on the development of a high order compact scheme for exhibiting the switching and dissipative solution natures in the camassa holm equation in this paper a two step iterative solution algorithm for solving the camassa holm equation which involves only the first order derivative term is presented in each set of the u p and u m differential equations one is governed by the inviscid nonlinear convection reaction equation for the time evolving fluid velocity component along the horizontal direction the other equation is known as the inhomogeneous helmholtz equation the resulting reduction of differential order facilitates us to develop the flux discretization scheme in a stencil with comparatively fewer points for accurately predicting the unidirectional propagation of the shallow water wave the modified equation analysis for eliminating several leading discretization error terms and the fourier analysis for minimizing a particular type of wave like error are employed in this study the fifth order spatially accurate combined compact upwind scheme is developed in a three point stencil for approximating the first order derivative term for the purpose of retaining a long term accurate hamiltonian and multi symplectic geometric structures in camassa holm equation the time integrator or time stepping scheme chosen in this study should conserve symplecticity another main emphasis of conducting the present calculation of camassa holm equation is to shed light on the conservation of hamiltonians up to the time before wave breaking we also intended to elucidate the switching scenario by virtue of the peakon peakon interaction problem and the dissipative scenario after the time of head on collision in the peakon antipeakon interaction problem

non linear rough heat equations this article is devoted to define and solve an evolution equation of the form dyt yt dt dx t yt wherestands for the laplace operator on a space of the form l p r n and x is a finite dimensional noisy nonlinearity whose typical form is given by xt n i 1 x i fi where each x x 1 x n is a holder function generating a rough path and each fi is a smooth enough function defined on l p r n the generalization of the usual rough path theory allowing to cope with such

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

joint space frequency segmentation using balanced wavelet packet trees for least cost image representation we examine the question of how to choose a space varying filterbank tree representation that minimizes some additive cost function for an image the idea is that for a particular cost function e g energy compaction or quantization distortion some tree structures perform better than others while the wavelet tree represents a good choice for many signals it is generally outperformed by the best tree from the library of wavelet packet frequency selective trees the double tree library of bases performs better still by allowing different wavelet packet trees over all binary spatial segments of the image we build on this foundation and present efficient new pruning algorithms for both one and two dimensional 1 d and 2 d trees that will find the best basis from a library that is many times larger than the library of the single tree or double tree algorithms the augmentation of the library of bases overcomes the constrained nature of the spatial variation in the double tree bases and is a significant enhancement in practice use of these algorithms to select the least cost expansion for images with a rate distortion cost function gives a very effective signal adaptive compression scheme this scheme is universal in the sense that without assuming a model for the signal or making use of training data it performs very well over a large class of signal types in experiments it achieves compression rates that are competitive with the best training based schemes

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

a compact finite difference scheme on a non equidistant mesh abstract a fourth order compact finite difference scheme is presented for second order partial differential equations the scheme is derived for a one dimensional non equidistant mesh which makes it particularly useful in problems with sharp boundary layers the inclusion of general boundary conditions does not reduce the order of the scheme for the boundary points the scheme is tested on a representative model equation we shortly discuss its stability in the simplest time dependent heat flow problem and its use in more dimensions

an energy method for rough partial differential equations abstract we present a well posedness and stability result for a class of nondegenerate linear parabolic equations driven by geometric rough paths more precisely we introduce a notion of weak solution that satisfies an intrinsic formulation of the equation in a suitable sobolev space of negative order weak solutions are then shown to satisfy the corresponding energy estimates which are deduced directly from the equation existence is obtained by showing compactness of a suitable sequence of approximate solutions whereas uniqueness relies on a doubling of variables argument and a careful analysis of the passage to the diagonal our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical pdes theory

on the development of a dispersion relation preserving dual compact upwind scheme for convection diffusion equation in this paper a dual compact scheme which accommodates a better dispersion relation for the convective terms shown in the transport equation is proposed to enhance the convective stability of the convection diffusion equation by virtue of the increased dispersive accuracy the dispersion relation preserving compact scheme has been rigorously developed within the three stencil point framework through the dispersion and dissipation analyses to verify the proposed method several problems that are amenable to the exact and benchmark solutions will be investigated the results with good rates of convergence are demonstrated for all the investigated problems

an entropy coded lattice vector quantizer for transform and subband image coding a lattice based vector quantizer vq and noiseless code are proposed for transform and subband image coding the quantization is simple to implement and no vector codebooks need to be stored the noiseless code enumerates lattice codevectors based on their weighted l sub 1 norm a software implementation is able to handle lattice codebooks of size 2 sup 256 the image coding performance is shown to be comparable or superior to the best encoding methods reported in the literature

long time asymptotic solution structure of camassa holm equation subject to an initial condition with non zero reflection coefficient of the scattering data in this article we numerically revisit the long time solution behavior of the camassa holm equation ut uxxt 2ux 3uux 2uxuxx uuxxx the finite difference solution of this integrable equation is sought subject to the newly derived initial condition with delta function potential our underlying strategy of deriving a numerical phase accurate finite difference scheme in time domain is to reduce the numerical dispersion error through minimization of the derived discrepancy between the numerical and exact modified wavenumbers additionally to achieve the goal of conserving hamiltonians in the completely integrable equation of current interest a symplecticity preserving time stepping scheme is developed based on the solutions computed from the temporally symplecticity preserving and the spatially wavenumber preserving schemes the long time asymptotic ch solution characters can be accurately depicted in distinct regions of the space time domain featuring with their own quantitatively very different solution behaviors we also aim to numerically confirm that in the two transition zones their long time asymptotics can indeed be described in terms of the theoretically derived painleve transcendents another attempt of this study is to numerically exhibit a close connection between the presently predicted finite difference solution and the solution of the painleve ordinary differential equation of type ii in two different transition zones

truncation error and oscillation property of the combined compact difference scheme in this work we study a sixth order combined compact difference scheme for a one dimensional convection diffusion equation we derive the truncation error representation and analyze its oscillation property numerical experiments are performed to illustrate and support our analysis

pyramidal lattice vector quantization for multiscale image coding introduces a new image coding scheme using lattice vector quantization the proposed method involves two steps biorthogonal wavelet transform of the image and lattice vector quantization of wavelet coefficients in order to obtain a compromise between minimum distortion and bit rate we must truncate and scale the lattice suitably to meet this goal we need to know how many lattice points lie within the truncated area we investigate the case of laplacian sources where surfaces of equal probability are spheres for the l sup 1 metric pyramids for arbitrary lattices we give explicit generating functions for the codebook sizes for the most useful lattices like z sup n d sub n e sub s spl and sub 16

derivative optimized empirical mode decomposition for the hilbert huang transform in the empirical mode decomposition emd for the hilbert huang transform hht a nonlinear and non stationary signal is adaptively decomposed by an hht into a series of intrinsic mode functions imfs with the lowest one as the trend at each step of the emd the low frequency component is mainly determined by the average of the upper envelope consisting of local maxima and the lower envelope consisting of local minima the high frequency component is the deviation of the signal relative to the low frequency component the fact that no local maximum and minimum can be determined at the two end points leads to detrend uncertainty and in turn causes uncertainty in the hht to reduce such uncertainty hermitian polynomials are used to obtain the upper and lower envelopes with the first derivatives at the two end points q l q r as parameters which are optimally determined on the base of minimum temporal variability of the low frequency component in the each step of the decomposition this well posed mathematical system is called the derivative optimized emd demd with the demd the end effect and detrend uncertainty are drastically reduced and scales are separated naturally without any a priori subjective selection criterion

essentially compact schemes for unsteady viscous incompressible flows a new fourth order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced the scheme is based on the vorticity stream function formulation it is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions it is also very efficient at every time step or runge kutta stage only two poisson like equations have to be solved the poisson like equations are amenable to standard fast poisson solvers usually designed for second order schemes detailed comparison with the second order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow this efficient fourth order scheme also made it possible to compute the driven cavity flow at reynolds number 106on a 10242grid at a reasonable cost fourth order convergence is proved under mild regularity requirements this is the first such result to our knowledge

convergence of the compact finite difference method for second order elliptic equations abstract in this paper we give a fourth order compact finite difference scheme for the general forms of two point boundary value problems and two dimensional elliptic partial differential equations pde s by decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problems we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix thus we prove the compact finite difference scheme converges with the fourth order of accuracy to solve the discretized block tri diagonal matrix equations of the two dimensional elliptic pde s we develop an efficient iterative method full multigrid method in our numerical experiments we compare the compact finite difference method with the finite element method and crank nicolson finite difference method the results show that the compact finite difference scheme is a highly efficient and accurate method

combined compact difference scheme for the time fractional convection diffusion equation with variable coefficients combined compact finite difference scheme for solving the time fractional convection diffusion reaction equation is given the primary unknown and its derivative are solved simultaneously the coefficients of the fractional partial differential equation are variable both dirichlet and robin boundary conditions are discussed and the algorithm gives high order convergence results fourth order combined compact finite difference scheme is given for solving the time fractional convection diffusion reaction equation with variable coefficients we introduce the flux as a new variable and transform the original equation into a system of two equations compact difference is used as a high order approximation for spatial derivatives of integer order in the coupled partial differential equations the caputo fractional derivative is discretized by a high order approximation both dirichlet and robin boundary conditions are discussed convergence analysis is given for the problem of integer order with constant coefficients under some assumption numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

adaptive and nonadaptive hermitian operator methods for combustion phenomena adaptive and nonadaptive three point fourth order accurate compact or hermitian operator methods are developed and used to study one dimensional combustion phenomena the nonadaptive hermitian operator methods are based on the time linearization of the nonlinear partial differential equations and employ an approximate factorization technique to reduce a three dimensional reactiondiffusion operator to a sequence of three one dimensional linear second order differential operators in space the three adaptive hermitian operator techniques presented in this paper are based on the equidistribution of the arc length of the vector of dependent variables and use a subequidistribution principle to obtain smooth grids the first adaptive technique uses quasilinearization and yields a block tridiagonal matrix for the values of the dependent variables at each iteration the second technique employs partial quasilinearization and yields a system of uncoupled linear algebraic equations for each dependent variable at each iteration the third technique employs a predictor corrector method to predict the grid point locations and a time linearization procedure to obtain the values of the dependent variables it is shown that the efficiency and accuracy of adaptive hermitian operator methods depend on the time step and number of grid points used in the calculations it is also shown that adaptive methods which use a crank nicolson scheme in time may yield oscillatory solutions and that nonadaptive hermitian operator methods require a much larger number of grid points than nonadaptive techniques if the solution of the governing equations is characterized by fast ignition phenomena and or steep fast moving flame fronts

spectral compact difference hybrid computation of passive scalar in isotropic turbulence an accurate and efficient hybrid numerical method is developed for direct numerical simulation of passive scalar in homogeneous turbulence for the schmidt number 1 and 50 the hybrid method uses the standard fourier spectral method for the incompressible navier stokes equation and the combined compact difference scheme for the passive scalar transport equation accuracy of the method is carefully examined by comparing with the full spectral method regarding the spectra probability density function field structure of the passive scalar and is found to be very satisfactory the computational time for the hybrid method is decreased by 26 for the schmidt number 1 when compared to the full spectral method and by 77 for the schmidt number 50 when the number of grid points for the velocity field is reduced under the scale separation

compact finite difference schemes for ocean models 1 ocean waves abstract a general framework is derived which leads to generic expressions of discrete dispersion relationships for inertia gravity and rossby waves valid for every finite difference scheme and every type of grid these relationships are used to investigate the performance of fourth order and sixth order compact implicit schemes on arakawa grids a e it is shown that the use of compact schemes leads to very clear improvement in approximating frequency and group velocity of inertia gravity waves on the other hand increasing the order of the schemes does not necessarily improve the accuracy of the discrete dispersion relationship in the case of rossby waves globally the fourth order family is found to be a good compromise which improves significantly the quality of the approximation of the dispersion properties with regard to conventional second order centered schemes

three point combined compact difference schemes for time fractional advection diffusion equations with smooth solutions a ccd method with the l1 formula is proposed for solving time fractional advection diffusion equations the method is globally at least fifth order accurate in space and 2 th order accurate in time the unconditional stability is proved by means of fourier analysis numerical experiments are conducted to demonstrate the efficiency the present method preserves the higher order accuracy for advection dominated problems with the smooth solutions in this paper we study the numerical solution for the time fractional advection diffusion equation tfade a high order accurate three point combined compact difference scheme with the l1 formula is proposed to solve a class of tfades the method is globally 2 th order accurate in time and at least fifth order accurate in space for the constant coefficient tfades subject to periodic boundary conditions where is the order of time fractional derivative in the governing equation the unconditional stability and the high order convergence are proved using the fourier analysis method the proposed method can be extended to treat the variable coefficient tfades subject to other local boundary conditions several numerical examples are computed to validate the numerical accuracy effectiveness and robustness of the present method for tfades with smooth solutions and different boundary conditions moreover the proposed algorithm preserves the higher order accuracy for advection dominated problems with the smooth solutions

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

a conservative phase field method for solving incompressible two phase flows in this paper a conservative phase field method based on the work of sun and beckermann y sun c beckermann sharp interface tracking using the phase field equation j comput phys 220 2007 626 653 for solving the two and three dimensional two phase incompressible navier stokes equations is proposed the present method can preserve the total mass as the cahn hilliard equation but the calculation and implementation are much simpler than that the dispersion relation preserving schemes are utilized for the advection terms while the helmholtz smoother is applied to compute the surface tension force term to verify the proposed method several benchmarks are examined and shown to have good agreements with previous results it also shows that the satisfactions of mass conservations are guaranteed

high order particle method for solving incompressible navier stokes equations within a mixed lagrangian eulerian framework abstract owing to the fact that the poisson equation of pressure for incompressible fluid flow is purely elliptic it is therefore computationally improper to compute pressure on irregularly distributed moving particles as addressed in the recent moving particle with embedded pressure mesh mppm method in the current work a modified mppm method known as the mixed lagrangian eulerian mle method is proposed for solving the incompressible navier stokes equations in the current velocity pressure formulation the momentum and continuity equations are approximated on the moving particles lagrangian and the uniform cartesian grid points respectively meanwhile the total derivative of velocity terms appeared in the momentum equations are estimated by simply advecting the moving particles thereby eliminating the convection stability problem and increasing the flow accuracy without introducing false diffusion error in the conventional moving particle semi implicit mps and mppm methods numerical accuracies of the laplacian and gradient operators are strongly dependent on the regularity of the particle distribution in some implicit schemes the gradient and laplacian terms are of second order and first order accuracy respectively in the current work the second order accuracies of these differential terms exhibited on moving particles are realized by interpolating the derivative values from the uniform cartesian grids calculated by using the high order combined compact difference ccd scheme from the numerical results of laplacian term approximation by using various numerical schemes it is shown that the new mle scheme is at least second order accurate the proposed mixed lagrangian eulerian mle method can be easily applied to simulate fluid flow problems ranging from low to high reynolds number it is found that the numerical results compare well with the benchmark solutions moreover it is more accurate than the recently proposed mppm method

analysis of a new high resolution upwind compact scheme a new high resolution upwind compact scheme is presented and analyzed along with several previously proposed schemes for numerical dispersion dissipation anisotropy phase damping dispersion relation preservation property and numerical stability the schemes are tested on problems of the propagation of a initially discontinuous wave and of the transport of a sharp scalar cone a benchmark problem of aeroacoustics is solved with the present scheme giving satisfactory resolution the scheme compares well with the existing schemes for the range of properties considered desirable in high resolution schemes

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

efficient and accurate numerical simulation of acoustic wave propagation in a 2d heterogeneous media abstract in this paper a compact fourth order finite difference scheme is derived to solve the 2d acoustic wave equation in heterogenous media the pade approximation is used to obtain fourth order accuracy in both temporal and spatial dimensions and the alternating direction implicit adi technique is used to reduce the computational cost due to the non constant wave velocity the conventional adi method is hard to implement as the algebraic manipulation cannot be used here a novel numerical strategy is proposed in this work so that the compact scheme still maintains fourth order accuracy in time and space the fourth order convergence order was firstly proved by theoretical error analysis then was confirmed by numerical examples it was shown that the proposed method is conditionally stable with a courant friedrichs lewy cfl condition that is comparable to other existing finite difference schemes several numerical examples were solved to demonstrate the efficiency and accuracy of the new algorithm

rough evolution equations we generalize lyons rough paths theory in order to give a pathwise meaning to some nonlinear infinite dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise as an illustration we discuss a class of linear and nonlinear 1d spdes driven by a space time gaussian noise with singular space covariance and brownian time dependence

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

high order implicit finite difference schemes for the two dimensional poisson equation abstract in this paper a new family of high order finite difference schemes is proposed to solve the two dimensional poisson equation by implicit finite difference formulas of 2 m 1 operator points the implicit formulation is obtained from taylor series expansion and wave plane theory analysis and it is constructed from a few modifications to the standard finite difference schemes the approximations achieve 2 m 4 order accuracy for the inner grid points and up to eighth order accuracy for the boundary grid points using a successive over relaxation method the high order implicit schemes have faster convergence as m is increased compensating the additional computation of more operator points thus the proposed solver results in an attractive method easy to implement with higher order accuracy but nearly the same computation cost as those of explicit or compact formulation in addition particular case m 1 yields a new compact finite difference schemes of sixth order of accuracy numerical experiments are presented to verify the feasibility of the proposed method and the high accuracy of these difference schemes

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

further improvement and analysis of ccd scheme dissipation discretization and de aliasing properties in this paper we further analyze a combined compact difference ccd scheme proposed recently t k sengupta v lakshmanan v v s n vijay a new combined stable and dispersion relation preserving compact scheme for non periodic problems j comput phys 228 8 2009 3048 3071 for its dissipation discretization properties to show that its superiority also helps in controlling aliasing error for a benchmark internal flow however application of the same ccd method to study the receptivity of a boundary layer experiencing adverse pressure gradient is not successful this is traced to the nature of the equilibrium flow where the better dissipation property is not helpful in the inviscid part of the flow where the aliasing problems continue to persist a further modification is proposed to the ccd method here to solve complex physical problems requiring information on higher order disturbance quantities as in problems of flow receptivity and instability

a finite volume formulation of compact central schemes on arbitrary structured grids in the present paper a formulation allowing the use of compact schemes in the finite volume context on arbitrary meshes is presented the proposed formulation is based on the use of an implicit formula to evaluate the fluxes on the cell faces a series of numerical experiments for a 2d model convection equation a flat plate a subsonic vortical problem as well as the les simulation of channel flow has been carried out the results indicate an important improvement in obtained accuracy compared to a standard central finite volume formulation

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

wong zakai approximation for the stochastic landau lifshitz gilbert equations abstract in this work we study stochastic landau lifshitz gilbert equations sllges in one dimension with non zero exchange energy only firstly by introducing a suitable transformation we convert the sllges to a highly nonlinear time dependent partial differential equation with random coefficients which is not fully parabolic we then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property following regular approximation of the brownian motion and using reverse transformation we show existence of strong solution of sllges taking values in a two dimensional unit sphere s 2 in r 3 the construction of the solution and its corresponding convergence results are based on wong zakai approximation

the compact and crank nicolson adi schemes for two dimensional semilinear multidelay parabolic equations this paper deals with numerical solutions of initial boundary value problems of the two dimensional semilinear multidelay parabolic equations two types of alternating direction implicit adi schemes are suggested the unique solvability convergence and unconditional stability of the schemes are analyzed and hence the corresponding criteria are established especially by using the discrete energy method it is proven that the compact adi scheme can attain second order accuracy in time and fourth order accuracy in space and the crank nicolson adi scheme has second order accuracy in both time and space numerical experiments are performed to verify the efficiency and accuracy of the both schemes

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

an unconditionally stable linearized ccdadi method for generalized nonlinear schrdinger equations with variable coefficients in two and three dimensions in this paper we propose a three level linearly implicit combined compact difference method ccd together with alternating direction implicit method adi for solving the generalized nonlinear schrdinger equation nlse with variable coefficients in two and three dimensions the method is sixth order accurate in space variable and second order accurate in time variable fourier analysis shows that the method is unconditionally stable comparing to the nonlinear ccdpradi scheme for solving the 2d cubic nlse with constant coefficients li etal 2015 current method is a linear scheme which generally requires much less computational cost moreover current method can handle 3d problems with variable coefficients naturally finally numerical results for both 2d and 3d cases are presented to illustrate the advantages of the proposed method

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

partial differential equations driven by rough paths abstract we study a class of linear first and second order partial differential equations driven by weak geometric p rough paths and prove the existence of a unique solution for these equations this solution depends continuously on the driving rough path this allows a robust approach to stochastic partial differential equations in particular we may replace brownian motion by more general gaussian and markovian noise support theorems and large deviation statements all become easy corollaries of the corresponding statements of the driving process in the case of first order equations with gaussian noise we discuss the existence of a density with respect to the lebesgue measure for the solution

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

a new compact difference scheme for second derivative in non uniform grid expressed in self adjoint form a single parameter family of self adjoint compact difference sacd schemes is developed for discretizing the laplacian operator in self adjoint form developed implicit scheme is formally second order accurate with respect to truncation error with a free parameter a which helps control the numerical properties in the spectral plane the sacd scheme is analyzed in the spectral plane for its resolution properties for both periodic and non periodic problems using the matrix spectral analysis t k sengupta g ganeriwal s de analysis of central and upwind schemes j comput phys 192 2 2003 677 694 the major objective here is to identify the advantages of the new scheme over the traditional explicit second order cd2 scheme in discretizing the laplacian operator in self adjoint form this appears in navier stokes equation and in other transport equations and boundary value problems bvp expressed in orthogonal co ordinate systems either in physical or in transformed plane we also compare the developed method with the higher order compact schemes for non uniform grids to demonstrate the accuracy of sacd scheme we have tested it for i bi directional wave propagation problem given by the second order wave equation and ii an elliptic bvp as in the stommel ocean model for the stream function these examples help infer the properties of sacd scheme when solving different types of partial differential equations most importantly the effects of grid stretching and choice of value of the free parameter a are investigated here we also compare the present implicit compact method with explicit compact method known as the higher order compact hoc method also the practical applications of the sacd scheme are explored by solving the navier stokes equation for the cases of a a flow inside a lid driven cavity and b the receptivity and instability of an external adverse pressure gradient flow over a flat plate in the former unsteadiness of the flow is captured and in the latter the receptivity of the flow is studied in causing flow instability by triggering tollmien schlichting waves the new scheme shows a marked improvement over the explicit scheme for low reynolds number steady flow in lid driven cavity whereas for the flow in the same geometry at higher reynolds numbers efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer

adaptive wavelet packet based image coding with optimal entropy constrained lattice vector quantizer eclvq in this work wavelet basis and source coding are jointly optimized while specifying the source coding strategy as entropy constrained lattice vector quantizer eclvq the presented approach differs from previous works in which the choice of wavelet basis is quasioptimal but the quantizer set is optimally chosen

a fast solver of the shallow water equations on a sphere using a combined compact difference scheme the spectral like three point combined compact difference sp ccd scheme is developed for the shallow water models on a sphere which is originally proposed by chu and fan j comput phys 140 1998 370 a standard test set proposed by williamson et al j comput phys 102 1992 211 is used to evaluate the accuracy of the new method it is shown that the accuracy of the method is comparable to that of the spherical harmonics model as a whole the relative efficiency of the method with respect to the spherical harmonics model appears from the resolution of 512 256 and it becomes significant for the higher resolutions the test results suggest that the method could be extended to numerical atmospheric models

conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics the proposed scheme which is a conservative form of the interpolated differential operator scheme ido cf can provide high accurate solutions for both compressible and incompressible fluid equations spatial discretizations with fourth order accuracy are derived from interpolation functions locally constructed by both cell integrated values and point values these values are coupled and time integrated by solving fluid equations in the flux forms for the cell integrated values and in the derivative forms for the point values the ido cf scheme exactly conserves mass momentum and energy retaining the high resolution more than the non conservative form of the ido scheme a direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods benchmark tests of riemann problems and lid driven cavity flows show that the ido cf scheme is immensely promising in compressible and incompressible fluid dynamics studies

high order compact adi methods for parabolic equations in this paper we develop a sixth order compact scheme coupled with alternating direction implicit adi methods and apply it to parabolic equations in both 2 d and 3 d unconditional stability is proved for linear diffusion problems with periodic boundary conditions numerical examples supporting our theoretical analysis are provided

a priori estimates for rough pdes with application to rough conservation laws we introduce a general framework to study pdes driven by rough paths we develop new a priori estimates based on a rough gronwall lemma argument for weak solutions to rough pdes this will allow us to follow standard pde strategies to obtain existence and uniqueness results in particular our approach does not rely on any sort of transformation formula flow transformation feynman kac representation formula etc and is therefore rather flexible as an application we study conservation laws driven by rough paths establishing well posedness for the corresponding kinetic formulation

a three point sixth order nonuniform combined compact difference scheme a three point nonuniform combined compact difference nccd scheme with sixth order accuracy is proposed for numerical models the nccd scheme is a generalization of the previously proposed combined compact difference ccd scheme with a global hermitan polynomial spline and has major improved features such as error and computational cpu time reduction for nonperiodic boundaries additional sixth or fifth order nonuniform boundary conditions are proposed the nccd scheme with either sixth or fifth order additional boundary conditions can increase the accuracy and decrease the cpu time about 1 2 orders of magnitude compared to the ccd scheme

an efficient fourth order low dispersive finite difference scheme for a 2 d acoustic wave equation in this paper we propose an efficient fourth order compact finite difference scheme with low numerical dispersion to solve the two dimensional acoustic wave equation combined with the alternating direction implicit adi technique and pade approximation the standard second order finite difference scheme can be improved to fourth order and solved as a sequence of one dimensional problems with high computational efficiency however such compact higher order methods suffer from high numerical dispersion to suppress numerical dispersion the compact and non compact stages are interlinked to produce a hybrid scheme in which the compact stage is based on pade approximation in both y and temporal dimensions while the non compact stage is based on pade approximation in y dimension only stability analysis shows that the new scheme is conditionally stable and superior to some existing methods in terms of the courant friedrichs lewy cfl condition the dispersion analysis shows that the new scheme has lower numerical dispersion in comparison to the existing compact adi scheme and the higher order locally one dimensional lod scheme three numerical examples are solved to demonstrate the accuracy and efficiency of the new method

compact θ method for the generalized delay diffusion equation abstract the generalized diffusion equation with a delay has inherent complex nature because its analytical solutions are hard to obtain therefore one has to seek numerical methods especially the high order accurate ones for their approximate solutions in this paper we have established the results of the numerical asymptotic stability of the compact method for the generalized delay diffusion equation it shows that the compact method is asymptotically stable if and only if k r t h 2 10 cos h 12 1 cos h 1 2 for 0 1 2 and is unconditionally asymptotically stable for 1 2 1 respectively the convergent results in the maximum norm are studied according to the consistency analysis and lax theorem in the end a series of numerical tests on stability and convergence are carried out to support our theoretical results

a compact high order alternating direction implicit method for three dimensional acoustic wave equation with variable coefficient abstract efficient and accurate numerical simulation of seismic wave propagation is important in various geophysical applications such as seismic full waveform inversion fwi problem however due to the large size of the physical domain and requirement on low numerical dispersion many existing numerical methods are inefficient for numerical modelling of seismic wave propagation in an heterogeneous media despite the great efforts that have been devoted during the past decades it still remains a challenging task in the development of efficient and accurate finite difference method for multi dimensional acoustic wave equation with variable velocity in this paper we proposed a pade approximation based finite difference scheme for solving the acoustic wave equation in three dimensional heterogeneous media the new method is obtained by combining the pade approximation and a novel algebraic manipulation the efficiency of the new algorithm is further improved through the alternative directional implicit adi method the stability of the new algorithm has been theoretically proved by energy method the new method is conditionally stable with a better courant friedrichs lewy condition cfl condition which has been verified numerically extensive numerical examples have been solved which demonstrated that the new method is accurate efficient and stable

global spectral analysis for convection diffusion reaction equation in one and two dimensions effects of numerical anti diffusion and dispersion abstract convection diffusion reaction cdr equation plays a central role in many disciplines of engineering science and finances as a consequence importance of analysis of numerical methods for the accurate solution of cdr equation has motivated the present research we have used the global spectral analysis to characterize all the three important physical processes in terms of the non dimensional numerical parameters namely the non dimensional wavenumber kh courant friedrich lewy cfl number n c the peclet number pe and the damkohler number da for the purpose of illustration we have focused on two space time discretization schemes known for accuracy and robustness the basic properties relate to numerical issues arising for the numerical amplification factor numerical diffusion coefficient numerical phase speed and numerical group velocity with the help of model one dimensional 1d and two dimensional 2d cdr equations we have reported the numerical property charts for the cases i when all the processes of convection diffusion and reaction are of same order with critical numerical behaviour enforcing low values of da for the 1d cdr equation studied here ii the 2d cdr equation considered is diffusion reaction dominated and as a consequence this enforces da to be larger we have thoroughly analyzed these cases to identify the essential roles of anti diffusion on the critical n c and pe values which in turn decides admissible space and time steps to be used with the discretization schemes the property charts have been used to calibrate the analysis with two model equations one of which has an exact solution for a 1d cdr equation and the second case for the 2d cdr equation has numerical solution available in the literature these cases help to identify the importance of such analysis in explaining the utility of the choice one can exercise in fixing the numerical parameters this also identifies and explains some hitherto unknown numerical problems for cdr equation and their alleviation techniques

numerical modelling of linear and nonlinear diffusion equations by compact finite difference method in this work accurate solutions to linear and nonlinear diffusion equations were introduced a combination of a sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been used for treatment of these equations the computed results with the use of this technique have been compared with the exact solution to show the accuracy of it here the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation the present method is seen to be a very good alternative method to some existing techniques for realistic problems

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

compact mixed methods for convection diffusion type problems abstract this paper presents a class of fourth order compact finite difference technique for solving two dimensional convection diffusion equation the equation is recasted as a first order mixed system introducing a conservation and flux equations since flux appears explicitly in the mixed formulation we search a fourth order compact approximation of the primary solution field and flux based on taylor series expansion the proposed compact mixed formulation generalizes the work of carey and spotz g f carey w f spotz higher order compact mixed methods commun numer meth eng 13 1997 we show that their fourth order formulation corresponds to a particular case of our presented scheme and we extend their work to variable diffusion and convection coefficients some numerical experiments are performed to demonstrate the fourth order effective convergence rate

multiple coarse grid acceleration for multiscale multigrid computation the multiscale multigrid method uses an iterative refinement procedure with the richardson extrapolation technique to obtain a higher order solution the computational cost for the iterative refinement procedure may be significant for some ill conditioned coefficient matrices in this paper we proposed an alternative strategy using multiple coarse grids to eliminate the iterative refinement procedure and thus accelerate the multiscale multigrid computation numerical investigations show that our multiple coarse grid computing strategy is more efficient and scalable than the iterative refinement procedure the multiscale multigrid method with multiple coarse grid strategy is used to solve two dimensional 2d poisson equation and convection diffusion equation but the idea can be used to solve other partial differential equations

solution of linearized rotating shallow water equations by compact schemes with different grid staggering strategies high accuracy solution of pdes requires proper error analysis previous analysis for a non dispersive system t k sengupta a dipankar p sagaut error dynamics beyond von neumann analysis j comput phys 226 2007 1211 1218 identified sources of error correctly here the aim is to extend the spectral analysis for the model linearized rotating shallow water equations lrswe as an example of dispersive system we perform the analysis when high accuracy compact schemes are used to solve the lrswe relevant to geophysical fluid dynamics using different grid arrangements proposed in mesinger and arakawa f mesinger a arakawa numerical methods used in atmospheric models garp publ ser no 17 vol 1 wmo geneva 1976 pp 43 64 and randall d a randall geostrophic adjustment and the finite difference shallow water equations mon wea rev 122 1994 1371 1377 compact schemes are used for fluid dynamical problem as these afford near spectral accuracy in solving non periodic problems however higher accuracy methods also suffer from errors those are often filtered by low order methods for example dispersion error is present in all numerical methods and extreme form of it leads to q waves which appear at higher wavenumbers for compact schemes as compared to lower order method we also evaluate a compact scheme specifically designed for use with staggered grids here two and four time level temporal discretization methods have been compared for solving lrswe by considering classical fourth order four stage runge kutta rk 4 two time level forward backward fb and four time level generalized fb temporal integration schemes

existence uniqueness and stability of semi linear rough partial differential equations abstract we prove well posedness and rough path stability of a class of linear and semi linear rough pde s on r d using the variational approach this includes well posedness of possibly degenerate linear rough pde s in l p r d and then based on a new method energy estimates for non degenerate linear rough pde s we accomplish this by controlling the energy in a properly chosen weighted l 2 space where the weight is given as a solution of an associated backward equation these estimates then allow us to extend well posedness for linear rough pde s to semi linear perturbations

numerical solutions to a two component camassa holm equation abstract in the present paper a three step iterative algorithm for solving a two component camassa holm 2ch equation is presented in the first step the time dependent equation for the horizontal fluid velocity with nonlinear convection is solved then an inhomogeneous helmholtz equation is solved finally the equation for modeling the transport of density is solved in the third step the differential order of 2ch equation has been reduced in order to facilitate numerical scheme development in a comparatively smaller grid stencil in this study a fifth order spatially accurate upwinding combined compact difference scheme uccd5 which differs from that in sheu et al 2011 is developed in a four point grid stencil for approximating the first order derivative term for the purpose of retaining long time hamiltonians in the 2ch equation the time integrator or time stepping scheme chosen is symplectic various numerical experiments such as the single peakon peakon antipeakon interaction and dam break problems are conducted to illustrate the effectiveness of the proposed numerical method it is shown that both the hamiltonians and casimir functions are conserved well for all problems

a high order finite difference method with richardsonextrapolation for 3d convection diffusion equation in this paper we extend the sun and zhang s 24 work on high order finite difference method which is based on the richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case firstly we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids then we use the richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid finally we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid during this process we use alternating direction implicit adi method to solve the resulting linear systems numerical experiments are conducted to verify the accuracy and effectiveness of the present method

accuracy study of the ido scheme by fourier analysis the numerical accuracy of the interpolated differential operator ido scheme is studied with fourier analysis for the solutions of partial differential equations pdes advection diffusion and poisson equations the ido scheme solves governing equations not only for physical variable but also for first order spatial derivative spatial discretizations are based on hermite interpolation functions with both of them in the fourier analysis for the ido scheme the fourier coefficients of the physical variable and the first order derivative are coupled by the equations derived from the governing equations the analysis shows the ido scheme resolves all the wavenumbers with higher accuracy than the fourth order finite difference fd and compact difference cd schemes for advection equation in particular for high wavenumbers the accuracy is superior to that of the sixth order combined compact difference ccd scheme the diffusion and poisson equations are also more accurately solved in comparison with the fd and cd schemes these results show that the ido scheme guarantees highly resolved solutions for all the terms of fluid flow equations

the standard upwind compact difference schemes for incompressible flow simulations compact difference schemes have been used extensively for solving the incompressible navier stokes equations however the earlier formulations of the schemes are of central type called central compact schemes ccs which are dispersive and susceptible to numerical instability to enhance stability of ccs the optimal upwind compact schemes oucs are developed recently by adding high order dissipative terms to ccs in this paper it is found that oucs are essentially not of the upwind type because they do not use upwind biased but central type of stencils furthermore oucs are not the most optimal since orders of accuracy of oucs are at least one order lower than the maximum achievable orders new upwind compact schemes called standard upwind compact schemes sucs are developed in this paper in contrast to oucs sucs are constructed based completely on upwind biased stencils and hence can gain adequate numerical dissipation with no need for introducing optimization calculations furthermore sucs can achieve the maximum achievable orders of accuracy and hence be more compact than oucs more importantly sucs have prominent advantages on combining the stable and high resolution properties which are demonstrated from the global spectral analyses and typical numerical experiments