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1.
C.V. Pao 《Numerische Mathematik》1995,72(2):239-262
Summary.
Two block monotone iterative schemes for a nonlinear
algebraic system, which is a finite difference approximation of a
nonlinear elliptic boundary-value problem, are presented and are
shown to converge monotonically either from above or from below to
a solution of the system. This monotone convergence result yields
a computational algorithm for numerical solutions as well as an
existence-comparison theorem of the system, including a sufficient
condition for the uniqueness of the solution. An advantage of the
block iterative schemes is that the Thomas algorithm can be used to
compute numerical solutions of the sequence of iterations in the
same fashion as for one-dimensional problems. The block iterative
schemes are compared with the point monotone iterative schemes of
Picard, Jacobi and Gauss-Seidel, and various theoretical comparison
results among these monotone iterative schemes are given. These
comparison results demonstrate that the sequence of iterations from
the block iterative schemes converges faster than the corresponding
sequence given by the point iterative schemes. Application of the
iterative schemes is given to a logistic model problem in ecology
and numerical ressults for a test problem with known analytical
solution are given.
Received
August 1, 1993 / Revised version received November 7, 1994 相似文献
2.
Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential
equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK
(MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and
MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C
1 continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter
families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria
are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained
result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific
optimal schemes are then obtained. New MIRK and CMIRK schemes are presented which are superior to those currently available.
Numerical examples are provided to demonstrate the practical improvements that can be obtained by employing the optimal schemes.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
3.
M. M. Khalsaraei F. Khodadosti 《Computational Mathematics and Mathematical Physics》2016,56(2):235-242
When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, a class of nonstandard 2-stage Runge–Kutta methods of order two (we call it nonstandard RK2) is considered. The preservation of some qualitative properties by this class of methods are discussed. In order to illustrate our results, we provide some numerical examples. 相似文献
4.
5.
The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well. 相似文献
6.
In this paper, we present some results of a study, specifically within the framework of symplectic geometry, of difference schemes for numerical solution of the linear Hamiltonian systems. We generalize the Cayley transform with which we can get different types of symplectic schemes. These schemes are various generalizations of the Euler centered scheme. They preserve all the invariant first integrals of the linear Hamiltonian systems. 相似文献
7.
Lalit Kumar Sivaji Ganesh Sista Konijeti Sreenadh 《Mathematical Methods in the Applied Sciences》2020,43(15):9129-9150
The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results. 相似文献
8.
Faruk Civan 《Numerical Methods for Partial Differential Equations》2009,25(2):347-379
Methodology for development of compact numerical schemes by the practical finite‐analytic method (PFAM) is presented for spatial and/or temporal solution of differential equations. The advantage and accuracy of this approach over the conventional numerical methods are demonstrated. In contrast to the tedious discretization schemes resulting from the original finite‐analytic solution methods, such as based on the separation of variables and Laplace transformation, the practical finite‐analytical method is proven to yield simple and convenient discretization schemes. This is accomplished by a special universal determinant construction procedure using the general multi‐variate power series solutions obtained directly from differential equations. This method allows for direct incorporation of the boundary conditions into the numerical discretization scheme in a consistent manner without requiring the use of artificial fixing methods and fictitious points, and yields effective numerical schemes which are operationally similar to the finite‐difference schemes. Consequently, the methods developed for numerical solution of the algebraic equations resulting from the finite‐difference schemes can be readily facilitated. Several applications are presented demonstrating the effect of the computational molecule, grid spacing, and boundary condition treatment on the numerical accuracy. The quality of the numerical solutions generated by the PFAM is shown to approach to the exact analytical solution at optimum grid spacing. It is concluded that the PFAM offers great potential for development of robust numerical schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
9.
侯延仁 《数学物理学报(B辑英文版)》2003,23(2)
Several kind of new numerical schemes for the stationary Navier-Stokes equations based on the virtue of Inertial Manifold and Approximate Inertial Manifold, which we call them inertial algorithms in this paper, together with their error estimations are presented. All these algorithms are constructed under an uniform frame, that is to construct some kind of new projections for the Sobolev space in which the true solution is sought. It is shown that the proposed inertial algorithms can greatly improve the convergence rate of the standard Galerkin approximate solution with lower computing effort. And some numerical examples are also given to verify results of this paper. 相似文献
10.
1.引 言本文我们将考虑非线性Cahn—Hilliard方程的初边值问题 相似文献
11.
Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional (2-D) and three dimensional (3-D) space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis. 相似文献
12.
该文讨论用Legendre拟谱方法数值求解非线性Cahn Hilliard方程的Dirichlet问题.建立了其半离散和全离散逼近格式,它们保持原问题能量耗散的性质.证明了离散解的存在唯一性,并给出了最佳误差估计.数值实验也证实了我们的结果. 相似文献
13.
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. 相似文献
14.
Alexander G. Ramm Alexandra B. Smirnova 《Numerical Functional Analysis & Optimization》2013,34(3-4):317-332
A two-step iterative process for the numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization parameter is introduced. A convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments practical recommendations for the choice of the regularization parameter are given. Some other iterative schemes are considered. 相似文献
15.
Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions 总被引:3,自引:0,他引:3
In this work, the method of radial basis functions is used for finding the solution of an inverse problem with source control parameter. Because a much wider range of physical phenomena are modelled by nonclassical parabolic initial-boundary value problems, theoretical behavior and numerical approximation of these problems have been active areas of research. The radial basis functions (RBF) method is an efficient mesh free technique for the numerical solution of partial differential equations. The main advantage of numerical methods which use radial basis functions over traditional techniques is the meshless property of these methods. In a meshless method, a set of scattered nodes are used instead of meshing the domain of the problem. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes. 相似文献
16.
S. L. Yang 《计算数学(英文版)》1996,14(4):311-318
A simple one-dimensional $2\times 2$ hyperbolic system is considered in the paper. The model contains a linear hyperbolic equation, as well as a hyperbolic equation of which the coefficients are about the solution of the linear one. The exact solution is presented and discussed, then numerical experiments are given by TVD (or MmB) type schemes for Riemann problems. From the results, we know that the solutions do have $\delta-$waves for some suitable initial data. 相似文献
17.
F. Costabile A. Napoli 《Applied Numerical Mathematics》2011,61(2):160-169
Economical Runge-Kutta schemes for the numerical solution of Stratonovich stochastic differential equations are proposed. The methods have strong global order 1. Numerical stability is studied and some examples are presented to support the theoretical results. 相似文献
18.
C. V. Pao 《Numerical Methods for Partial Differential Equations》1998,14(3):339-351
In this article we use the monotone method for the computation of numerical solutions of a nonlinear reaction-diffusion-convection problem with time delay. Three monotone iteration processes for a suitably formulated finite-difference system of the problem are presented. It is shown that the sequence of iteration from each of these iterative schemes converges from either above or below to a unique solution of the finite-difference system without any monotone condition on the nonlinear reaction function. An analytical comparison result among the three processes of iterations is given. Also given is the application of the iterative schemes to some model problems in population dynamics, including numerical results of a model problem with known analytical solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 339–351, 1998 相似文献
19.
In this article, three difference schemes of the Ginzburg‐Landau Equation in two dimensions are presented. In the three schemes, the nonlinear term is discretized such that nonlinear iteration is not needed in computation. The plane wave solution of the equation is studied and the truncation errors of the three schemes are obtained. The three schemes are unconditionally stable. The stability of the two difference schemes is proved by induction method and the time‐splitting method is analysized by linearized analysis. The algebraic multigrid method is used to solve the three large linear systems of the schemes. At last, we compute the plane wave solution and some dynamics of the equation. The numerical results demonstrate that our schemes are reliable and efficient. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011py; 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011 相似文献
20.
C. V. Pao 《Numerical Methods for Partial Differential Equations》2001,17(4):347-368
The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001 相似文献