共查询到20条相似文献,搜索用时 15 毫秒
1.
Darae Jeong 《Journal of Computational and Applied Mathematics》2010,234(2):613-623
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. 相似文献
2.
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. 相似文献
3.
We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with
a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular
behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the
error is of order k
2 + h
2, where k and h are the parameters for the time and space meshes, respectively. 相似文献
4.
In this article, we develop an explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter. The parameter is used for the optimization of the method in order to solve efficiently the Schrödinger equation and related oscillatory problems. We evaluate the local truncation error and the interval of periodicity as functions of the parameter. We reveal a direct relationship between the periodicity interval and the local truncation error. We also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature. The analysis and the numerical results help us to determine the optimal values of the parameter, which render the new method highly efficient. 相似文献
5.
C. De Luigi 《Journal of Computational and Applied Mathematics》2010,234(1):181-191
We describe how to use new reduced size polynomial approximations for the numerical solution of the Poisson equation over hypercubes. Our method is based on a non-standard Galerkin method which allows test functions which do not verify the boundary conditions. Numerical examples are given in dimensions up to 8 on solutions with different smoothness using the same approximation basis for both situations. A special attention is paid on conditioning problems. 相似文献
6.
We establish the local well-posedness for the generalized Camassa–Holm equation. We also prove that the equation has smooth solutions that blow up in finite time. 相似文献
7.
Bosco García-Archilla 《Numerische Mathematik》1992,61(1):291-310
Summary A finite-difference method for the integration of the Korteweg-de Vries equation on irregular grids is analyzed. Under periodic boundary conditions, the method is shown to be supraconvergent in the sense that, though being inconsistent, it is second order convergent. However, such a convergence only takes place on grids with an odd number of points per period. When a grid with an even number of points is used, the inconsistency of the method leads to divergence. Numerical results backing the analysis are presented. 相似文献
8.
Summary. To solve 1D linear integral equations on bounded intervals with nonsmooth input functions and solutions, we have recently
proposed a quite general procedure, that is essentially based on the introduction of a nonlinear smoothing change of variable
into the integral equation and on the approximation of the transformed solution by global algebraic polynomials. In particular,
the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation
for an airfoil with a flap. In these cases we have obtained arbitrarily high orders of convergence through the solution of
very-well conditioned linear systems. In this paper, to enlarge the domain of applicability of our technique, we show how
the above procedure can be successfully used also to solve the classical Symm's equation on a piecewise smooth curve. The
collocation method we propose, applied to the transformed equation and based on Chebyshev polynomials of the first kind, has
shown to be stable and convergent. A comparison with some recent numerical methods using splines or trigonometric polynomials
shows that our method is highly competitive.
Received October 1, 1998 / Revised version received September 27, 1999 / Published online June 21, 2000 相似文献
9.
Based on an integral equation formulation, we present numerical methods for the inverse problem of recovering part of the domain boundary from boundary measurements of solutions to the Laplace equation on an accessible part of the boundary. 相似文献
10.
In this paper, we study the smoothness effect of Cauchy problem for the spatially homogeneous Landau equation in the hard potential case and the Maxwellian molecules case. We obtain the analytic smoothing effect for the solutions under rather weak assumptions on the initial datum. 相似文献
11.
We study the domain of existence of a solution to a Riemann problem for the pressure gradient equation in two space dimensions. The Riemann problem is the expansion of a quadrant of gas of constant state into the other three vacuum quadrants. The global existence of a smooth solution was established in Dai and Zhang [Z. Dai, T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal. 155 (2000) 277-298] up to the free boundary of vacuum. We prove that the vacuum boundary is the coordinate axes. 相似文献
12.
13.
A. Melman 《Numerische Mathematik》1995,69(4):483-493
Summary.
A method is proposed for the solution of a secular equation, arising
in modified symmetric eigenvalue problems and in several other
areas.
This equation has singularities which make the application of standard root-finding
methods difficult.
In order to solve the equation, a class of transformations of variables is
considered, which transform the equation into one for which Newton's method
converges from any point in a certain given interval. In addition, the form of the
transformed equation suggests a convergence accelerating modification of
Newton's method. The same ideas are applied to the secant
method and numerical results are presented.
Received July 1, 1994 相似文献
14.
We consider global solutions of a dynamical equation in ferrimagnet. We show that it admits a global weak solution by using the penalty method. By the energy estimates method we show there exists a unique global smooth solution. Finally we establish the relationship between this equation and wave maps. 相似文献
15.
Toshiyuki Koto 《Numerische Mathematik》1999,84(2):233-247
Summary. This paper deals with stability properties of Runge-Kutta (RK) methods applied to a non-autonomous delay differential equation
(DDE) with a constant delay which is obtained from the so-called generalized pantograph equation, an autonomous DDE with a
variable delay by a change of the independent variable. It is shown that in the case where the RK matrix is regular stability
properties of the RK method for the DDE are derived from those for a difference equation, which are examined by similar techniques
to those in the case of autonomous DDEs with a constant delay. As a result, it is shown that some RK methods based on classical
quadrature have a superior stability property with respect to the generalized pantograph equation. Stability of algebraically
stable natural RK methods is also considered.
Received May 5, 1998 / Revised version received November 17, 1998 / Published online September 24, 1999 相似文献
16.
17.
Summary For the linear advection equation we consider explicit multi-time-level schemes of highest order which are one step in space direction only. If a stencil involvesk time steps we show that it is stable in theL
2-sense for Courant numbers in the interval (0, 1/k). Since the order is 2k–1 one can use these schemes for high order discretization of the boundary conditions in hyperbolic initial value problems.Part of this work has been performed in the project Mehrschritt-Differenzenschemata of the Schwerpunktprogramm Finite Approximationen in der Strömungsmechanik which has been supported by the DFG 相似文献
18.
《Quaestiones Mathematicae》2013,36(2):199-214
AbstractIn this paper we study the combined sinh-cosh-Gordon equation, which arises in mathematical physics and has a wide range of scientific applications that range from chemical reactions to water surface gravity waves. We employ Lie symmetry analysis along with the simplest equation method to obtain exact solutions based on the optimal systems of one-dimensional subalgebras for the combined sinh-cosh-Gordon equation. Furthermore, conservation laws for the combined sinh-cosh-Gordon equation are derived by employing two different methods; the direct method and new conservation theorem. 相似文献
19.
We prove a Harnack inequality for a degenerate parabolic equation using proper estimates based on a suitable version of the
Rayleigh quotient.
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday 相似文献
20.
Well-posedness,global solutions and blowup phenomena for a nonlinearly dispersive wave equation 总被引:1,自引:0,他引:1
We establish the local well-posedness for a new nonlinearly dispersive wave equation which has solutions that exist for indefinite times as well as solutions that blowup infinite time. We also derive an explosion criterion for the equation, and we give a sharp estimate of the existence time for solutions with smooth initial data. 相似文献