time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

efficient cell vertex multigrid scheme for the three dimensional navier stokes equations a cell vertex scheme for the three dimensional navier stokes equations which is based on central difference approximations and runge kutta time stepping is described by using local time stepping implicit residual smoothing with locally varying coefficients a multigrid method and carefully controlled dissipative terms very good convergence rates are obtained for two and three dimensional flows details of the acceleration techniques which are important for convergence on meshes with high aspect ratio cells are discussed emphasis is placed on the analysis of the implicit smoothing of the explicit residuals with coefficients which depend on cell aspect ratios

unconditionally stable numerical method for a nonlinear partial integro differential equation the paper presents an unconditionally stable numerical scheme to solve a nonlinear integro differential equation which arises in mathematical modeling of the penetration of a magnetic field into a substance if the temperature is kept constant throughout the material numerical scheme comprises of the galerkin finite element method jangveladze and kiguradze 2011 for the spatial discretization followed by an implicit finite difference scheme for the time stepping we extended the results for stability estimates to a non homogeneous problem and derived optimal order error estimates for the semidiscretized and fully discretized equations using h 0 1 projection further to show the efficiency the proposed numerical method is demonstrated via numerical example

implicit time stepping methods for the navier stokes equations a novel two factor approximate factorization for three dimensional flows is described the method uses a preconditioned conjugate gradient solver for one of the factors and this solution is based on a method developed for two dimensional flows this two dimensional method is described and its performance is summarized for steady and unsteady flows the two factor method is then given larger time steps can be used with this method than is possible with the more conventional three factor method test results for the transonic steady flow over the onera m6 wing are given

finite difference approximation of a nonlinear integro differential system finite difference approximation of the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied the convergence of the finite difference scheme is proved the rate of convergence of the discrete scheme is given the decay of the numerical solution is compared with the analytical results proven earlier

linearly localized difference schemes for the nonlinear maxwell model of a magnetic field into a substance abstract a linearly localized difference scheme with the first order time approximation is proposed for solving a nonlinear maxwell model associated with the penetration of a magnetic field into a substance the new scheme is computationally efficient since the resulting algebra equations are linear and can be computed by the fast thomas algorithm without any newton type inner iterations it is also local in time that is only numerical solutions in one previous time level are necessary to update the current solutions such that it requires much less storage compared with the fully implicit method furthermore the exponential decaying behavior of difference solution which is analogous to that of the continuous solution is obtained to improve the time accuracy we apply the crank nicolson type time discretization to construct a second order linearly localized method numerical examples are presented to support our theoretical results

flux difference split algorithm for unsteady thin layer navier stokes solutions an efficient flux difference split scheme for calculating steady and unsteady thin layer navier stokes solutions for airfoils and wings is presented steady solutions for a flat plate laminar boundary layer profile are correctly modeled with only three grid cells internal to the boundary layer the flux difference split algorithm was used to evaluate the scheme for unsteady viscous calculations the diffusive terms are time lagged in the solution process the scheme is second order accurate in space and first order accurate in time due to the explicit treatment or the diffusive terms

on fully discrete galerkin approximations for partial integro differential equations of parabolic type the subject of this work is the application of fully discrete galerkin finite element methods to initial boundary value problems for linear partial integro differential equations of parabolic type we investigate numerical schemes based on the pade discretization with respect to time and associated with certain quadrature formulas to approximate the integral term a preliminary error estimate is established which contains a term related to the quadrature rule to be specified in particular we consider quadrature rules with sparse quadrature points so as to limit the storage requirements without sacrificing the order of overall convergence for the backward euler scheme the crank nicolson scheme and a third order pade type scheme the specific quadrature rules analyzed are based on the rectangular the trapezoidal and simpson s rule for all the schemes studied optimal order error estimates are obtained in the case that the solution of the problem is smooth enough since this is important for our error analysis we also discuss the regularity of the exact solutions of our equations high order regularity results with respect to both space and time are given for the solution of problems with smooth enough data 17 refs

multigrid convergence of an implicit symmetric relaxation scheme the multigrid method has been applied to an existing three dimensional compressible euler solver to accelerate the convergence of the implicit symmetric relaxation scheme this lower upper symmetric gauss seidel implicit scheme is shown to be an effective multigrid driver in three dimensions a grid refinement study is performed including the effects of large cell aspect ratio meshes performance figures of the present multigrid code on cray computers including the new c90 are presented a reduction of three orders of magnitude in the residual for a three dimensional transonic inviscid flow using 920 k grid points is obtained in less than 4 min on a cray c90 23 refs

convergence analysis for h1 galerkin mixed finite element approximation of one nonlinear integro differential model in this paper we investigate h 1 galerkin mixed finite element approximation of one nonlinear integro differential equation this method possesses some advantages such as approximating the unknown function and its gradient simultaneously as well as the finite element spaces being free of lbb condition a priori error estimates of the unknown function and its gradient are derived for both semi discrete and fully discrete schemes a numerical example is presented to illustrate the theoretical findings

spurious behavior of a symplectic integrator in this article we study the existence and behavior of spurious solutions of symplectic euler method for some hamiltonian systems it is shown that the symplectic integrator applied to hamiltonian systems in general doesn t avoid spurious behavior even spurious period two solutions the numerical results are presented

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

large time behavior of solutions to a nonlinear integro differential system abstract asymptotic behavior of solutions as t to the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied initial boundary value problems with two kinds of boundary data are considered the first with homogeneous conditions on whole boundary and the second with non homogeneous boundary data on one side of lateral boundary the rates of convergence are given too

preservation of hopf bifurcation for neutral delay differential equations by θ methods this paper considers the preservation of hopf bifurcation of some neural delay differential equations by q method by analyzing the dynamics of the numerical discrete system derived by q method we show that q method could inherit the hopf bifurcation and the asymptotical stability for sufficiently small stepsize h 1 m where m is a positive integer in particular for q 1 2 the result holds for any stepsize h 1 m furthermore the stability of the closed invariant curve is established finally some numerical examples are illustrated to support the analytic results

long time numerical solution of a parabolic equation with memory long time stability and convergence properties of two time discreti zation methods for an integro differential equation of parabolic type are studied the methods are based on the standard backward euler and second order back ward differencing methods the memory term is approximated by a quadrature rule with emphasis on such rules with reduced computational memory require ments discretization of the spatial partial differential operators by the finite element method is also considered

a new collocation approach for solving systems of high order linear volterra integro differential equations with variable coefficients this paper contributes an efficient numerical approach for solving the systems of high order linear volterra integro differential equations with variable coefficients under the mixed conditions the method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations the obtained matrix equation is based on the matrix forms of fibonacci polynomials and their derivatives by means of collocations in addition the method is presented with error numerical results with comparisons are given to demonstrate the applicability efficiency and accuracy of the proposed method the results of the examples indicated that the method is simple and effective and could provide an approximate solution with high accuracy or exact solution of the system of high order linear volterra integro differential equations

periodic solutions of a nonlinear parabolic equation associated with the penetration of a magnetic field into a substance abstract we prove the existence and uniqueness of a t periodic weak solution of the nonlinear parabolic equation u t 1 r r a 0 t u r r s 2 ds r u r f u 0 r t 0 1 x o t u r 0 t u r 1 t h t u 1 t 0 in a sobolev space with weight in the proof the galerkin method is employed numerical results are given

galerkin finite element method for one nonlinear integro differential model galerkin finite element method for the approximation of a nonlinear integro differential equation associated with the penetration of a magnetic field into a substance is studied first type initial boundary value problem is investigated the convergence of the finite element scheme is proved the rate of convergence is given too the decay of the numerical solution is compared with the analytical results

finite difference scheme for one system of nonlinear partial integro differential equations abstract system of maxwell equations is considered reduction to the integro differential form is given existence uniqueness and large time behavior of solutions of the initial boundary value problem for integro differential model with two component and one dimensional case are studied finite difference scheme is investigated wider class of nonlinearity is studied than one has been investigated before freefem realization code and results of numerical experiments are given

finite element methods for parabolic and hyperbolic partial integro differential equations on formule plusieurs methodes de galerkin a temps discret pour des equations integrodifferentielles paraboliques

unsteady aerodynamic and aeroelastic calculations for wings using euler equations a procedure to solve simultaneously the euler flow equations and modal structural equations of motion is presented for computing aeroelastic responses of wings the euler flow equations are solved by a finite difference scheme with dynamic grids the coupled aeroelastic equations of motion are solved using the linear acceleration method the aeroelastic configuration adaptive dynamic grids are time accurately generated using the aeroelastically deformed shape of the wing the unsteady flow calculations are validated with the experiment both for a semi infinite wing and a wall mounted cantilever rectangular wing aeroelastic responses are computed for a rectangular wing using the modal data generated by the finite element method the robustness of the present approach in computing unsteady flows and aeroelastic responses that are beyond the capability of earlier approaches using potential equations are demonstrated

implicit flux split schemes for the euler equations progress in the development of implicit algorithms for the euler equations using the flux vector splitting method is described comparisons of the relative efficiency of relaxation and spatially split approximately factored methods on a vector processor for transonic and supersonic twodimensional channel flows are made a hybrid threedimensional algorithm is developed that uses relaxation in one coordinate direction and approximate factorization in the crossflow plane the scheme is completely vectorizable and recovers conventional space marching schemes for fully supersonic flows the method is applied to a forebody shape in supersonic flow with an embedded pocket of subsonic flow

preservation of stability properties near fixed points of linear hamiltonian systems by symplectic integrators abstract based on reasonable testing model problems we study the preservation by symplectic runge kutta method srk and symplectic partitioned runge kutta method sprk of structures for fixed points of linear hamiltonian systems the structure preservation region provides a practical criterion for choosing step size in symplectic computation examples are given to justify the investigation

error estimates for semidiscrete finite element methods for parabolic integro differential equations application des methodes d elements finis aux equations integro differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d ordre 2

solver for unfactored implicit schemes conclusions when allowed to rotate freely a splitter plate shorter than five cylinder diameters does not align itself with the freestream instead it migrates to a stable position on one side of the wake or the other with the splitter plate angle a strong function of splitter plate length but independent of reynolds number in the range tested the wake of a freely rotatable splitter plate smaller than about two cylinder diameters is not significantly altered by the presence of the plate when comparing vortex formation length and karman vortex street shedding frequency in the near wake a cylinder with a free splitter plate behaves much differently than does a cylinder with a rigidly mounted splitter plate of the same length on the other hand a freely rotatable splitter plate larger than about two cylinder diameters although positioning itself at a nonzero angle affects the wake in nearly the same manner as does a rigid plate of the same length the inverse relationship between vortex formation length and vortex shedding frequency first suggested by gerrard is apparently valid for both fixed and free splitter plates

finite differences versus finite elements for solving nonlinear integro differential equations consider the nonlinear integro differential equation ut x t 0t a t x ux x d f x t 0 x 0 t t with appropriate initial and boundary conditions this problem serves as a model for one dimensional heat flow in materials with memory the numerical solution via finite elements was discussed in b neta j math anal appl 89 1982 598 611 in this paper we compare the results obtained there with finite difference approximation from the point of view of accuracy and computer storage it turns out that the finite difference method yields comparable results for the same mesh spacing using less computer storage

a unified approach to spurious solutions introduced by time discretization part i basic theory the asymptotic states of numerical methods for initial value problems are examined in particular spurious steady solutions solutions with period 2 in the timestep and spurious invariant curves

an integro differential formulation for magnetic induction in bounded domains boundary element finite volume method simulations of magnetohydrodynamic mhd flows in bounded domains using spectral methods suffer from a number of serious limitations alternative methods based on local discretization raise the problem of how to implement non local boundary conditions for the magnetic field we have developed a new strategy for the numerical solution of mhd problems in bounded domains which combines the flexibility of a local discretization with a rigorous formulation of magnetic boundary conditions next to an insulator in arbitrary geometries in accordance with the character of underlying equations we apply a global integral approach at the boundary and a differential approach inside the conducting domain the formulation of the boundary problem in terms of primitive variables allows us to combine these approaches and propose a mixed finite volume and boundary element method we illustrate its efficiency on magnetic diffusion problems in a sphere and in a finite cylinder

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

numerical dynamics of a nonstandard finite difference method for a class of delay differential equations abstract this paper proposes a nonstandard finite difference nsfd method for a class of delay differential equations we prove that for any step size h 1 m m z this numerical method could intrinsically preserve the qualitative behavior of the dynamical system including the local stability of equilibrium the existence and the direction of hopf bifurcation and the stability of bifurcating periodic solution finally to illustrate the analytic results the nsfd method is used for a model of the survival of red blood cells in animals

numerical solution of a nonlinear integro differential equation abstract galerkin s method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro differential equation u t x t 0 t a t x u x x d f x t 0 an error estimate in a suitable norm will be derived for the difference u uh between the exact solution u and the approximant uh it turns out that the rate of convergence of uh to u as h 0 is optimal this result was confirmed by the numerical experiments

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

finite element approximation of a parabolic integro differential equation with a weakly singular kernel we give error estimates for the numerical solution by means of the galerkin finite element method of an integro differential equation of parabolic type with a memory term containing a weakly singular kernel optimal order estimates are shown for spatially semidiscrete and completely discrete methods special attention is paid to the regularity of the exact solution

numerical solution of semilinear integrodifferential equations of parabolic type with nonsmooth data error estimates are derived for the discretization in space by finite elements and in time by the backward euler method of a parabolic equation with a semilinear memory term special attention is paid to the case of nonsmooth data and the computation of the memory term by quadrature

finite element methods of the two nonlinear integro differential equations abstract transformation of dependent variables such as the kirchhoff transformation is a classical tool for solving nonlinear partial differential equations in larsson et al this approach was used in connection with the finite element method zlamal used it in galerkin approximation in space the euler method in time and applied it to the solution of nonlinear heat equations here we give justification of the methods for the two nonlinear integro heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit euler method in time with quadrature rules which require the storage of relatively few values of the solution the estimate of the discretization error in the maximum norm is introduced the results depend on superconvergence in the gradient for a related elliptic problem which is shown to hold only for very special meshes