共查询到20条相似文献,搜索用时 0 毫秒
1.
Xia CuiGuang-wei Yuan Jing-yan Yue 《Journal of Computational and Applied Mathematics》2011,236(2):253-264
A nonlinear iteration method named the Picard-Newton iteration is studied for a two-dimensional nonlinear coupled parabolic-hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization-discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard-Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard-Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard-Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy. 相似文献
2.
Tingchun Wang Tao Nie Luming Zhang 《Journal of Computational and Applied Mathematics》2009,231(2):745-759
In general, proofs of convergence and stability are difficult for symplectic schemes of nonlinear equations. In this paper, a symplectic difference scheme is proposed for an initial-boundary value problem of a coupled nonlinear Schrödinger system. An important lemma and an induction argument are used to prove the unique solvability, convergence and stability of numerical solutions. An iterative algorithm is also proposed for the symplectic scheme and its convergence is proved. Numerical examples show the efficiency of the symplectic scheme and the correction of our numerical analysis. 相似文献
3.
Wensheng ZhangEric T. Chung 《Journal of Computational and Applied Mathematics》2011,236(6):1343-1353
In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt4+h4) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method. 相似文献
4.
Tingchun Wang 《Journal of Computational and Applied Mathematics》2011,235(14):4237-4250
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. 相似文献
5.
In this article, a class of nonlinear evolution equations – reaction–diffusion equations with time delay – is studied. By
combining the domain decomposition technique and the finite difference method, the results for the existence, convergence
and the stability of the numerical solution are obtained in the case of subdomain overlap and when the time-space is completely
discretized.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
6.
In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion
problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by
combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference
scheme constructed on a tailored mesh. We prove that the resulting scheme is ε-uniformly convergent of second order in time
and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of
uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon,
which is usually produced in the time integration process when the boundary conditions are time dependent.
This research has been partially supported by the project MEC/FEDER MTM2004-01905 and the Diputación General de Aragón. 相似文献
7.
In this paper, we analyze a first-order time discretization scheme for a nonlinear geodynamo model and carry out the convergence analysis of this numerical scheme. It is concluded that our numerical scheme converges with first-order accuracy in the sense of L2-norm with respect to the velocity field u and the magnetic field B and with half-order accuracy in time for the total kinematic pressure P. 相似文献
8.
9.
We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.This work has been supported by funds from ACI JC 1041 “Mouvements d’interfaces avec termes non-locaux”, from ACI-JC 1025 “Dynamique des dislocations” and from ONERA, Office National d’Etudes et de Recherches. The second author was also supported by the ENPC-Région Ile de France. 相似文献
10.
Guodong Wang 《Journal of Computational and Applied Mathematics》2011,235(17):4966-4977
An Engquist-Osher type finite difference scheme is derived for dealing with scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. The new monotone difference scheme is based on introducing a new interface numerical flux function, which is called a generalized Engquist-Osher flux. By means of this scheme, the existence and uniqueness of weak solutions to the scalar conservation laws are obtained and the convergence theorem is established. Some numerical examples are presented and the corresponding numerical results are displayed to illustrate the efficiency of the methods. 相似文献
11.
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. 相似文献
12.
Semi-Lagrangian semi-implicit (SLSI) method is currently one of the most efficient approaches for numerical solution of the atmosphere dynamics equations. In this research we apply splitting techniques in the context of a two-time-level SLSI scheme in order to simplify the treatment of the slow physical modes and optimize the solution of the elliptic equations related to implicit part of the scheme. The performed numerical experiments show the accuracy and computational efficiency of the scheme. 相似文献
13.
High-order compact finite difference scheme for option pricing in stochastic volatility models 总被引:1,自引:0,他引:1
We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth order accurate in space and second order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the analysis becomes too involved we validate our findings by a numerical study. Numerical experiments for the European option pricing problem are presented. We observe fourth order convergence for non-smooth payoff. 相似文献
14.
In this paper, we propose a new method to compute the numerical flux of a finite volume scheme, used for the approximation
of the solution of the nonlinear partial differential equation ut+div(qf(u))−ΔΦ(u)=0 in a 1D, 2D or 3D domain. The function Φ is supposed to be strictly increasing, but some values s such that Φ′(s)=0 can exist. The method is based on the solution, at each interface between two control volumes, of the nonlinear elliptic
two point boundary value problem (qf(υ)+(Φ(υ))′)′=0 with Dirichlet boundary conditions given by the values of the discrete approximation in both control volumes. We prove
the existence of a solution to this two point boundary value problem. We show that the expression for the numerical flux can
be yielded without referring to this solution. Furthermore, we prove that the so designed finite volume scheme has the expected
stability properties and that its solution converges to the weak solution of the continuous problem. Numerical results show
the increase of accuracy due to the use of this scheme, compared to some other schemes. 相似文献
15.
New conservative finite difference schemes for certain classes of nonlinear wave equations are proposed. The key tool there is “discrete variational derivative”, by which discrete conservation property is realized. A similar approach for the target equations was recently proposed by Furihata, but in this paper a different approach is explored, where the target equations are first transformed to the equivalent system representations which are more natural forms to see conservation properties. Applications for the nonlinear Klein–Gordon equation and the so-called “good” Boussinesq equation are presented. Numerical examples reveal the good performance of the new schemes. 相似文献
16.
Takayasu Matsuo 《Journal of Computational and Applied Mathematics》2010,234(4):1258-243
A new Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation is presented. The scheme has an additional welcome feature that in exact arithmetic it is unconditionally stable in the sense that the solution is always bounded. Numerical examples that confirm the theory and the effectiveness of the scheme are also given. 相似文献
17.
J. Becker 《BIT Numerical Mathematics》1998,38(4):644-662
The numerical solution of a parabolic problem is studied. The equation is discretized in time by means of a second order two
step backward difference method with variable time step. A stability result is proved by the energy method under certain restrictions
on the ratios of successive time steps. Error estimates are derived and applications are given to homogenous equations with
initial data of low regularity. 相似文献
18.
In this work a system of two parabolic singularly perturbed equations of reaction–diffusion type is considered. The asymptotic behaviour of the solution and its partial derivatives is given. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution we consider the implicit Euler method for time stepping and the central difference scheme for spatial discretization on a special piecewise uniform Shishkin mesh. We prove that this scheme is uniformly convergent, with respect to the diffusion parameters, having first-order convergence in time and almost second-order convergence in space, in the discrete maximum norm. Numerical experiments illustrate the order of convergence proved theoretically. 相似文献
19.
Summary. This paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of Hamilton-Jacobi
type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical
limit for the Schr?dinger equation. We first state an existence and uniqueness result in the framework of viscosity and duality
solutions. Then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions
to ensure convergence. As an illustration, some practical computations are provided.
Received December 6, 1999 / Revised version received August 2, 2000 / Published online June 7, 2001 相似文献
20.
Robert Eymard Thierry Gallouït Raphaèle Herbin Anthony Michel 《Numerische Mathematik》2002,92(1):41-82
Summary. One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation by a piecewise constant function using a discretization in space and time and a finite volume scheme. The convergence of to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on are used to prove the convergence, up to a subsequence, of to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process
solution is proved, yielding the strong convergence of to{\it u}. Some on a model equation are shown.
Received September 27, 2000 / Published online October 17, 2001 相似文献