new conservative schemes with discrete variational derivatives for nonlinear wave equations new conservative finite difference schemes for certain classes of nonlinear wave equations are proposed the key tool there is discrete variational derivative by which discrete conservation property is realized a similar approach for the target equations was recently proposed by furihata but in this paper a different approach is explored where the target equations are first transformed to the equivalent system representations which are more natural forms to see conservation properties applications for the nonlinear klein gordon equation and the so called good boussinesq equation are presented numerical examples reveal the good performance of the new schemes

a hamiltonian preserving discontinuous galerkin method for the generalized korteweg de vries equation the invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations in order to minimize phase and amplitude errors after long time simulation in this paper we design analyze and numerically validate a hamiltonian preserving discontinuous galerkin method for solving the korteweg de vries kdv equation for the generalized kdv equation the semi discrete formulation is shown to preserve both the first and the third conserved integrals and approximately preserve the second conserved integral for the linearized kdv equation all the first three conserved integrals are preserved and optimal error estimates are obtained for polynomials of even degree the preservation properties are also maintained by the fully discrete dg scheme our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized kdv equation we also show that the shape of the solution after long time simulation is well preserved due to the hamiltonian preserving property

on the numerical experiments of the cauchy problem for semi linear klein gordon equations in the de sitter spacetime abstract the computational analysis of the cauchy problem for semi linear klein gordon equations in the de sitter spacetime is considered several simulations are performed to show the time global behaviors of the solutions of the equations in the spacetime based on the structure preserving scheme it is remarked that the sufficiently large hubble constant yields the strong diffusion effect which gives the long and stable simulations for the defocusing semi linear terms the reliability of the simulations is confirmed by the preservation of the numerically modified hamiltonian of the equations

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

variational r adaption in elastodynamics we develop a variational r adaptive finite element framework for solid dynamic applications and explore its conceptual links with the theory of dynamic configurational forces the central idea underlying the proposed approach is to allow hamilton s principle of stationary action to determine jointly the evolution of the displacement field and the discretization of the reference configuration of the body this is accomplished by rendering the action stationary with respect to the material and spatial nodal coordinates simultaneously however we find that a naive consistent galerkin discretization of the action leads to intrinsically unstable solutions remarkably we also find that this unstable behavior is eliminated when a mixed multifield version of hamilton s principle is adopted additional features of the proposed numerical implementation include the use of uncoupled space and time discretizations the use of independent space interpolations for velocities and deformations the application of these interpolations over a continuously varying adaptive mesh and the application of mixed variational integrators with independent time interpolations for velocities and nodal parameters the accuracy robustness and versatility of the approach are assessed and demonstrated by way of convergence tests and selected examples

generic formalism and discrete variational derivative method for the two dimensional vorticity equation the vorticity equation for two dimensional incompressible viscous flows is formulated within the generic formalism for non equilibrium thermodynamics the laws of conservation of energy and increasing entropy derived from the generic formulation are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method the law of increasing entropy corresponds to the dissipation of enstrophy for the vorticity equation some numerical experiments have been done to examine the usefulness of the proposed method

langevin equations and surface growth departamento de fi sica universidad nacional de san luis chacabuco 917 5700 san luis argentina received 9 january 1997 the three approaches usually used to study surface growth are i master equation ii stochastic langevinequation and iii microscopic models all of them give the same scaling exponents recently vvedenskyet al phys rev e 48 852 1993 derived a stochastic langevin equation from a master equation of the birthand death type for the epitaxial growth demonstrating the equivalence of both approaches in this paper astochastic langevin equation is derived from a discrete model the results are the same as those obtained byvvedensky et al demonstrating that the three approaches are equivalent as a nontrivial example our proce dure is used to derive the kardar parisi zhang kpz equation from the ballistic deposition process thismodel with vacancies and overhangs is very dif cult to handle due to the algebraic complications that arisewhen the master equation approach is used s1063 651x 97 11905 5 pacs number s 05 40 1ji introduction

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

a multilevel wavelet collocation method for solving partial differential equations in a finite domain abstract a multilevel wavelet collection method for the solution of partial differential equations is developed two different approaches of treating general boundary conditions are suggested both are based on the wavelet interpolation technique developed in the present research the first approach uses wavelets as a basis and results in a differential algebraic system of equations where the algebraic part arises from boundary conditions the second approach utilizes extended wavelets which satisfy boundary conditions exactly this approach results in a system of coupled ordinary differential equations the method is tested on the one dimensional burgers equation with small viscosity the solutions are compared with those resulting from the use of other numerical algorithms the present results indicate that the method is competitive with well established numerical algorithms

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

path integral formulation of stochastic processes for exclusive particle systems we present a systematic formalism to derive a path integral formulation for hard core particle systems far from equilibrium writing the master equation for a stochastic process of the system in terms of the annihilation and creation operators with mixed commutation relations we find the kramers moyal coefficients for the corresponding fokker planck equation fpe and the stochastic differential equation sde is derived by connecting these coefficients in the fpe to those in the sde finally the sde is mapped onto field theory using the path integral giving the field theoretic action which may be analyzed by the renormalization group method we apply this formalism to a two species reaction diffusion system with drift finding a universal decay exponent for the long time behavior of the average concentration of particles in arbitrary dimension

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

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

discrete stochastic evolution rules and continuum evolution equations abstract the continuum evolution equations are derived from updating rules for three classes of stochastic models the first class corresponds to models whose stochastic continuum equations are of the langevin type obtained after carrying out a local average known as coarse graining the second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations this average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables in which case a hierarchy of ordinary differential equations is obtained the third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice this double average generates a hierarchy of deterministic ordinary differential equations the second and third classes of equations are truncated using a mean field m n closure approximation in order to obtain a finite set of equations illustrative examples of every class are given

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 robust numerical integrator for the short pulse equation near criticality abstract the short pulse equation was introduced by schafer wayne 2004 for modeling the propagation of ultrashort optical pulses while it can describe a wide range of solutions its ultrashort pulse solutions with a few cycles which the conventional nonlinear schrodinger equation does not possess have drawn much attention in such a region the solutions can become quite close to a singularity and thus existing numerical methods cease to work stably or even if they do they require high computational cost in this paper we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure preserving methods the resulting scheme successfully works even when the singularity occurs and thus gives a highly robust method for resolving near singular solutions of interest it is confirmed by numerical experiments and some new insights about derivative blow up are obtained

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

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

conservative numerical schemes for the ostrovsky equation the ostrovsky equation describes gravity waves under the influence of coriolis force it is known that solutions of this equation conserve the l 2 norm and an energy function that is determined non locally in this paper we propose four conservative numerical schemes for this equation a finite difference scheme and a pseudospectral scheme that conserve the norm and the same types of schemes that conserve the energy a numerical comparison of these schemes is also provided which indicates that the energy conservative schemes perform better than the norm conservative schemes

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

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

dissipative conservative galerkin method using discrete partial derivatives for nonlinear evolution equations a new method is proposed for designing galerkin schemes that retain the energy dissipation or conservation properties of nonlinear evolution equations such as the cahn hilliard equation the korteweg de vries equation or the nonlinear schrodinger equation in particular as a special case dissipative or conservative finite element schemes can be derived the key device there is the new concept of discrete partial derivatives as examples of the application of the present method dissipative or conservative galerkin schemes are presented for the three equations with some numerical experiments

an evaluation of inferential procedures for adaptive clinical trial designs with pre specified rules for modifying the sample size summary many papers have introduced adaptive clinical trial methods that allow modifications to the sample size based on interim estimates of treatment effect there has been extensive commentary on type i error control and efficiency considerations but little research on estimation after an adaptive hypothesis test we evaluate the reliability and precision of different inferential procedures in the presence of an adaptive design with pre specified rules for modifying the sampling plan we extend group sequential orderings of the outcome space based on the stage at stopping likelihood ratio statistic and sample mean to the adaptive setting in order to compute median unbiased point estimates exact confidence intervals and p values uniformly distributed under the null hypothesis the likelihood ratio ordering is found to average shorter confidence intervals and produce higher probabilities of p values below important thresholds than alternative approaches the bias adjusted mean demonstrates the lowest mean squared error among candidate point estimates a conditional error based approach in the literature has the benefit of being the only method that accommodates unplanned adaptations we compare the performance of this and other methods in order to quantify the cost of failing to plan ahead in settings where adaptations could realistically be pre specified at the design stage we find the cost to be meaningful for all designs and treatment effects considered and to be substantial for designs frequently proposed in the literature

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

the fourier collocation method for the cahn hilliard equation abstract in this paper a fourier collocation method for numerically solving cahn hilliard equations with periodic boundary conditions is developed we establish their semidiscrete and fully discrete schemes that inherit the energy dissipation property and mass conservation property from the associated continuous problem we prove existence and uniqueness of the numerical solution and derive the optimal error bounds we perform some numerical experiments which confirm our results

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

entropy energy stable schemes for evolutionary dispersal models in this paper we propose some entropy energy stable finite difference schemes for the reaction diffusion advection equation arising in the evolution of biased dispersal of population dynamics the peculiar feature of these active dispersal models is that the transient solution converges to the stable steady state when time goes to infinity for the numerical method to capture the long time pattern of persistence or extinction we use the relative entropy when the resource potential is logarithmic and explore the usual energy for other resource potentials the present schemes are shown to satisfy three important properties of the continuous model for the population density i positivity preserving ii equilibrium preserving and iii entropy or energy satisfying these ensure that our schemes provide a satisfying long time behavior thus revealing the desired dispersal pattern moreover we present several numerical results which confirm the second order accuracy for various resource potentials and underline the efficiency to preserve the large time asymptotic

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

a hamiltonian conserving galerkin scheme for the camassa holm equation a new hamiltonian conserving galerkin scheme for the camassa holm equation is presented the scheme has an additional welcome feature that in exact arithmetic it is unconditionally stable in the sense that the solution is always bounded numerical examples that confirm the theory and the effectiveness of the scheme are also given

exact confidence bounds following adaptive group sequential tests summary we provide a method for obtaining confidence intervals point estimates and p values for the primary effect size parameter at the end of a two arm group sequential clinical trial in which adaptive changes have been implemented along the way the method is based on applying the adaptive hypothesis testing procedure of muller and schafer 2001 to a sequence of dual tests derived from the stage wise adjusted confidence interval of tsiatis rosner and mehta 1984 in the non adaptive setting this confidence interval is known to provide exact coverage in the adaptive setting exact coverage is guaranteed provided the adaptation takes place at the penultimate stage in general however all that can be claimed theoretically is that the coverage is guaranteed to be conservative nevertheless extensive simulation experiments supported by an empirical characterization of the conditional error function demonstrate convincingly that for all practical purposes the coverage is exact and the point estimate is median unbiased no procedure has previously been available for producing confidence intervals and point estimates with these desirable properties in an adaptive group sequential setting the methodology is illustrated by an application to a clinical trial of deep brain stimulation for parkinson s disease

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

conservative finite difference schemes for the degasperis procesi equation we consider the numerical integration of the degasperis procesi equation which was recently introduced as a completely integrable shallow water equation for the equation we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi hamiltonian form of the equation at the same time we also prove the unique solvability of the schemes and show some numerical examples

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

matching in studies of classification accuracy implications for analysis efficiency and assessment of incremental value in case control studies evaluating the classification accuracy of a marker controls are often matched to cases with respect to factors associated with the marker and disease status in contrast with matching in epidemiologic etiology studies matching in the classification setting has not been rigorously studied in this article we consider the implications of matching in terms of the choice of statistical analysis efficiency and assessment of the incremental value of the marker over the matching covariates we find that adjustment for the matching covariates is essential as unadjusted summaries of classification accuracy can be biased in many settings matching is the most efficient covariate dependent sampling scheme and we provide an expression for the optimal matching ratio however we also show that matching greatly complicates estimation of the incremental value of the marker we recommend that matching be carefully considered in the context of these findings

stochastic equations for simple discrete models of epitaxial growth we derive continuous stochastic equations governing the asymptotic behavior of growth from a master equation for discrete growth models with local relaxation we consider several simple models of epitaxial growth the family the wolf villain and the das sarma tamborenea models and their modifications i n 1 11 dimensions we derive for each model the corresponding langevin equation and identify leading terms that determine the universality class our results for models with local relaxation are in agreement with recent computer simulations the applicability of the method in 211 dimensions is demonstrated in the case of the family model problems of the procedure in particular regularization in the continuous limit are discussed s1063 651x 96 00710 6

adaptive group sequential designs for clinical trials combining the advantages of adaptive and of classical group sequential approaches a general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing this allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs the concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans

a fast adaptive wavelet collocation algorithm for multidimensional pdes a fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed the computational cost of the algorithm is independent of the dimensionality of the problem and is o n where n is the total number of collocation points the method can handle general boundary conditions the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where singularities or sharp transitions occur numerical results demonstrate the ability of the method to resolve localized structures such as shocks which change their location and steepness in space and time the present results indicate that the method has clear advantages in comparison with well established numerical algorithms

point estimation in adaptive enrichment designs adaptive enrichment designs are an attractive option for clinical trials that aim at demonstrating efficacy of therapies which may show different benefit for the full patient population and a prespecified subgroup in these designs based on interim data either the subgroup or the full population is selected for further exploration when selection is based on efficacy data this introduces bias to the commonly used maximum likelihood estimator for the situation of two stage designs with a single prespecified subgroup we present six alternative estimators and investigate their performance in a simulation study the most consistent reduction of bias over the range of scenarios considered was achieved by a method combining the uniformly minimum variance conditionally unbiased estimator with a conditional moment estimator application of the methods is illustrated by a clinical trial example

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

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

non markovian stochastic evolution equations abstract non markovian continuum stochastic and deterministic equations are derived from a set of discrete stochastic and deterministic evolution equations examples are given of discrete evolution equations whose updating rules depend on two or more previous time steps among them the continuum stochastic evolution equation of the newton second law the stochastic evolution equation of a wave equation the stochastic evolution equation for the scalar meson field etc are obtained as special cases extension to systems of evolution equations and other extensions are considered and examples are given the concept of isomorphism and almost isomorphism are introduced in order to compare the coefficients of the continuum evolution equations of two different smoothing procedures that arise from two different approaches usually these discrepancies arising from two sources on the one hand the use of different representations of the generalized functions appearing in the models and on the other hand the different approaches used to describe the models these new concept allows to overcome controversies that were appearing during decades in the literature

symplectic wavelet collocation method for hamiltonian wave equations this paper introduces a novel symplectic wavelet collocation method for solving nonlinear hamiltonian wave equations based on the autocorrelation functions of daubechies compactly supported scaling functions collocation method is conducted for the spatial discretization which leads to a finite dimensional hamiltonian system then appropriate symplectic scheme is employed for the integration of the hamiltonian system under the hypothesis of periodicity the properties of the resulted space differentiation matrix are analyzed in detail conservation of energy and momentum is also investigated various numerical experiments show the effectiveness of the proposed method

group sequential designs using a family of type i error probability spending functions performing planned or unplanned interim analyses on accumulating data in clinical trials is a frequent practice in this paper we propose a general one parameter family of type i error probability spending functions to construct customized group sequential boundaries with unequal increments in information time this proposed family generalized the spending functions of lan and demets1 and kim and demets 2 we give an example to illustrate the use of this family

exact inference for adaptive group sequential designs methods for controlling the type 1 error of an adaptive group sequential trial were developed in seminal papers by cui hung and wang biometrics 1999 lehmacher and wassmer biometrics 1999 and muller and schafer biometrics 2001 however corresponding solutions for the equally important and related problem of parameter estimation at the end of the adaptive trial have not been completely satisfactory in this paper a method is provided for computing a two sided confidence interval having exact coverage along with a point estimate that is median unbiased for the primary efficacy parameter in a two arm adaptive group sequential design the possible adaptations are not only confined to sample size alterations but also include data dependent changes in the number and spacing of interim looks and changes in the error spending function the procedure is based on mapping the final test statistic obtained in the modified trial into a corresponding backward image in the original trial this is an advance on previously available methods which either produced conservative coverage and no point estimates or provided exact coverage for one sided intervals only

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

a note on the adaptive conservative dissipative discretization for evolutionary partial differential equations an adaptive conservative or dissipative numerical method for nonlinear partial differential equations is established the method not only inherits the conservation or dissipation property of the equation but also uses suitable non uniform grids at each time step our numerical experiments indicate that the method is useful especially for localized solutions such as solitary wave solutions

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

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

a least squares finite element method for the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line in this article we discuss the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall the method we employ combines a finite element space approximation with a time discretization by operator splitting to solve the cahn hilliard part of the problem we use a least squares conjugate gradient method we also show that the scheme has the total energy decaying in time property under certain conditions our numerical experiments indicate that the method discussed here is accurate stable and efficient

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

modification of sample size in group sequential clinical trials in group sequential clinical trials sample size reestimation can be a complicated issue when it allows for change of sample size to be influenced by an observed sample path our simulation studies show that increasing sample size based on an interim estimate of the treatment difference can substantially inflate the probability of type i error in most practical situations a new group sequential test procedure is developed by modifying the weights used in the traditional repeated significance two sample mean test the new test has the type i error probability preserved at the target level and can provide a substantial gain in power with the increase of sample size generalization of the new procedure is discussed

uniform stabilization of numerical schemes for the critical generalized korteweg de vries equation with damping this work is devoted to the analysis of a fully implicit numerical scheme for the critical generalized korteweg de vries equation gkdv with p 4 in a bounded domain with a localized damping term the damping is supported in a subset of the domain so that the solutions of the continuous model issuing from small data are globally defined and exponentially decreasing in the energy space based in this asymptotic behavior of the solution we introduce a finite difference scheme which despite being one of the first order has the good property to converge in l 4 strong combining this strong convergence with discrete multipliers and a contradiction argument we show that the smallness of the initial condition leads to the uniform with respect to the mesh size exponential decay of the energy associated to the scheme numerical experiments are provided to illustrate the performance of the method and to confirm the theoretical results

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

probing universality classes in solid on solid deposition we consider several stochastic processes corresponding to the same physical solid on solid deposition problem simplified models presenting the same conditional mean and variance for each process are also introduced as well as generalizations in terms of the deposition of blobs and probabilistic deposition rules we compare the evolution of the roughness as a function of time for a three parameter family that includes as limit cases the family model and the edwards wilkinson equation showing that in all cases the derived models with the same mean and variance are indistinguishable from the originating models in terms of the evolution of the roughness finally we show that although all the models studied belong to the same universality class some relevant features such as the final surface roughness are reproduced only for models within a restricted class determined by sharing the same conditional mean and variance

derivation of continuum stochastic equations for discrete growth models we present a formalism to derive the stochastic differential equations sdes for several solid on solid growth models our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion we then write the master equations for these corresponding particle systems and find the sdes for the particle densities finally by connecting the particle densities with the growth heights we derive the sdes for the height variables applying this formalism to discrete growth models we find the edwards wilkinson equation for the symmetric body centered solid on solid bcsos model the kardar parisi zhang equation for the asymmetric bcsos model and the generalized restricted solid on solid rsos model and the villain lai das sarma equation for the conserved rsos model in addition to the consistent forms of equations for growth models we also obtain the coefficients associated with the sdes

dissipative or conservative finite difference schemes for complex valued nonlinear partial differential equations abstract we propose a new procedure for designing finite difference schemes that inherit energy conservation or dissipation property from complex valued nonlinear partial differential equations pdes such as the nonlinear schrodinger equation the ginzburg landau equation and the newell whitehead equation the procedure is a complex version of the procedure that furihata has recently presented for real valued nonlinear pdes furthermore we show that the proposed procedure can be modified for designing linearly implicit finite difference schemes that inherit energy conservation or dissipation property

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

exact confidence intervals following a group sequential test a numerical method is used to compute confidence intervals which have exact coverage probabilities for the mean of a normal distribution following a group sequential test this method which uses an ordering of the sample space similar to that employed by siegmund 1978 biometrika 65 341 349 is contrasted with the usual confidence interval for the mean

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

new conservative difference schemes for a coupled nonlinear schrodinger system abstract in this paper two conservative difference schemes for solving a coupled nonlinear schrodinger cnls system are numerically analyzed firstly a nonlinear implicit two level finite difference scheme for cnls system is studied then a linear three level difference scheme for cnls system is presented an induction argument and the discrete energy method are used to prove the second order convergence and unconditional stability of the linear scheme numerical examples show the efficiency of the new scheme

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

adaptive sample size calculations in group sequential trials a method for group sequential trials that is based on the inverse normal method for combining the results of the separate stages is proposed without exaggerating the type i error rate this method enables data driven sample size reassessments during the course of the study it uses the stopping boundaries of the classical group sequential tests furthermore exact test procedures may be derived for a wide range of applications the procedure is compared with the classical designs in terms of power and expected sample size

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

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

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

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 second order accurate non linear difference scheme for the n component cahn hilliard system a b s t r a c t we consider a second order conservative nonlinear numerical scheme for the ncomponent cahn hilliard system modeling the phase separation of a n component mixture the scheme is based on a crank nicolson finite difference method and is solved by an efficient and accurate nonlinear multigrid method we numerically demonstrate the second order accuracy of the numerical scheme we observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results we also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture we investigate the effects of a concentration dependent mobility on phase separation

fokker planck equation for lattice deposition models an asymptotically exact fokker planck equation for the height fluctuations of lattice deposition models is derived from a van kampen expansion of the master equation using an edwards wilkinson type model as an example the solution of the equivalent langevin equation reproduces the surface roughness and lateral height correlations obtained with kinetic monte carlo kmc simulations our discrete equations of motion thereby provide an exact analytic and computational alternative to kmc simulations of these models