a dynamically consistent method to solve nonlinear multidimensional advection reaction equations with fractional diffusion abstract this work is motivated by an extension of both the burgers fisher and the burgers huxley equations in multiple dimensions considering riesz fractional diffusion initial boundary conditions which are positive and bounded are imposed on a closed and bounded set and a finite difference method is proposed to approximate the solutions of the fractional model the methodology is a linear and implicit technique which is based on fractional centered differences we show in this manuscript that the method can be expressed in vector form using a minkowski matrix under suitable conditions the main properties of minkowski matrices are used then to establish the existence and the uniqueness of the solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness additionally we show that the method is a consistent technique which is stable and convergent with first order of convergence in time and second order in space some illustrative simulations show that the scheme is capable of preserving the positivity and the boundedness of the numerical approximations

conservative nonlinear difference scheme for the cahn hilliard equation ii abstract numerical solutions for the cahn hilliard equation is considered using the crank nicolson type finite difference method existence of the solution for the difference scheme has been shown by brouwer fixed point theorem stability convergence and error analysis of the scheme are shown we also show that the scheme preserves the discrete mass even though the linearized scheme in 1 is conditionally stable and does not preserve the mass

bracket formulations and energy and helicity preserving numerical methods for the three dimensional vorticity equation abstract the vorticity equation for three dimensional viscous incompressible fluid flows is formulated within different bracket formalisms using the poisson or nambu bracket together with a dissipative bracket the budgets of kinetic energy helicity and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method in particular energy and helicity are conserved precisely in inviscid flow computations the energy and enstrophy dissipate properly owing to viscosity in viscous flow computations and the enstrophy is appropriately produced by the vortex stretching effect in both inviscid and viscous flow computations the relationships between the stream function velocity and vorticity as well as the solenoidal conditions on the velocity and vorticity fields are also inherited numerical experiments on a periodic array of rolls that permits analytical solutions have been done to examine the properties and usefulness of the proposed method

numerically induced phase shift in the kdv soliton abstract when using a finite difference scheme to study the motion of a kdv solition a shift in the position of the solition from the exact solution is detected in this paper we retain the lowest order terms in the truncation error and treat them analytically as a perturbation of the kdv equation it is found that perturbation theory can be used to determine the numerically induced shift

a numerical energy conserving method for the dnls equation an implicit numerical energy conserving method is developed for the derivative nonlinear schrodinger dnls equation for periodic boundary conditions we find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the dnls equation if the modulation is small and k0 a22 2 t k0 is the wavenumber and a the amplitude the numerical scheme is used to follow the nonlinear behavior of the dnls modulational instability the numerical code is also tested by the evolution for one soliton initial data these tests show that if the modulation is not small compared to the background wave amplitude new nonlinear numerical instabilities are introduced

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

the fourier pseudospectral method with a restrain operator for the korteweg de vries equation abstract in this paper we develop a fourier pseudospectral method with a restrain operator for the korteweg de vries equation we prove the generalized stability of the schemes and give convergence estimations depending on the smoothness of the solution of the p d e

efficient local energy dissipation preserving algorithms for the cahn hilliard equation abstract in this paper we show that the cahn hilliard equation possesses a local energy dissipation law which is independent of boundary conditions and produces much more information of the original problem to inherit the intrinsic property we derive three novel local structure preserving algorithms for the 2d cahn hilliard equation by the concatenating method in particular when the nonlinear bulk potential f in the equation is chosen as the ginzburg landau double well potential the method discussed by zhang and qiao 2012 50 is a special case of our scheme ii thanks to the leibnitz rules and properties of operators the three schemes are rigorously proven to conserve the discrete local energy dissipation law in any local time space region under periodic boundary conditions the schemes are proven to possess the discrete mass conservation and total energy dissipation laws numerical experiments are conducted to show the performance of the proposed schemes

kolmogorov s contributions to information theory and algorithmic complexity briefly information theory says a random object x p x has complexity entropy h xp x log p x with the attendant interpretation that h bits are sufficient to describe x on the average algorithmic complexity says an object x has a complexity k x equal to the length of the shortest binary program that describes x it is a beautiful fact that these ideas are much the same in fact it is roughly true that ek x h moreover if we let pu x pr u prints x be the probability that a given computer u prints x when given a random program it can be shown that log 1 pu x k x for all x thus establishing a vital link between the universal probability measure pu and the universal complexity k more on this later the relationship of these ideas to probability theory was summed up in kolmogorov s 1983 paper which was based on his 1970 talk in nice perhaps only

distortion program size complexity with respect to a fidelity criterion and rate distortion function a novel concept of distortion kolmogorov chaitin complexity is proposed bearing analogy to ordinary kolmogorov chaitin complexity that represents the information content of a single message distortion kolmogorov chaitin complexity can be regarded as the amount of information about a single message that must be conveyed in order to reproduce it with a bounded distortion this explanation is justified by the properties of distortion kolmogorov chaitin complexity and the equivalences between distortion kolmogorov chaitin complexity and the rate distortion function which are analogous to those between ordinary kolmogorov chaitin complexity and the shannon entropy some implications for universal almost sure data compression are also discussed

a second order accurate linearized difference scheme for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is of fourth order in space in this paper a linearized finite difference scheme is derived by the method of reduction of order it is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete l2 norm the coefficient matrix of the difference system is symmetric and positive definite so many well known iterative methods e g gauss seidel sor can be used to solve the system

notes on finite difference schemes to a parabolic elliptic system modelling chemotaxis we consider numerical solutions to a nonlinear parabolic elliptic system in mathematical biology an important property of solutions to the system is the conservation of l 1 norm applying the upwind technique we make a finite difference scheme satisfying the conservation of a discrete l 1 norm we also describe a remark on time discretization from the view point of lyapunov s property which is another important feature of the system several numerical examples clarify the nature of our scheme

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

lulu operators and discrete pulse transform for multidimensional arrays the lulu operators for sequences that is ln un and their compositions are extended to multidimensional arrays in a way which preserves their essential properties e g consistent separation total variation and shape preservation the power of the operators is demonstrated by deriving the discrete pulse transform dpt which is a hierarchical decomposition of the input array into pulses similar to its 1 d counterpart this transform satisfies a basic consistency property

direct discretization of planar div curl problems a control volume method is proposed for planar div curl systems the method is independent of potential and least squares formulations and works directly with the div curl system the novelty of the technique lies in its use of a single local vector field component and two control volumes rather than the other way round a discrete vector field theory comes quite naturally from this idea and is developed in the paper error estimates are proved for the method and other ramifications investigated

exact and asymptotic distributions of lulu smoothers this paper considers a class of non linear smoothers called lulu smoothers introduced by rohwer in the late eighties in the mathematics literature and since then investigated fairly extensively by a number of authors for its mathematical properties they have been successfully applied in various engineering and scientific problems however to date their distribution theory has not received any attention in the literature in this paper we derive their exact as well as asymptotic distributions and show their relationship to the upper order statistics

multisymplectic geometry and multisymplectic preissmann scheme for the kdv equation the multisymplectic structure of the kdv equation is presented directly from the variational principle from the numerical view point we give a multisymplectic twelve points scheme which is equivalent to the multisymplectic preissmann scheme finally we test the twelve points scheme on solitary waves over long time intervals

locally monotone robust approximation of sequences abstract the study of nonlinear smoothers for the removal of impulsive noise in a sequence leads to the problem of the unique and stable allocation of a particular root to a given noise corrupted sequence an established procedure is iteration of a median smoother till convergence this process is neither well defined nor well posed recursive smoothing used to circumvent these intuitively perceived problems is not satisfactory either simple equivalent smoothing operators exist that the allocate a class of sequences that are roots of the median smoother as well as a whole collection of equivalent smoothers this class is an interval in the usual lattice on sequences the allocation is well defined and well posed and blends into the framework of interval arithmetic

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

rate distortion and denoising of individual data using kolmogorov complexity we examine the structure of families of distortion balls from the perspective of kolmogorov complexity special attention is paid to the canonical rate distortion function of a source word which returns the minimal kolmogorov complexity of all distortion balls containing that word subject to a bound on their cardinality this canonical rate distortion function is related to the more standard algorithmic rate distortion function for the given distortion measure examples are given of list distortion hamming distortion and euclidean distortion the algorithmic rate distortion function can behave differently from shannon s rate distortion function to this end we show that the canonical rate distortion function can and does assume a wide class of shapes unlike shannon s we relate low algorithmic mutual information to low kolmogorov complexity and consequently suggest that certain aspects of the mutual information formulation of shannon s rate distortion function behave differently than would an analogous formulation using algorithmic mutual information we explore the notion that low kolmogorov complexity distortion balls containing a given word capture the interesting properties of that word which is hard to formalize in shannon s theory and this suggests an approach to denoising

an extension of the discrete variational method to nonuniform grids the discrete variational method is a method used to derive finite difference schemes that inherit the conservation dissipation property of the original equations although this method has mainly been developed for uniform grids we extend this method to multidimensional nonuniform meshes

the discrete variational derivative method based on discrete differential forms as is well known for pdes that enjoy a conservation or dissipation property numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice lately furihata and matsuo have developed the so called discrete variational derivative method that automatically constructs energy preserving or dissipative finite difference schemes although this method was originally developed on uniform meshes the use of non uniform meshes is of importance for multi dimensional problems on the other hand the theories of discrete differential forms have received much attention recently these theories provide a discrete analogue of the vector calculus on general meshes in this paper we show that the discrete variational derivative method and the discrete differential forms by bochev and hyman can be combined applications to the cahn hilliard equation and the klein gordon equation on triangular meshes are provided as demonstrations we also show that the schemes for these equations are h1 stable under some assumptions in particular one for the nonlinear klein gordon equation is obtained by combination of the energy conservation property and the discrete poincare inequality which are the temporal and spacial structures that are preserved by the above methods

continuous finite element methods which preserve energy properties for nonlinear problems abstract continuous finite elements are used to develop numerical methods which preserve energy properties for nonlinear ordinary and partial differential equations schemes are constructed that give high orders of accuracy time discretization algorithms for the cahn hilliard klein gordon korteweg de vries and schrodinger equations are presented comparisons are made with existing schemes

integrable discretization of time and its application on the fourier pseudospectral method to the korteweg de vries equation this paper presents a new integrable discretization of the korteweg de vries kdv equation different from other discrete analogues we discretize the variable time and obtain an integrable differential difference system this system has the original kdv equation as a standard limit when the step size tends to zero the main idea is based on hirota s bilinear method and backlund transformation and can be applied to other integrable systems by applying the fourier pseudospectral method to the space variable we derive a new numerical scheme for the kdv equation numerical results are found to agree with the exact solution and the first five conservation quantities are preserved quite well

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

finite state adaptive block to variable length noiseless coding of a nonstationary information source the author associates with a given nonstationary finite alphabet information source a certain class of stationary processes that he terms the stationary hull of the given source he shows that the optimum average rate at which the given source can be noiselessly coded by means of a finite state adaptive block to variable length coding schemes is the largest entropy rate among those processes in the stationary hull he explains what he means by an adaptive block to variable length coding scheme

fixed slope universal lossy data compression corresponding to any lossless codeword length function l three universal lossy data compression schemes are presented one is with a fixed rate another is with a fixed distortion and a third is with a fixed slope the former two universal lossy data compression schemes are the generalization of yang kieffer s see ibid vol 42 no 1 p 239 45 1995 results to the general case of any lossless codeword length function l whereas the third is new in the case of fixed slope spl lambda 0 our universal lossy data compression scheme works as follows for any source sequence x sup n of length n the encoder first searches for a reproduction sequence y sup n of length n which minimizes a cost function n sup 1 l y sup n spl lambda spl rho sub n x sup n y sup n over all reproduction sequences of length n and then encodes x sup n into the binary codeword of length l y sup n associated with y sup n via the lossless codeword length function l where spl rho sub n x sup n y sup n is the distortion per sample between x sup n and y sup n under some mild assumptions on the lossless codeword length function l it is shown that when this fixed slope data compression scheme is applied to encode a stationary ergodic source the resulting encoding rate per sample and the distortion per sample converge with probability one to r sub spl lambda and d sub spl lambda respectively where d sub spl lambda r sub spl lambda is the point on the rate distortion curve at which the slope of the rate distortion function is spl lambda this result holds particularly for the arithmetic codeword length function and lempel ziv codeword length function the main advantage of this fixed slope universal lossy data compression scheme over the fixed rate fixed distortion universal lossy data compression scheme lies in the fact that it converts the encoding problem to a search problem through a trellis and then permits one to use some sequential search algorithms to implement it simulation results show that this fixed slope universal lossy data compression scheme combined with a suitable search algorithm is promising

discretization of nonlinear evolution equations over associative function algebras abstract a general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces defined over a regular lattice of points in such a way that both their symmetry and integrability properties are preserved a class of discrete kdv equations is introduced also new hierarchies of discrete evolution equations of gelfand dickey type are defined

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

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

simple universal lossy data compression schemes derived from the lempel ziv algorithm two universal lossy data compression schemes one with fixed rate and the other with fixed distortion are presented based on the well known lempel ziv algorithm in the case of fixed rate r the universal lossy data compression scheme works as follows first pick a codebook b sub n consisting of all reproduction sequences of length n whose lempel ziv codeword length is spl les nr and then use b sub n to encode the entire source sequence n block by n block this fixed rate data compression scheme is universal in the sense that for any stationary ergodic source or for any individual sequence the sample distortion performance as n spl rarr spl infin is given almost surely by the distortion rate function a similar result is shown in the context of fixed distortion lossy source coding

sample converses in source coding theory the rate and distortion performance of a sequence of codes along a sample sequence of symbols generated by a stationary ergodic information source are studied two results are obtained 1 the source sample sequence is encoded by an arbitrary sequence of block codes which operate at a fixed rate level r and a sample converse is obtained which states that with probability one the lower limit of the code sample distortions is lower bounded by d r the value of the distortion rate function at r 2 the source sample sequence is encoded by an arbitrary sequence of variable rate codes which operate at a fixed distortion level d and a sample converse is obtained which states that with probability one the lower limit of the code sample rates is lower bounded by r d the value of the rate distortion function at d a novel ergodic theorem is used to obtain both sample converses

numerical method satisfying the first two conservation laws for the korteweg de vries equation in this paper we develop a finite volume scheme for the kdv equation which conserves both the momentum and energy the main ingredient of the method is a numerical device we developed in recent years that enables us to construct numerical method for a pde that also simulates its related equations in the method numerical approximations to both the momentum and energy are conservatively computed the operator splitting approach is adopted in constructing the method in which the conservation and dispersion parts of the equation are alternatively solved our numerical device is applied in solving the conservation part of the equation the feasibility and stability of the method is discussed which involves an important property of the method the so called jensen condition the truncation error of the method is analyzed which shows that the method is second order accurate finally several numerical examples including the zabusky kruskal s example are presented to show the good stability property of the method for long time numerical integration

an on line universal lossy data compression algorithm via continuous codebook refinement i basic results a new on line universal lossy data compression algorithm is presented for finite memoryless sources with unknown statistics its performance asymptotically approaches the fundamental rate distortion limit the codebook is generated on the fly and continuously adapted by simple rules there is no separate codebook training or codebook transmission candidate codewords are randomly generated according to an arbitrary and possibly suboptimal distribution through a carefully designed gold washing or information theoretic sieve mechanism good codewords and only good codewords are promoted to permanent status with high probability we also determine the rate at which our algorithm approaches the fundamental limit

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

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

exponential finite difference method applied to korteweg de vries equation for small times the korteweg de vries equation is numerically solved by using the exponential finite difference technique the accuracy of computed solutions is examined by comparison with other numerical and analytical solutions using two examples the close results agreement between the current results and the exact solutions confirms that the proposed finite difference procedure is an effective technique for the solution of the korteweg de vries equation at the small times

local discontinuous galerkin methods for nonlinear dispersive equations we develop local discontinuous galerkin dg methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions the so called compactons the schemes we present extend the previous works of yan and shu on approximating solutions for linear dispersive equations and for certain kdv type equations we present two classes of dg methods for approximating solutions of such pdes first we generate nonlinearly stable numerical schemes with a stability condition that is induced from a conservation law of the pde an alternative approach is based on constructing linearly stable schemes i e schemes that are linearly stable to small perturbations the numerical simulations we present verify the desired properties of the methods including their expected order of accuracy in particular we demonstrate the potential advantages of using dg methods over pseudospectral methods in situations where discontinuous fronts and rapid oscillations co exist in a solution

a direct discontinuous galerkin method for the generalized korteweg de vries equation energy conservation and boundary effect a direct discontinuous galerkin method for solving the generalized korteweg de vries kdv equation with both periodic data and non homogeneous boundary data is developed a class of numerical fluxes is identified so that the method preserves both momentum and energy for the initial value problem the method with such a property is numerically shown robust with less error in both phase and amplitude after long time simulation numerical examples are given to confirm the theoretical result and the capacity of this method for capturing soliton wave phenomena and various boundary wave patterns

design of linear combination of weighted medians this paper introduces a novel nonlinear filtering structure the linear combination of weighted medians lcwm the proposed filtering scheme is modeled on the structure and design procedure of the linear phase fir highpass hp filter in that the linear phase fir hp filter can be obtained by changing the sign of the filter coefficients of the fir lowpass lp filter in the odd positions the hp filter can be represented as the difference between two lp subfilters that have all positive coefficients this representation of the fir hp filter is analogous to the difference of estimates doe such as the difference of medians dom the dom is essentially a nonlinear hp filter that is commonly used in edge detection based on this observation we introduce a class of lcwm filters whose output is given by a linear combination of weighted medians of the input sequence we propose a method of designing the 1 d and 2 d lcwm filters satisfying required frequency specifications the proposed method adopts a transformation from the fir filter to the lcwm filter we show that the proposed lcwm filter can offer various frequency filtering characteristics including lp bandpass bp and hp responses

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

rate distortion optimization for h 264 interframe coding a general framework and algorithms rate distortion rd optimization for h 264 interframe coding with complete baseline decoding compatibility is investigated on a frame basis using soft decision quantization sdq rather than the standard hard decision quantization we first establish a general framework in which motion estimation quantization and entropy coding in h 264 for the current frame can be jointly designed to minimize a true rd cost given previously coded reference frames we then propose three rd optimization algorithms a graph based algorithm for near optimal sdq in h 264 baseline encoding given motion estimation and quantization step sizes an algorithm for near optimal residual coding in h 264 baseline encoding given motion estimation and an iterative overall algorithm to optimize h 264 baseline encoding for each individual frame given previously coded reference frames with them embedded in the indicated order the graph based algorithm for near optimal sdq is the core given motion estimation and quantization step sizes it is guaranteed to perform optimal sdq if the weak adjacent block dependency utilized in the context adaptive variable length coding of h 264 is ignored for optimization the proposed algorithms have been implemented based on the reference encoder jm82 of h 264 with complete compatibility to the baseline profile experiments show that for a set of typical video testing sequences the graph based algorithm for near optimal sdq the algorithm for near optimal residual coding and the overall algorithm achieve on average 6 8 and 12 respectively rate reduction at the same psnr ranging from 30 to 38 db when compared with the rd optimization method implemented in the h 264 reference software

numerical solution of korteweg de vries equation by galerkin b spline finite element method a finite element solution of the kdv equation is presented to demonstrate the efficiency of the method two test problems are considered the numerical solutions of the kdv equation are compared with both the exact solutions and other numerical solutions in the literature the numerical solutions are found to be in good agreement with the exact solutions

accuracy and conservation properties in numerical integration the case of the korteweg de vries equation when numerically integrating time dependent differential equations it is often recommended to employ methods that preserve some of the invariant quantities mass energy etc of the problem being considered this recommendation is usually justified on the grounds that conservation of invariant quantities may ensure that the numerical solution possesses some important qualitative features however there are cases where schemes that preserve invariants are also advantageous in that they possess favourable error propagation mechanisms that render them superior from a quantitative point of view in the present paper we consider the korteweg de vries equation as a case study we show rigorously that for soliton problems and at leading order the error of conservative schemes consists of a phase error that grows linearly with time plus a complementary term that is bounded in the h 1 norm uniformly in time for general nonconservative schemes the error involves a linearly growing amplitude error a quadratically growing phase error and a complementary term that grows linearly in the h 1 norm numerical experiments are presented

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

a semi discrete modified kdv equation in this paper we present an integrable semi discretization of the modified korteweg devries mkdv equation we discretize the time variable of the mkdv equation and get an integrable differential difference system under a standard limit the differential difference system converges to the continuous mkdv equation by hirota s bilinear method we find some explicit solutions including solitons and breather solutions from the semi discrete system we design a numerical scheme to the mkdv equation and carry out several numerical experiments with the 3 soliton solution and breather solution