high order difference schemes for unsteady one dimensional diffusion convection problems abstract for unsteady 1d diffusion convection problems this paper develops an extensive analysis of two level three point finite difference schemes of order 2 in time and 4 in space this general class of fds includes several schemes independently proposed by different authors one main objective is the identification of difference schemes yielding satisfactory numerical results for strongly convective problems i e when the cell reynolds number b h 2 is greater than unity the stability and the oscillatory behaviour of the schemes are carefully studied and the analyses are completed by some numerical experiments we outline some key points i the great difficulty to obtain accurate numerical results for large values of ii the possibility of virtually optimum schemes is essentially theoretical and requires in practice careful experiments iii for strongly convective problems some second order explicit schemes are almost as efficient and less costly than implicit fourth order schemes

a difference scheme for a nonlinear partial integrodifferential equation a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in proving consistency the verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature a stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper

canonical transformations in quantum mechanics a canonically invariant path integral a particular form of the path integral is presented that allows the implementation of general canonical transformations in quantum mechanics as well as a consistent quantization procedure

upper semicontinuity of attractors for approximations of semigroups and partial differential equations abstract suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined applications are given to parabolic and hyperbolic partial differential equations

a note on exponential rosenbrock euler method for the finite element discretization of a semilinear parabolic partial differential equation abstract in this paper we consider the numerical approximation of a general second order semilinear parabolic partial differential equation equations of this type arise in many contexts such as transport in porous media using finite element method for space discretization and the exponential rosenbrock euler method for time discretization we provide a convergence proof in space and time under only the standard lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solutions this is in contrast to restrictive assumptions made in the literature where the authors have considered only approximation in time so far in their convergence proofs the main result reveals how the convergence orders in both space and time depend heavily on the regularity of the initial data in particular the method achieves optimal convergence order o h 2 t 2 when the initial data belongs to the domain of the linear operator numerical simulations to sustain our theoretical result are provided

error estimates for semidiscrete galerkin type approximations to semilinear evolution equations with nonsmooth initial data error estimates for the semidiserete galerkin method for abstract semilinear evolution equations with non smooth initial data are given in concrete cases almost optimal order of convergence for linear finite elements results

superconvergence and asymptotic expansion for semidiscrete bilinear finite volume element approximation of the parabolic problem abstract we first derive the asymptotic expansion of the bilinear finite volume element for the linear parabolic problem by employing the energy embedded method on uniform grids and then obtain a high accuracy combination pointwise formula of the derivatives for the finite volume element approximation based on the above asymptotic expansion furthermore we prove that the approximate derivatives have the convergence rate of order two numerical experiments confirm the theoretical results

transient radiation and scattering from fluid loaded elastic cylinders two dimensional radiation and scattering from fluid loaded circular elastic cylinders are addressed using two different time domain techniques in the first method the normal velocity of the shell is expanded as a fourier series of angle with time dependent coefficients whereas in the second method the velocity field of the shell is expressed as an in vacuo modal vector expansion with time dependent coefficients in both methods the velocity coefficients are expressed as the solution of convolution integral equations and the field pressures are then expressed as a fourier series in angle with range and time dependent coefficients that are convolution integrals involving the velocity coefficients recursive and iterative methods of solving the convolution integral equations are presented and discussed finally numerical results are presented and discussed for the normal velocity and surface pressure resulting from a transient line force excitation and an incident transient plane wave excitation

wave propagation in a hollow cylinder due to prescribed velocity at the boundary analyzed is transient response of a hollow cylinder to time dependent radial and axial velocities prescribed at the cylinder s inner boundary modal and static solutions are superimposed for solving transient response axial dependence is expressed by two orthogonal sets of periodic functions one set satisfies vanishing axial stress at the cylinder ends and applies to the radial traction problem and the other set satisfies vanishing shear stress at the ends and applies to the axial traction problem the mixed boundary value problem with velocity prescribed over part of the boundary and vanishing stress prescribed over the remaining part is analyzed by the method of influence coefficients this method superimposes response from several external annular traction segments of unit intensity with time dependent weights yielding a combined response equal to the prescribed instantaneous velocity

acoustical resonances of solid elastic cylinders parametric study and introduction to the inverse problem in underwater acoustics target recognition is of major importance in order to reach such a goal one must solve the inverse problem to deduce from the received echoes the signature of the target a rigorous method would require an inversion of the exact direct problem which is at the present time not feasible because of the complexity of the analytical expressions involved even for an isotropic homogeneous elastic cylinder after trying different approximate alternative methods of resolution of such problems in this lab a new method is presented in this paper its basis is a parametric study of resonance frequencies and widths and some simple approximate expressions deduced from this work are used to solve the inverse problem this study leads to a new classification of most resonances connected with the polarization of the corresponding waves the frequencies of such resonances are given by very simple equations that can be very easily inverted after carrying out this procedure this model is ap

3d vibration analysis of solid and hollow circular cylinders via chebyshev ritz method a general approach is presented for solving the free vibration of solid and hollow circular cylinders the analysis procedure is based on the small strain linear and exact elasticity theory by taking the chebyshev polynomial series multiplied by a boundary function to satisfy the geometric boundary conditions as the admissible functions the ritz method is applied to derive the frequency equation of the cylinder according to the axisymmetric geometrical property of a circular cylinder the vibration modes are divided into three distinct categories axisymmetric vibration torsional vibration and circumferential vibration moreover for a cylinder with the same boundary conditions at the two ends the vibration modes can be further divided into antisymmetric and symmetric ones in the length direction convergence and comparison studies demonstrate the high accuracy and small computational cost of the present method a significant advantage over other ritz solutions is that the present method can guarantee stable numerical operation even when a large number of terms of admissible functions are used not only the lower order but also the higher order frequencies can be obtained by using a few terms of the chebyshev polynomials finally the first several frequencies of circular cylinders with different boundary conditions with respect to various parameters such as the length radius ratio and the inside outside radius ratio are given 2003 elsevier science b v all rights reserved

an analysis for the convergence order of gradient schemes for semilinear parabolic equations gradient schemes are numerical methods which can be conforming and nonconforming have been recently developed in droniou et al 2013 droniou et al 2015 eymard et al 2012 and references therein to approximate different types of partial differential equations they are written in a discrete variational formulation and based on the approximation of functions and gradients the aim of the present paper is to provide gradient schemes along with an analysis for the convergence order of these schemes for semilinear parabolic equations in any space dimension we present three gradient schemes the first two schemes are nonlinear whereas the third one is linear the existence and uniqueness of the discrete solutions for the first two schemes is proved thanks to the use of the method of contractive mapping under the assumption that the mesh size of the time discretization k is small whereas the existence and uniqueness of the discrete solution for the third scheme is proved for arbitrary k we provide a convergence rate analysis in discrete semi norms of l h 1 and w 1 2 l 2 and in the norm of l l 2 we prove that the order in space is the same one proved in eymard et al 2012 when approximating elliptic equations and one or two in time the existence uniqueness and the convergence results stated above do not require any relation between spacial and temporal discretizations as an application of these results we focus on the gradient schemes which use the discrete gradient introduced recently in the sushi method eymard et al 2010 and we provide some numerical tests

transient response of a hollow cylinder with radial and axial material inhomogeneity the transient response of a hollow finite cylinder with radial and axial material inhomogeneity to a pulse of short duration is analyzed the problem is solved in two steps radial inhomogeneity is treated first by dividing the cylinder into coaxial segments each with a constant modulus e which is allowed to vary from segment to segment transfer matrices relating variables at the two radial interfaces of a segment combine to satisfy continuity of stress and displacement at these interfaces the galerkin method is then utilized to treat the general case with both radial and axial inhomogeneity adopting the eigenfunctions of the radially inhomogeneous cylinder as trial functions features of both static and transient responses resemble that for a weakening material along the axis displacement increases and stress reduces in proportion to the reduction in modulus spatial attenuation of transient stress cannot be replicated when geometric dimensionality is reduced from 3 d axisymmetric to 2 d plane strain

an analysis of a second order time accurate scheme for a finite volume method for parabolic equations on general nonconforming multidimensional spatial meshes the present work is an extension of our previous work bradji and fuhrmann 2011 4 which dealt with error analysis of a finite volume scheme of first order both in time and space for parabolic equations on general nonconforming multidimensional spatial meshes introduced recently in eymard et al 2010 12 we aim in this paper to get a higher order time accurate scheme for a finite volume method for parabolic equations using the same class of spatial generic meshes stated above we derive a finite volume scheme approximating the heat equation as a model for parabolic equations in which the discretization in time is performed using the crank nicolson method we derive an a priori estimate for the discrete problem and we prove that the error estimate of the finite volume scheme is of order two in time and it is of optimal order in space the error estimate is analysed in several norms which allow us to derive approximations for the exact solution and its first derivatives both spatial and temporal whose the convergence order is two in time and it is optimal in space we prove in particular when the discrete flux is calculated using a stabilized discrete gradient that the convergence order is k 2 h d where h d resp k is the mesh size of the spatial resp time discretization these estimates are valid under the regularity assumption u c 3 0 t c 2 w for the exact solution u the proof of these error estimates is based essentially on a new a priori estimate for the discrete problem and a comparison between the finite volume approximate solution and an auxiliary finite volume approximation

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

the trivialization of constraints in quantum theory working in a general gauge parametrization on trouve une facon de trivialiser et ainsi de traiter les contraintes d une maniere invariante par transformation de jauge

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

the long time behavior of finite element approximations of solutions of semilinear parabolic problems error estimates for finite element approximations of the solutions to semilinear parabolic problems are proved under the hypothesis that the exact solution is asymptotically stable as t to infty error estimates of optimal order that hold uniformly on the unbounded time interval 0 leqq t infty are obtained both semidiscrete and completely discrete approximations are considered

high order difference methods for heat equations in polar cylindrical coordinates abstract fourth order difference methods for the solution of poisson equations in cylindrical polar coordinates are proposed the same technique is then applied to obtain o k 2 h 4 two level unconditionally stable adi methods for the solution of the heat equation in two dimensional polar coordinates and three dimensional cylindrical coordinates numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

error estimates for spatially discrete approximations of semilinear parabolic equations with initial data of low regularity etude des methodes d elements finis semi discretes appliquees aux equations paraboliques semi lineaires en r d avec d 3 certains resultats concernant le taux de convergence sont ameliores

on the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel ii nonsmooth initial data abstract backward euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in error analyses the stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the galerkin finite element methods in the space variables

transient waves in an inhomogeneous hollow infinite cylinder abstract effects on transient waves of circumferential and radial inhomogeneity are studied in a plane strain hollow cylinder a periodic circumferential inhomogeneity modulating a constant value is analyzed adopting the galerkin method where trial functions are chosen as the axisymmetric and asymmetric modes of the homogeneous cylinder a periodic radial inhomogeneity is analyzed by dividing the cylinder into annular segments of constant width a stepwise variation in modulus is assumed where modulus is constant over each segment adopting transfer matrices continuity of state variables at interfaces of segments establishes the global dynamic equilibrium of the segmented cylinder the static dynamic superposition method is employed to solve for transient response

weak order for the discretization of the stochastic heat equation in this paper we study the approximation of the distribution of x t hilbert valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as dx t ax t dt q 1 2 d w t quad x 0 x in h quad t in 0 t driven by a gaussian space time noise whose covariance operator q is given we assume that a alpha is a finite trace operator for some alpha 0 and that q is bounded from h into d a beta for some beta geq 0 it is not required to be nuclear or to commute with a the discretization is achieved thanks to finite element methods in space parameter h 0 and implicit euler schemes in time parameter delta t t n we define a discrete solution x n h and for suitable functions varphi defined on h we show that e varphi x n h e varphi x t o h 2 gamma delta t gamma noindent where gamma 1 alpha beta let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations

biquadratic finite volume element methods based on optimal stress points for parabolic problems in this paper the semi discrete and full discrete biquadratic finite volume element schemes based on optimal stress points for a class of parabolic problems are presented optimal order error estimates in h 1 and l 2 norms are derived in addition the superconvergences of numerical gradients at optimal stress points are also discussed a numerical experiment confirms some results of theoretical analysis

a finite element model for the time dependent joule heating problem we study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body we prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables d 3 we establish the existence of solutions with the required regularity over arbitrarily long intervals of time when d 2

axisymmetric free vibrations of finite poroelastic bone axisymmetric free vibrations of poroelastic finite cylindrical bone which behaves as transversely isotropic material are investigated both curved and plane end surfaces of the solid cylinder are free from mechanical stresses and average fluid stresses two sets of basic solutions are derived to the equations of motion and poroelastic equation due to biot s theory by applying variable separable technique from the shear stress free boundary conditions eigenvalues for wave numbers are found using the basic solutions and eigen wave numbers solutions to the mechanical displacements and the fluid velocities are developed in series form the series form solutions satisfy the shear stress free boundary conditions exactly term by term remaining boundary conditions are satisfied by an orthogonalization procedure using trigonometric functions and first kind bessel functions natural frequencies of vibrations are calculated for human bone by varying the number of terms in the series that are tabulated the

generalized green s functions and the effective domain of influence one well known approach to a posteriori analysis of finite element solutions of elliptic problems estimates the error in a quantity of interest in terms of residuals and a generalized green s function the generalized green s function solves the adjoint problem with data related to a quantity of interest and measures the effects of stability including any decay of influence characteristic of elliptic problems we show that consideration of the generalized green s function can be used to improve the efficiency of the solution process when the goal is to compute multiple quantities of interest and or to compute quantities of interest that involve globally supported information such as average values and norms in the latter case we introduce a solution decomposition in which we solve a set of problems involving localized information and then recover the desired information by combining the local solutions by treating each computation of a quantity of interest independently the maximum number of elements required to achieve the desired accuracy can be decreased significantly

high order finite element methods for singularly perturbed elliptic and parabolic problems we develop a framework for applying high order finite element methods to singularly perturbed elliptic and parabolic differential systems that utilizes special quadrature rules to confine spurious effects such as excess diffusion and nonphysical oscillations to boundary and interior layers this approach is more suited for use with adaptive mesh refinement and order variation techniques than other problem dependent methods quadrature rules developed for two point convection diffusion and reaction diffusion problems are used with finite element software to solve examples involving ordinary and partial differential equations numerical artifacts are confined to layers for all combinations of meshes orders and singular perturbation parameters that were tested radau or lobatto quadrature used with the finite element method to solve respectively convection and reaction diffusion problems provide the benefits of the specialized quadrature formulas and are simpler to implement

eigenforms and eigenfrequency spectrum of finite elastic cylinder the reflection of elastic waves from free boundaries is accompanied by the effects of transformation of longitudinal waves to transverse ones and vice versa the transformation is the physical basis for interesting and practically important peculiarities in eigenfrequency spectrum and characteristics of eigenforms the cases of finite cylinder r l are considered to illustrate these peculiarities the main idea of the boundary problem solution method is described the method provides a way to get the eigenmode characteristics accurately in a wide frequency range the general conclusion of the study is that it is not possible to give a qualitative explanation of the eigenfrequency spectrum and eigenform properties in the scope of the concept of standing waves with respect to propagating ones in long elastic cylinder and infinite layer the special propagating evanescent waves are important in the eigenmode forming process the influence of this kind of wave results for example in occurrence of eigenfrequencies value which increases when dimensions of the cylinder increase specific features of corresponding eigenforms are discussed comparison of the numerical and experimental data is presented

postprocessing the finite element method for semilinear parabolic problems in this paper we consider postprocessing of the finite element method for semilinear parabolic problems the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the convergence rate is increased at almost no additional computational cost this procedure was introduced and analyzed in garcia archilla and titi siam j numer anal 37 2000 pp 470 499 we extend the analysis to the fully discrete case and prove error estimates for both space and time discretization the analysis is based on error estimates for the approximation of time derivatives by difference quotients

transient response of a thin cylinder with local material inhomogeneity analyzed is transient response of a thin cylinder with a local material inhomogeneity to a pulse of short duration the galerkin method is utilized to solve the inhomogeneous dynamic equations with the eigenfunctions of the homogeneous cylinder serving as trial functions also treated is response of the limiting cases of a disk and of a ring with the same local inhomogeneity all limiting cases yield to analysis when modulus e is approximated by segments of constant e along the radius for the disk and circumference for the ring transfer matrices relating variables at the two ends of a segment combine to satisfy continuity of variables at interfaces of segments curvature and axial dependence make the cylinder unique in response properties that neither disk nor ring possess

error estimates on a new nonlinear galerkin method based on two grid finite elements a new nonlinear galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations the new scheme is based on two different finite element spaces defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h nonlinearity and time dependence are both treated on the coarse space and only a fixed stationary equation needs to be solved on the fine space at each time with linear finite element discretizations it is proved that the difference between the new nonlinear galerkin solution and the standard galerkin solution in h 1 omega norm is of the order of h 3

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data on considere les approximations spatialement discretes continues par rapport au temps des problemes mixtes par la methode des elements finis de galerkin pour les equations paraboliques semi lineaires avec des donnees initiales irregulieres ou incompatibles on etablit des estimations d erreur

a full analysis of a new second order finite volume approximation based on a low order scheme using general admissible spatial meshes for the unsteady one dimensional heat equation abstract the present work is an extension of our previous works 3 4 5 which dealt with first order both in time and space and second order time accurate second order in time and first order in space implicit finite volume schemes for parabolic equations we aim in this work and some forthcoming studies at getting higher order both in time and space finite volume approximations for the exact solution of parabolic equations using the class of spatial generic meshes introduced recently in 13 we focus in the present contribution on the one dimensional heat equation and its implicit finite volume scheme described in 3 the implicit finite volume scheme approximating the one dimensional heat equation we consider hereafter referred to as the basic finite volume scheme yields linear systems to be solved successively the matrices involved in these linear systems are tridiagonal the finite volume approximate solution is of order h k where h resp k is the mesh size of the spatial resp time discretization we construct a new finite volume approximation of order h k 2 in several discrete norms which allows us to get approximations of order two for the exact solution and its first derivatives this new high order approximation can be computed using the same linear systems involved in the basic finite volume scheme while the right hand sides are corrected the construction of these right hand sides includes the approximations of the second third and fourth spatial derivatives of the exact solution the computation of the approximation of these high order derivatives can be performed using the same matrices stated above with another two tridiagonal matrices the manner by which this new high order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above these high order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of 12 without any restrictive condition on the spatial mesh a full analysis for the stated theoretical results as well as some numerical examples supporting the theory is presented the results obtained in the present study are based essentially on two facts the first fact is the use of the results provided in 3 which state the convergence order of the finite volume approximate solution in several norms the second fact is the comparison between the stated new higher order approximations and suitable auxiliary finite volume approximations

eine aussage zurl stabilitat und zur genauen konvergenzordnung derh 0 0 projektionen in this paper we settle a question concerning the necessity of the logh factor appearing in error estimates of linear finite element solutions let 2 be a bounded convex polygon and t h a regular family of subdivisions of into triangles be furthers 0 h h 0 1 the corresponding linear finite element spaces andp h h 0 1 s 0 h theh 0 1 projections with respect to the uniform norm asymptotically estimates of the form parallel u p h u parallel infty leqq c logh dist u s 0 h u in c bar omega cap h 0 1 omega hold as pointed out by schatz 1980 for functionsu with bounded second derivatives this implies parallel u p h u parallel infty leqq ch 2 log h u 2 infty a result which was proved by nitsche 1975 and scott 1976 with different methods it was an open question if the factor logh is necessary or only due to the methods of proof in this paper it is shown by an example that the cited estimates are sharp in the very strong sense that in general they are no longer valid if logh is replaced by a termo logh

error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions semidiscrete finite element error estimates for linear parabolic equations are derived under minimal regularity with the help of l2 projectors then analogous minimal regularity semidiscrete error estimates for semilinear parabolic equations are derived

approximations to the solution of cauchy problem for a linear evolution equation via the space shift operator second order equation example abstract we present a general method of solving the cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two one and zero the method is based on the chernoff approximation procedure applied to a specially constructed shift operator it is proven that approximations converge uniformly to the exact solution

special functions from quantum canonical transformations quantum canonical transformations are used to derive the integral representations and kummer solutions of the confluent hypergeometric and hypergeometric equations integral representations of the solutions of the nonperiodic three body toda equation are also found the derivation of these representations motivate the form of a two dimensional generalized hypergeometric equation which contains the nonperiodic toda equation as a special case and whose solutions may be obtained by quantum canonical transformation

on the discretization in time of semilinear parabolic equations with nonsmooth initial data single step discretization methods are considered for equations of the form u au f t u where a is a linear positive definite operator in a hubert space h it is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize even if the initial data v are only in h but also that in contrast to the situation in the linear homogeneous case higher order convergence is not possible in general without further assumptions on v

stability under galerkin truncation of a stable runge kutta discretizations in time we consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces we approximate their semiflow by an implicit a stable runge kutta discretization in time and a spectral galerkin truncation in space we show regularity of the galerkin truncated semiflow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value we also prove convergence of the space time discretization without any condition that couples the time step to the spatial resolution then we estimate the galerkin truncation error for the semiflow of the evolution equation its runge kutta discretization and their respective derivatives showing how the order of the galerkin truncation error depends on the smoothness of the initial data our results apply in particular to the semilinear wave equation and to the nonlinear schr odinger equation

high order difference methods for linear variable coefficient parabolic equation abstract in this paper we derive o k 2 h 4 two level three point finite difference methods for the solution of the general linear variable coefficient parabolic equation in one dimension u t a x u xx b x u x c x u 0 x 1 t 0 under suitable initial and boundary conditions the stability of the new schemes is examined using a linear stability analysis in particular we derive unconditionally stable o k 2 h 4 methods for the solution of the convection diffusion type equation in cylindrical and spherical coordinates viz u t u xx x u x cu 0

approximations for weakly nonlinear evolution equations convergence of approximations for a large class of weakly nonlinear parabolic and hyperbolic equations is proven the main emphasis is on proving convergence of finite element and spectral galerkin approximations of solutions to the weakly nonlinear wave equation u t au t f t u t u t u 0 xo u 0 yo under minimal assumptions on the linear operator a and on the approximation spaces a can be a very general elliptic operator not just of 2nd order and not necessarily in a bounded domain a can also be very singular and degenerate the results apply also to systems of equations verification of the hypotheses is completely elementary for a large class of problems 1 overview approximations of solutions to initial value problems of the form 1 u t au t f t u u o xo are studied the basic results in rather abstract form are presented in section 2 they can be used to prove convergence of finite difference finite element spectral etc approximations of weakly nonlinear hyperbolic and parabolic equations throughout the paper it is assumed that the nonlinear operator f is lipschitz continuous in u and for this reason 1 is said to be weakly nonlinear a is the generator of a strongly continuous semigroup the approach used to prove these results has origins in works of kato 17 trotter 29 segal 27 section 3 contains a version of the results of section 2 that is suitable for proving convergence of finite element approximations of solutions to the weakly nonlinear wave equation u t au t f t u t u t u o xo u o yo theorem 3 5 and to weakly nonlinear parabolic problems theorem 3 3 theorem 3 5 is new the main emphasis is on requiring as little as possible on the linear operator a and on the approximation spaces a can be a very general elliptic operator not just of 2nd order and not necessarily in a bounded domain a can also be very singular and degenerate theorem 4 2 of section 4 is perhaps the most interesting new result it is a consequence of theorem 3 5 and of the remarkable fact that sectorial operators have friedrichs extensions by using the friedrichs extension one can bypass the energy space that is used in variational methods and this makes it possible to state the assumptions of theorem 4 2 in a form that can be verified in a completely elementary way for a very general wave equation received july 11 1988 1980 mathematics subject classification 1985 revision primary 65m60 65j15 35l70 35l80 47b44 1989 american mathematical society 0025 5718 89 1 00 25 per page

error estimates for the semidiscrete galerkin approximations of the fitzhugh nagumo equations l 2 error estimates are obtained for the semidiscrete approximation of the fitzhugh nagumo equations with nonsmooth initial data the equations consist of a nonlinear parabolic equation coupled with an ordinary differential equation smoothing due to the parabolic equation is used to show increased accuracy for the approximation for positive time