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.     

Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson(C-N) scheme for solving decoupled FBSDEs. Based on the Taylor and It-Taylor expansions, the Malliavin calculus theory(e.g.,the multiple Malliavin integration-by-parts formula), and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.     

Smoothing of Crank-Nicolson scheme for the two-dimensional diffusion with an integral condition

The Parabolic partial differential equations (PDEs) with nonlocal boundary conditions model various physical phenomena, e.g. chemical diffusion, thermoelasticity, heat conduction process, control theory and medicine science. This paper deals with the smoothing of the Crank-Nicolson numerical scheme for two-dimensional parabolic PDEs with nonlocal boundary conditions. We use the numerical scheme based on Padé approximations of the matrix exponential. The graphs of numerical results demonstrate the successful smoothing of the Crank-Nicolson numerical scheme.     

A modified Crank-Nicolson scheme with incremental unknowns for convection dominated diffusion equations

A modified Crank-Nicolson scheme based on one-sided difference approximations is proposed for solving time-dependent convection dominated diffusion equations in two-dimensional space. The modified scheme is consistent and unconditionally stable. A priori L² error estimate for the fully discrete modified scheme is derived. With the use of the incremental unknowns preconditioner at each time step, a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified Crank-Nicolson scheme.     

Stabilized finite element method based on the Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations

This paper provides an error analysis for the Crank-Nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two-dimensional time-dependent Navier-Stokes problem, where the finite element space pair for the approximation of the velocity and the pressure is constructed by the low-order finite element: the quadrilateral element or the triangle element with mesh size . Error estimates of the numerical solution to the exact solution with are derived.

     

Maximum norm error estimates of the Crank-Nicolson scheme for solving a linear moving boundary problem

The Crank-Nicolson scheme is considered for solving a linear convection-diffusion equation with moving boundaries. The original problem is transformed into an equivalent system defined on a rectangular region by a linear transformation. Using energy techniques we show that the numerical solutions of the Crank-Nicolson scheme are unconditionally stable and convergent in the maximum norm. Numerical experiments are presented to support our theoretical results.     

Weak Galerkin finite element methods combined with Crank-Nicolson scheme for parabolic interface problems

This article is devoted to the a priori error estimates of the fully discrete Crank-Nicolson approximation for the linear parabolic interface problem via weak Galerkin finite element methods (WG-FEM). All the finite element functions are discontinuous for which the usual gradient operator is implemented as distributions in properly defined spaces. Optimal order error estimates in both $L^{\infty}(H^1)$ and $L^{\infty}(L^2)$ norms are established for lowest order WG finite element space $({\cal P}_{k}(K),\;{\cal P}_{k-1}(\partial K),\;\big[{\cal P}_{k-1}(K)\big]^2)$. Finally, we give numerical examples to verify the theoretical results.     

On the set visited once by a random walk

Summary In this paper we prove the following statement. Given a random walk ,n=1, 2, ... where ₁, ₂ ... are i.i.d. random variables, let (n) denote the number of points visited exactly once by this random walk up to timen. We show that there exists some constantC, 0 <C < , such that with probability 1. The proof applies some arguments analogous to the techniques of the large deviation theory.Research supported by the Hungarian National Foundation for Scientific Research, Grant N^o # 819/1     

Crank-Nicolson method for the numerical solution of models of excitability

We analyze a Crank-Nicolson scheme for a family of nonlinear parabolic partial differential equations. These equations cover a wide class of models of excitability, in particular the Hodgkin Huxley equations. To do the analysis, we have in mind the general discretization framework introduced by López-Marcos and Sanz-Serna [in Numerical Treatment of Differential Equations, K. Strehemel, Ed., Teubner-Texte zur Mathematik, Leipzig, 1988, p. 216]. We study consistency, stability and convergence properties of the scheme. We use a technique of modified functions, introduced by Strang [Numer. Math. 6 , 37 (1964)], in the study of consistency. Stability is derived by means of the energy method. Finally we obtain existence and convergence of numerical approximations by means of a result due to Stetter (Analysis of Discretization Methods for Ordinary Differential Equations. Springer-Verlag, Berlin, 1973). We show that the method has optimal order of accuracy in the discrete H¹ norm. © 1994 John Wiley & Sons, Inc.     

求解广义KdV方程的线性化局部Crank-Nicolson方法

针对广义KdV方程,构造了基于局部Crank-Nicolson方法的一种线性化差分格式,格式是一个可以显式求解的隐格式.数值试验表明,格式能够较好地求解广义KdV方程.     

A posteriori error estimates for the Crank-Nicolson method for parabolic equations

We derive optimal order a posteriori error estimates for time discretizations by both the Crank-Nicolson and the Crank-Nicolson-Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank-Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank-Nicolson method.

     

On the F-fundamental group scheme

Let X be a smooth projective variety defined over a perfect field k of positive characteristic, and let FX be the absolute Frobenius morphism of X. For any vector bundle E→X, and any polynomial g with non-negative integer coefficients, define the vector bundle using the powers of FX and the direct sum operation. We construct a neutral Tannakian category using the vector bundles with the property that there are two distinct polynomials f and g with non-negative integer coefficients such that . We also investigate the group scheme defined by this neutral Tannakian category.     

On the strong monotonicity of the CABARET scheme

The strong monotonicity of the CABARET scheme with single flux correction is analyzed as applied to the linear advection equation. It is shown that the scheme is strongly monotone (has the NED property) at Courant numbers r ∈ (0,0,5), for which it is monotone. Test computations illustrating this property of the CABARET scheme are presented.     

On the reducibility of the postulation Hilbert scheme

We give a condition in terms of the possible graded Betti numbers compatible with a given Hilbert functionH of 0-dimensional subschemes of ℙ ⁿ which implies the reducibility of the postulation Hilbert scheme and of its subscheme which parametrizes reduced subschemes with Hilbert functionH.     

On the horocyclic coordinate for the Teichmüller space of once punctured tori

This paper deals with analytic and geometric properties of the Maskit embedding of the Teichmüller space of once punctured tori. We show that the image of this embedding has an inward-pointing cusp and study the boundary behavior of conformal automorphisms. These results are proved using Y.N. Minsky's Pivot Theorem.

     

On a simulation scheme for the Boltzmann equation

A scheme for the simulation of solutions of the Boltzmann equation derived by Nanbu is investigated. Rigorous results concerning questions of justification, the computation effort and the energy fluctuations are presented.