Maximum norm error estimates of efficient difference schemes for second-order wave equations

The three-level explicit scheme is efficient for numerical approximation of the second-order wave equations. By employing a fourth-order accurate scheme to approximate the solution at first time level, it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two. Since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters (time step and spacings), an unusual Richardson extrapolation formula is needed in promoting the second-order solution to fourth-order accuracy. Extensions of our technique to the classical ADI scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation. Numerical experiments are presented to support our theoretical results.     

A new fourth‐order numerical algorithm for a class of three‐dimensional nonlinear evolution equations

In this article, a new compact alternating direction implicit finite difference scheme is derived for solving a class of 3‐D nonlinear evolution equations. By the discrete energy method, it is shown that the new difference scheme has good stability and can attain second‐order accuracy in time and fourth‐order accuracy in space with respect to the discrete H¹ ‐norm. A Richardson extrapolation algorithm is applied to achieve fourth‐order accuracy in temporal dimension. Numerical experiments illustrate the accuracy and efficiency of the extrapolation algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Maximum norm error bound of a linearized difference scheme for a coupled nonlinear Schrödinger equations

In this article, a linearized conservative difference scheme for a coupled nonlinear Schrödinger equations is studied. The discrete energy method and an useful technique are used to analyze the difference scheme. It is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm. Numerical experiments are presented to support the theoretical results.     

An algorithm using the finite volume element method and its splitting extrapolation

This paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation. This method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation, such as a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than a Richardson extrapolation. Because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition.     

A characteristics-mixed covolume method for a convection-dominated transport problem

In this paper, we propose a characteristics-mixed covolume method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term. The velocity and press are approximated by the lowest order Raviart-Thomas mixed finite element space on rectangles. The projection of a mixed covolume element is introduced. We prove its first order optimal rate of convergence for the approximate velocities in the L² norm as well as for the approximate pressures in the L² norm.     

Adaptive techniques for Landau–Lifshitz–Gilbert equation with magnetostriction

In this paper we propose a time–space adaptive method for micromagnetic problems with magnetostriction. The considered model consists of coupled Maxwell's, Landau–Lifshitz–Gilbert (LLG) and elastodynamic equations. The time discretization of Maxwell's equations and the elastodynamic equation is done by backward Euler method, the space discretization is based on Whitney edge elements and linear finite elements, respectively. The fully discrete LLG equation reduces to an ordinary differential equation, which is solved by an explicit method, that conserves the norm of the magnetization.     

High order compact Alternating Direction Implicit method for the generalized sine-Gordon equation

High order compact Alternating Direction Implicit scheme is given for solving the generalized sine-Gordon equation in a two-dimensional rectangular domain. We apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives, and we give a linearized three time level algorithm for solving the original nonlinear equation. Error estimate is given by the energy method. Numerical results are provided to verify the accuracy and efficiency of this algorithm.     

A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping

An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.     

Locally linearized fractional step methods for nonlinear parabolic problems

This work deals with the efficient numerical solution of a class of nonlinear time-dependent reaction-diffusion equations. Via the method of lines approach, we first perform the spatial discretization of the original problem by applying a mimetic finite difference scheme. The system of ordinary differential equations arising from that process is then integrated in time with a linearly implicit fractional step method. For that purpose, we locally decompose the discrete nonlinear diffusion operator using suitable Taylor expansions and a domain decomposition splitting technique. The totally discrete scheme considers implicit time integrations for the linear terms while explicitly handling the nonlinear ones. As a result, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled parallelizable linear subsystems. The convergence of the proposed methods is illustrated by numerical experiments.     

Conservative difference methods for the Klein–Gordon–Zakharov equations

Firstly an implicit conservative finite difference scheme is presented for the initial-boundary problem of the one space dimensional Klein–Gordon–Zakharov (KGZ) equations. The existence of the difference solution is proved by Leray–Schauder fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable and second order convergent for U   in _l∞$l_{\infty }$ norm, and for N   in _l2$l_2$ norm on the basis of the priori estimates. Then an explicit difference scheme is proposed for the KGZ equations, on the basis of priori estimates and two important inequalities about norms, convergence of the difference solutions is proved. Because it is explicit and not coupled it can be computed by a parallel method. Numerical experiments with the two schemes are done for several test cases. Computational results demonstrate that the two schemes are accurate and efficient.     

Time discretization via Laplace transformation of an integro-differential equation of parabolic type

We consider the discretization in time of an inhomogeneous parabolic integro-differential equation, with a memory term of convolution type, in a Banach space setting. The method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature, using the approach in recent work of López-Fernández and Palencia. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The method is combined with finite element discretization in the spatial variables to yield a fully discrete method. The paper is a further development of earlier work by the authors, which on the one hand treated purely parabolic equations and, on the other, an evolution equation with a positive type memory term. The authors acknowledge the support of the Australian Research Council.     

Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy

Summary A fully discrete finite element method for the Cahn-Hilliard equation with a logarithmic free energy based on the backward Euler method is analysed. Existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved. Two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented.     

Higher-order compact finite difference method for systems of reaction–diffusion equations

This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction–diffusion equations. This method has the accuracy of fourth-order in both space and time. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richardson extrapolation is used to achieve fourth-order accuracy in time. An application is given to an enzyme–substrate reaction–diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach.     

Stability and accuracy of time-extrapolated ADI-FDTD methods for solving wave equations

Some recent work on the ADI-FDTD method for solving Maxwell's equations in 3-D have brought out the importance of extrapolation methods for the time stepping of wave equations. Such extrapolation methods have previously been used for the solution of ODEs. The present context (of wave equations) brings up two main questions which have not been addressed previously: (1) when will extrapolation in time of an unconditionally stable scheme for a wave equation again feature unconditional stability, and (2) how will the accuracy and computational efficiency depend on how frequently in time the extrapolations are carried out. We analyze these issues here.     

Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations

Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth‐order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher‐order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A time- and spaceadaptive algorithm for the linear time-dependent Schrödinger equation

Summary. We prove an a posteriori error estimate for the linear time-dependent Schr?dinger equation in . From this, we derive a residual based local error estimator that allows us to adjust the mesh and the time step size in order to obtain a numerical solution with a prescribed accuracy. As a special feature, the error estimator controls localization and size of the finite computational domain in each time step. An algorithm is described to compute this solution and numerical results in one space dimension are included. Received March 17, 1995     

On the Galerkin finite element approximations to multi-dimensional differential and integro-differential parabolic equations

The maximum norm error estimates of the Galerkin finite element approximations to the solutions of differential and integro-differential multi-dimensional parabolic problems are considered. Our method is based on the use of the discrete version of the elliptic-Sobolev inequality and some operator representations of the finite element solutions. The results of the present paper lead to the error estimates of optimal or almost optimal order for the case of simplicial Lagrangian piecewise polynomial elements.     

Spectral methods for initial boundary value problems with nonsmooth data

Summary. In this paper we consider hyperbolic initial boundary value problems with nonsmooth data. We show that if we extend the time domain to minus infinity, replace the initial condition by a growth condition at minus infinity and then solve the problem using a filtered version of the data by the Galerkin-Collocation method using Laguerre polynomials in time and Legendre polynomials in space, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution. Received August 1, 1995 / Revised version received August 19, 1997     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.