithms can be used even when a is not the node edge incidence matrix of a network however they are particularly attractive for the network case in fact network basic solutions are efficiently manipulated when appropriated data structures are used in the computational implementation of an algorithm

on certain quantum deformations of gl n r in this article all deformations of the general linear group subject to certain restrictions which in particular ensure a smooth passage to the lie group limit are obtained representations are given in terms of certains sets of creation and annihilation operators these creation and annihilation operators may belong to a generalization of the q quark type or q hadronic type with q boson or q fermion behavior this leads to a natural definition of q direct sums of q algebras

quantum brst charge for quadratically nonlinear lie algebras we consider the construction of a nilpotent brst charge for extensions of the virasoro algebra of the form ta tb fabctc vabcdtctd classical algebras in terms of poisson brackets and ta tb habi fabctc vabcd tctd quantum algebras in terms of commutator brackets normal ordering of the product tctd is understood in both cases we assume that the set of generators ta splits into a set hi generating an ordinary lie algebra and remaining generators s such that only thevij are nonvanishing in the classical case a nilpotent brst charge can always be constructed for the quantum case we derive a condition which is necessary and sufficient for the existence of a nilpotent brst charge non trivial examples are the spin 3 algebra with central chargec 100 and theso n extended superconformal algebras with levels 2 n 3

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

adaptive method of lines solutions for the extended fifth order korteweg de vries equation in this paper we investigate the dynamics and interaction properties of recently discovered embedded solitons in an extended fifth order kdv model inspired by water waves in the presence of surface tension the dynamical behaviour of the solitons can be efficiently followed by using a moving or an adaptive finite difference mesh in combination with a suitable time integrator we will demonstrate this numerically for different types of wave solutions such as solitary waves multihumped waves and interacting waves

generalized micz kepler system duality polynomial and deformed oscillator algebras we present the quadratic algebra of the generalized micz kepler system in three dimensional euclidean space e3 and its dual the four dimensional singular oscillator in four dimensional euclidean space e4 we present their realization in terms of a deformed oscillator algebra using the daskaloyannis construction the structure constants are in these cases functions not only of the hamiltonian but also of other integrals commuting with all generators of the quadratic algebra we also present a new algebraic derivation of the energy spectrum of the micz kepler system on the three sphere s3 using a quadratic algebra these results point out also that results and explicit formula for structure functions obtained for quadratic cubic and higher order polynomial algebras in the context of two dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles

quantization and representation theory of finite w algebras in this paper we study the finitely generated algebras underlyingw algebras these so called finitew algebras are constructed as poisson reductions of kirillov poisson structures on simple lie algebras the inequivalent reductions are labeled by the inequivalent embeddings ofsl 2 into the simple lie algebra in question for arbitrary embeddings a coordinate free formula for the reduced poisson structure is derived we also prove that any finitew algebra can be embedded into the kirillov poisson algebra of a semi simple lie algebra generalized miura map furthermore it is shown that generalized finite toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finitew symmetry in the second part we brst quantize the finitew algebras the brst cohomology is calculated using a spectral sequence which is different from the one used by feigin and frenkel this allows us to quantize all finitew algebras in one stroke examples are given in the last part of the paper we study the representation theory of finitew algebras it is shown using a quantum version of the generalized miura transformation that the representations of finitew algebras can be constructed from the representations of a certain lie subalgebra of the original simple lie algebra as a byproduct of this we are able to construct the fock realizations of arbitrary finitew algebras

some observations on a static spatial remeshing method based on equidistribution principles this paper proposes a method of lines solution procedure with time and space adaptation for one dimensional systems of partial differential equations whose solutions display steep moving fronts the spatial remeshing algorithm which is a variation of the method published by sanz serna and christie and an extension suggested by revilla is a static remeshing method based on equidistribution principles the selection of the several algorithm components i e grid placement criterion spatial discretization scheme time integrator adaptation frequency and parameter tuning are investigated and illustrated with several test examples i e the cubic schrodinger equation a model of a single step reaction with diffusion and a model of flame propagation

the relation between polynomial deformations of sl 2 r and quasi exact solvability a general method based on the polynomial deformations of the lie algebra sl 2 r is proposed in order to exhibit the quasi exact solvability of specific hamiltonians implied by quantum physical models this method using the finite dimensional representations and differential realizations of such deformations is illustrated on the sextic oscillator as well as on second harmonic generation

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

fast high dimensional node generation with variable density abstract we present an algorithm for producing discrete distributions with a prescribed nearest neighbor distance function our approach is a combination of quasi monte carlo q mc methods and weighted riesz energy minimization the initial distribution is a stratified q mc sequence with some modifications a suitable energy functional on the configuration space is then minimized to ensure local regularity the resulting node sets are good candidates for building meshless solvers and interpolants as well as for other purposes where a point cloud with a controlled separation covering ratio is required applications of a three dimensional implementation of the algorithm in particular to atmospheric modeling are also given