practical aspects of higher order numerical schemes for wave propagation phenomena this paper examines practical issues related to the use of compact difference based fourth and sixth order schemes for wave propagation phenomena with focus on maxwell s equations of electromagnetics an outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches transformed plane method and physical space differencing in the first technique the truncation error expansion for the sixth order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available physical space differencing formulas are derived for the five point stencil by expressing the coefficients in terms of local spacing ratios the order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes to ensure stability for a broad range of problems fourier analysis is employed to develop a single parameter family of up to tenth order tridiagonal based spatial filters the implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1 d cavity where they are employed to suppress a late time instability the paper concludes after demonstrating the application of the scheme to several 3 d canonical problems utilizing cartesian as well as curvilinear meshes published in 1999 by john wiley sons ltd this article is a u s government work and is in the public domain in the united states

transport and diffusion of material quantities on propagating interfaces via level set methods we develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework level set methods are computational techniques for tracking moving interfaces they work by embedding the propagating interface as the zero level set of a higher dimensional function and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes the traditional level set method works in the trace space of the evolving interface and hence disregards any parameterization in the interface description consequently material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework we develop model equations and algorithmic techniques to extend the level set method to include these problems we demonstrate the accuracy of our approach through a series of test examples and convergence studies

solving eigenvalue problems on curved surfaces using the closest point method eigenvalue problems are fundamental to mathematics and science we present a simple algorithm for determining eigenvalues and eigenfunctions of the laplace beltrami operator on rather general curved surfaces our algorithm which is based on the closest point method relies on an embedding of the surface in a higher dimensional space where standard cartesian finite difference and interpolation schemes can be easily applied we show that there is a one to one correspondence between a problem defined in the embedding space and the original surface problem for open surfaces we present a simple way to impose dirichlet and neumann boundary conditions while maintaining second order accuracy convergence studies and a series of examples demonstrate the effectiveness and generality of our approach

numerical solution of the helmholtz equation with high wavenumbers this paper investigates the pollution effect and explores the feasibility of a local spectral method the discrete singular convolution dsc algorithm for solving the helmholtz equation with high wavenumbers fourier analysis is employed to study the dispersive error of the dsc algorithm our analysis of dispersive errors indicates that the dsc algorithm yields a dispersion vanishing scheme the dispersion analysis is further confirmed by the numerical results for one and higher dimensional helmholtz equations the dsc algorithm is shown to be an essentially pollution free scheme furthermore for large scale computation the grid density of the dsc algorithm can be close to the optimal two grid points per wavelength the present study reveals that the dsc algorithm is accurate and efficient for solving the helmholtz equation with high wavenumbers copyright 2003 john wiley sons ltd

sum accelerated pseudospectral methods finite differences and sech weighted differences this work continues our previous studies of algorithms for accelerating the convergence of pseudospectral derivative series in order to obtain new differentiation schemes which have the sparsity and low cost of finite differences but the accuracy of spectral methods we develop a general theoretical framework for difference schemes finite differences are close to optimum but can be bettered by a new scheme we have dubbed sech weighted differences through numerical examples we show that sech weighted differences are effective in contrast non linear accelerations like pade approximants and the levin u transform so popular in other applications are inferior for approximating derivatives to linear accelerations like finite differences sech weighted differences and the euler method

a cell based particle method for modeling dynamic interfaces abstract we propose several modifications to the grid based particle method gbpm 21 for moving interface modeling there are several nice features of the proposed algorithm the new method can significantly improve the distribution of sampling particles on the evolving interface unlike the original gbpm where footpoints sampling points tend to cluster to each other the sampling points in the new method tend to be better separated on the interface moreover by replacing the grid based discretization using the cell based discretization we naturally decompose the interface into segments so that we can easily approximate surface integrals as a possible alternative to the local polynomial least square approximation we also study a geometric basis for local reconstruction in the resampling step we will show that such modification can simplify the overall implementations numerical examples in two and three dimensions will show that the algorithm is computationally efficient and accurate

integrable solutions of a generalized mixed type functional integral equation in this work we prove the existence of integrable solutions for the following generalized mixed type nonlinear functional integral equation x t g t t x t f t 0 t k t s u t s q x s d s t 0 our result is established by means of a krasnosel skii type fixed point theorem proved by taoudi 2009 in the last section we give an example to illustrate our result

a lag averaged generalization of euler s method for accelerating series abstract euler s method is an efficient accelerator for an alternating series that is one whose terms a j are such that sign a j sign a j 1 however its effectiveness deteriorates rapidly as the period p of the oscillations in degree j increases to reduce this nonuniformity we introduce an algorithm which begins with the recursive computation of a triangular array of numbers the diagonal elements define a generalization of euler s method each horizontal row defines a delayed euler method which is even more effective the key element is that the array is generated by repeatedly averaging partial sums of the original series with a lag that is a distance in degree between the partial sums that are combined which optimally is half the period p of the oscillations in the series we illustrate the algorithm with the fourier series for a function with a shock wave jump discontinuity the terms a j 1 j sin jx vary from strictly alternating p 2 at x 0 to monotonic p as x the lag averaged euler scheme with and without delay is much more efficient than its classical counterpart especially near the discontinuities we present an analytical convergence theory for both the standard and generalized euler methods

a fourier error analysis for radial basis functions and the discrete singular convolution on an infinite uniform grid part 1 on an infinite grid with uniform spacing h the cardinal basis cj x h for many spectral methods consists of translates of a master cardinal function cj x h c x h j the cardinal basis satisfies the usual lagrange cardinal condition cj mh jm where jm is the kronecker delta function all such shift invariant subspace master cardinal functions are of localized sinc form in the sense that c x sinc x s x for a localizer function s which is smooth and analytic on the entire real axis and the whittaker cardinal function is sinc x sin x x the localized sinc approximation to a general f x is f localized sinc x h j f j h s x j h h sinc x jh h in contrast to most radial basis function applications matrix factorization is unnecessary we prove a general theorem for the fourier transform of the interpolation error for localized sinc bases for exponentially convergent radial basis functions rbfs gaussians inverse multiquadrics etc and the basis functions of the discrete singular convolution dsc the localizer function is known exactly or approximately this allows us to perform additional error analysis for these bases we show that the error is similar to that for sinc bases except that the localizer acts like a diffusion in fourier space smoothing the sinc error

a grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion we develop numerical methods for solving partial differential equations pde defined on an evolving interface represented by the grid based particle method gbpm recently proposed in s leung h k zhao a grid based particle method for moving interface problems j comput phys 228 2009 7706 7728 in particular we develop implicit time discretization methods for the advection diffusion equation where the time step is restricted solely by the advection part of the equation we also generalize the gbpm to solve high order geometrical flows including surface diffusion and willmore type flows the resulting algorithm can be easily implemented since the method is based on meshless particles quasi uniformly sampled on the interface furthermore without any computational mesh or triangulation defined on the interface we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes as an interesting application we study locally inextensible flows governed by energy minimization we introduce tension force via a lagrange multiplier determined by the solution to a helmholtz equation defined on the evolving interface extensive numerical examples are also given to demonstrate the efficiency of the proposed approach

numerical simulation of dealloying by surface dissolution via the evolving surface finite element method in this article we formulate a numerical method for the simulation of dealloying of a binary alloy by the selective removal of one component via electrochemical dissolution such that there is phase separation of the other component at the solid electrolyte interface the evolution of the interface is modelled by a forced mean curvature flow strongly coupled to the solution of a cahn hilliard equation modelling surface phase separation the method is based on a triangulated hypersurface whose evolution is calculated as well as the solution of the cahn hilliard equation by the evolving surface finite element method esfem the numerical experiments simulate complex morphology and concentration patterns providing evidence that the mathematical model may describe the formation of nanoporosity

high order finite differences and the pseudospectral method on staggered grids finite difference approximations for the first derivative valid halfway between equidistant gridpoints are in general much more accurate than the corresponding approximations which are valid at gridpoints the pseudospectral fourier method can be viewed as the limit of finite difference approximations when the order of accuracy tends to infinity a fundamentally different and more accurate pseudospectral method is obtained from the halfway approximations this study derives these and related methods and discusses their accuracies

an rbf fd closest point method for solving pdes on surfaces abstract partial differential equations pdes on surfaces appear in many applications throughout the natural and applied sciences the classical closest point method ruuth and merriman 2008 17 is an embedding method for solving pdes on surfaces using standard finite difference schemes in this paper we formulate an explicit closest point method using finite difference schemes derived from radial basis functions rbf fd unlike the orthogonal gradients method piret 2012 22 our proposed method uses rbf centers on regular grid nodes this formulation not only reduces the computational cost but also avoids the ill conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm leung and zhao 2009 26 when compared to the standard finite difference discretization of the closest point method the proposed method requires a smaller computational domain surrounding the surface resulting in a decrease in the number of sampling points on the surface in addition higher order schemes can easily be constructed by increasing the number of points in the rbf fd stencil applications to a variety of examples are provided to illustrate the numerical convergence of the method

fixed point theorems for the sum of two weakly sequentially continuous mappings abstract in the present paper we prove some krasnosel skii leray schauder type fixed point theorems for weak topology some fixed point theorems for the sum of two weakly sequentially continuous mappings are also presented our results extend and improve on ones from several earlier works

optimized compact difference based finite volume schemes for linear wave phenomena high order compact difference based finite volume schemes are developed analyzed and implemented for linear wave propagation phenomena with particular emphasis on computational electromagnetics in the time domain the formulation combines the primitive function approach with five point spatially sixth and fourth order methods optimization in the semi discrete case is achieved by minimizing dispersion and isotropy error the fully discrete scheme is examined by adopting the classical fourth order runge kutta technique stability bounds are established and the coefficients for the spatial discretization are readjusted for optimum performance the scheme is then formally extended to multiple dimensions consistent boundary conditions are presented for the reconstruction operator as well as flux specification several calculations in one and three dimensions confirm the properties of the method

modeling and computation of two phase geometric biomembranes using surface finite elements biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions the modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces this leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints in this paper we propose a new computational tool for computing equilibria based on an l 2 relaxation flow for the total energy in which the line energy is approximated by a surface ginzburg landau phase field functional the relaxation dynamics couple a nonlinear fourth order geometric evolution equation of willmore flow type for the membrane with a surface allen cahn equation describing the lateral decomposition a novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface the mean curvature vector and the surface phase field function the resulting variational formulation uses h 1 spaces and we employ triangulated surfaces and h 1 conforming quadratic surface finite elements for approximating solutions together with a semi implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells discocytes and starfishes with lateral phase separation

a comparison of the spectral and the discrete singular convolution schemes for the kdv type equations the discrete singular convolution dsc algorithm is applied to the korteweg de vries kdv and modified kdv mkdv equations with a particular emphasis on a comparison with the fourier pseudospectral fps method the errors of the dsc algorithm with respect to the numerical resolution are investigated by using the discrete fourier analysis the numerical results indicate that the dsc method can be more accurate than the fps method for many nonlinear wave problems that are non bandlimited in nature

a grid based particle method for moving interface problems we propose a novel algorithm for modeling interface motions the interface is represented and is tracked using quasi uniform meshless particles these particles are sampled according to an underlying grid such that each particle is associated to a grid point which is in the neighborhood of the interface the underlying grid provides an eulerian reference and local sampling rate for particles on the interface it also renders neighborhood information among the meshless particles for local reconstruction of the interface the resulting algorithm which is based on lagrangian tracking using meshless particles with eulerian reference grid can naturally handle control topological changes moreover adaptive sampling of the interface can be achieved easily through local grid refinement with simple quad oct tree data structure extensive numerical examples are presented to demonstrate the capability of our new algorithm

finite difference schemes for long time integration a general method for constructing finite difference schemes for longtime integration problems is presented it is demonstrated for discretizations of first and second space derivatives however the approach is not limited to these cases the schemes are constructed so as to minimize the global truncation error taking into account the initial data the resulting second order compact schemes can be used for integration times fourfold or more longer than previously studied schemes with similar computational complexity a similar approach was used to obtain improved integration schemes

schwarz solvers and preconditioners for the closest point method the discretization of surface intrinsic elliptic partial differential equations pdes poses interesting challenges not seen in flat spaces the discretization of these pdes typically proceeds by e

krasnoselskii s fixed point theorem for weakly continuous maps abstract we consider the sum a b m x where m is a weakly compact and convex subset of a banach space x a m x is weakly continuous and b l x with bp 1 p 1 an alternative condition is given in order to guarantee the existence of fixed points in m for a b some illustrative applications are given

a new compact spectral scheme for turbulence simulations we propose a new kind of compact difference scheme for the computation of the first and second derivatives in the simulation of high reynolds number turbulent flows the scheme combines and truncates the pseudospectral representation of derivative for convergence acceleration comparison of the wave resolution property with available optimized compact schemes minimizing the prescribed wave resolution error reveals our scheme s superiority for the same size of stencils without introducing optimization parameters an accompanying boundary scheme is also proposed with the stability analysis the proposed scheme is tested for the evaluation of derivatives of a function that decays very slowly in the wavenumber space and for the simulation of three dimensional isotropic turbulence

a localized meshless method for diffusion on folded surfaces partial differential equations pdes on surfaces arise in a variety of application areas including biological systems medical imaging fluid dynamics mathematical physics image processing and computer graphics in this paper we propose a radial basis function rbf discretization of the closest point method the corresponding localized meshless method may be used to approximate diffusion on smooth or folded surfaces our method has the benefit of having an a priori error bound in terms of percentage of the norm of the solution a stable solver is used to avoid the ill conditioning that arises when the radial basis functions rbfs become flat

a grid based particle method for evolution of open curves and surfaces we propose a new numerical method for modeling motion of open curves in two dimensions and open surfaces in three dimensions following the grid based particle method we proposed in s leung h k zhao a grid based particle method for moving interface problems j comput phys 228 2009 2993 3024 we represent the open curve or the open surface by meshless lagrangian particles sampled according to an underlying fixed eulerian mesh the underlying grid is used to provide a quasi uniform sampling and neighboring information for meshless particles the key idea in the current paper is to represent and to track end points of the open curve and boundary points of the open surface explicitly and consistently with interior particles we apply our algorithms to several applications including spiral crystal growth modeling and image segmentation using active contours

pdes on moving surfaces via the closest point method and a modified grid based particle method partial differential equations pdes on surfaces arise in a wide range of applications the closest point method ruuth and merriman 2008 20 is a recent embedding method that has been used to solve a variety of pdes on smooth surfaces using a closest point representation of the surface and standard cartesian grid methods in the embedding space the original closest point method cpm was designed for problems posed on static surfaces however the solution of pdes on moving surfaces is of considerable interest as well here we propose solving pdes on moving surfaces using a combination of the cpm and a modification of the grid based particle method leung and zhao 2009 12 the grid based particle method gbpm represents and tracks surfaces using meshless particles and an eulerian reference grid our modification of the gbpm introduces a reconstruction step into the original method to ensure that all the grid points within a computational tube surrounding the surface are active we present a number of examples to illustrate the numerical convergence properties of our combined method experiments for advection diffusion equations that are strongly coupled to the velocity of the surface are also presented

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

the implicit closest point method for the numerical solution of partial differential equations on surfaces many applications in the natural and applied sciences require the solutions of partial differential equations pdes on surfaces or more general manifolds the closest point method is a simple and accurate embedding method for numerically approximating pdes on rather general smooth surfaces however the original formulation is designed to use explicit time stepping this may lead to a strict time step restriction for some important pdes such as those involving the laplace beltrami operator or higher order derivative operators to achieve improved stability and efficiency we introduce a new implicit closest point method for surface pdes the method allows for large stable time steps while retaining the principal benefits of the original method in particular it maintains the order of accuracy of the discretization of the underlying embedding pde it works on sharply defined bands without degrading the accuracy of the method and it applies to general smooth surfaces it also is very simple and may be applied to a rather general class of surface pdes convergence studies for the in surface heat equation and a fourth order biharmonic problem are given to illustrate the accuracy of the method we demonstrate the flexibility and generality of the method by treating flows involving diffusion reaction diffusion and fourth order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension

fourier analysis of numerical algorithms for the maxwell equations the fourier method is used to analyze the dispersive dissipative and isotropy errors of various spatial and time discretizations applied to the maxwell equations on multi dimensional grids both cartesian grids and non cartesian grids based on hexagons and tetradecahedra are studied and compared the numerical errors are quantitatively determined in terms of phase speed wavenumber propagation direction gridspacings and cfl number the study shows that centered schemes are more efficient and accurate than upwind schemes and the non cartesian grids yield superior isotropy than the cartesian ones for the centered schemes the staggered grids produce less errors than the unstaggered ones a new unstaggered algorithm which has all the best properties is introduced using an optimization technique to determine the nodal weights the new algorithm provides the highest accuracy among all the schemes discussed the study also demonstrates that a proper choice of time discretization can reduce the overall numerical errors due to the spatial discretization

wavenumber extended high order upwind biased finite difference schemes for convective scalar transport this paper proposes some new wavenumber extended high order upwind biased schemes the dispersion and dissipation errors of upwind biased finite difference schemes are assessed and compared by means of a fourier analysis of the difference schemes up to 11th order upwind biased schemes are analyzed it is shown that both the upwind biased scheme of order 2n 1 and the corresponding centered differencing scheme of order 2nhave the same dispersion characteristics thus the former can be considered to be the latter plus a correction that reduces the numerical dissipation the new second order wavenumber extended scheme is tested and compared with some well known schemes the range of wavenumbers that are accurately treated by the upwind biased schemes is improved by using additional constraints from the fourier analysis to construct the new schemes the anisotropic behavior of the dispersion and dissipation errors is also analyzed for both the conventional and the new wavenumber extended upwind biased finite difference schemes

high accuracy algorithms for computational aeroacoustics this paper presents an analysis of high bandwidth operators developed for use with an essentially nonoscillatory eno method the spatial operators of a sixth order eno code are modified to resolve waves with as few as 7 points per wavelength ppw by decreasing the formal order of the algorithm numerical and analytical solutions are compared for the model problems of plane wave propagation and sound generation by an oscillating sphere these problems involve linear propagation wave steepening and shock formation an analysis of the ppw required for sufficient accuracy shows that low order algorithms need an excessive number of grid points to produce acceptable solutions in contrast high order codes provide good predictions on relatively coarse grids the high bandwidth operators produce only modest improvements over the original sixth order operators for nonlinear problems in which wave steepening is significant however they clearly outperform the original operators for long distance linear propagation because the high bandwidth operators have the same stencil as the original sixth order operators these gains are achieved with no additional computational work

krasnosel skii type fixed point theorems under weak topology features abstract in this paper we prove the following krasnosel skii type fixed point theorem let m be a nonempty bounded closed convex subset of a banach space x suppose that a m x and b x x are two weakly sequentially continuous mappings satisfying i a m is relatively weakly compact ii b is a strict contraction iii x b x a y y m x m then a b has at least one fixed point in m this result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping the results presented in this paper encompass several earlier ones in the literature

a topological and geometric approach to fixed points results for sum of operators and applications abstract in this paper we establish a fixed point result of krasnoselskii type for the sum a b where a and b are continuous maps acting on locally convex spaces our results extend previous ones we apply such results to obtain strong solutions for some quasi linear elliptic equations with lack of compactness we also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a banach space in the last section we develop a sequentially weak continuity result for a class of operators acting on vector valued lebesgue spaces such a result is used together with a geometric condition as the main tool to provide an existence theory for nonlinear integral equations in l p e

a proof that the discrete singular convolution dsc lagrange distributed approximating function ldaf method is inferior to high order finite differences finite differences approximate the mth derivative of a function u x by a series j n nd j m u x j where the x j are the grid points the closely related discrete singular convolution dsc and lagrange distributed approximating function ldaf methods treated here as a single algorithm approximate derivatives in the same way as finite differences but with different numerical weights that depend upon a free parameter a by means of fourier analysis and error theorems we show that the dsc is worse than the standard finite differences in differentiating exp ikx for all k when a a f d where a f d 1 n 1 with n as the stencil width is the value of the dsc parameter that makes its weights most closely resemble those of finite differences for a

schaefer krasnoselskii fixed point theorems using a usual measure of weak noncompactness abstract we present some extension of a well known fixed point theorem due to burton and kirk t a burton c kirk a fixed point theorem of krasnoselskii schaefer type math nachr 189 1998 423 431 for the sum of two nonlinear operators one of them compact and the other one a strict contraction the novelty of our results is that the involved operators need not to be weakly continuous finally an example is given to illustrate our results

an analytic approximation to the cardinal functions of gaussian radial basis functions on an infinite lattice gaussian radial basis functions rbfs have been very useful in computer graphics and for numerical solutions of partial differential equations where these rbfs are defined on a grid with uniform spacing h as translates of the master function f x a h exp a 2 h 2 x 2 where a is a user choosable constant unfortunately computing the coefficients of f x jh a h requires solving a linear system with a dense matrix it would be much more efficient to rearrange the basis functions into the equivalent lagrangian or cardinal basis because the interpolation matrix in the new basis is the identity matrix the cardinal basis c j x a h is defined by the set of linear combinations of the gaussians such that c j kh 1 when k j and c j kh 0 for all integers k j we show that the cardinal functions for the uniform grid are c j x h a c x h j a where c x a a 2 p sin px sinh a 2x the relative error is only about 4exp email protected 2 a 2 as demonstrated by the explicit second order approximation it has long been known that the error in a series of gaussian rbfs does not converge to zero for fixed a as h 0 but only to an error saturation proportional to exp email protected 2 a 2 because the error in our approximation to the master cardinal function c x a is the square of the error saturation there is no penalty for using our new approximations to obtain matrix free interpolating rbf approximations to an arbitrary function f x the master cardinal function on a uniform grid in d dimensions is just the direct product of the one dimensional cardinal functions thus in two dimensions c x y a a 4 p 2 sin px sin py sinh a 2x sinh a 2y we show that the matrix free interpolation can be extended to non uniform grids by a smooth change of coordinates

comparison of the discrete singular convolution and three other numerical schemes for solving fisher s equation in this paper a discrete singular convolution dsc algorithm is introduced to solve fisher s equation to which obtaining an accurate and reliable traveling wave solution is a challenging numerical problem two novel numerical treatments a moving frame scheme and a new asymptotic scheme are designed to overcome the subtle difficulties involved in the numerical solution of fisher s equation the moving frame scheme is proposed to provide accurate and efficient spatial resolution of the problem the new asymptotic scheme is introduced to facilitate the correct prediction of the wave speed after long time integrations the resulting dsc algorithm is able to correctly predict long time traveling wave behavior the performance of the present algorithm is demonstrated through extensive numerical experiments as well as through a comparison with three other standard approaches i e methods of the fourier pseudospectral the accurate spatial derivatives and the crank nicolson schemes the spatial and temporal accuracies of all four methods are analyzed and numerically verified