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1.
Hong Wang Mohamed Al-Lawatia Aleksey S. Telyakovskiy 《Numerical Methods for Partial Differential Equations》1997,13(6):617-661
We develop two Runge–Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian–Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov–Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge–Kutta methods developed in this article. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 617–661, 1997 相似文献
2.
Runge–Kutta methods that require only two memory locations per variable and have strong local order γ=1.5 for non-commutative systems of stochastic differential equations driven by one Wiener process are devised in this paper. A first step in the derivation is to extend existing deterministic methods to the commutative stochastic case, for which higher accuracy is also obtained. Numerical results are presented to validate the approach. 相似文献
3.
This paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro-differential equations. The dissipativity results are obtained for algebraically stable Runge–Kutta methods when they are applied to above problems. 相似文献
4.
Generalized Kronecker product splitting iteration for the solution of implicit Runge–Kutta and boundary value methods 下载免费PDF全文
Hao Chen 《Numerical Linear Algebra with Applications》2015,22(2):357-370
This paper is concerned with a generalization of the Kronecker product splitting (KPS) iteration for solving linear systems arising in implicit Runge–Kutta and boundary value methods discretizations of ordinary differential equations. It is shown that the new scheme can outperform the standard KPS method in some situations and can be used as an effective preconditioner for Krylov subspace methods. Numerical experiments are presented to demonstrate the effectiveness of the methods. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
5.
In this paper we consider the analytical and numerical stability regions of Runge-Kutta methods for differential equations with piecewise continuous arguments with complex coefficients. It is shown that the analytical stability region contained in the numerical one is violated for a∉R by the geometric technique. And we give the conditions under which the analytical stability region is contained in the union of the numerical stability regions of two Runge-Kutta methods. At last, some experiments are given. 相似文献
6.
In this paper, we give a definition of pseudo almost periodic sequence and study the existence of pseudo almost periodic sequence to difference equation. Based on these, we investigate the existence of pseudo almost periodic solutions to differential equations with piecewise constant arguments which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems. 相似文献
7.
Alberto Cabada Juan Bosco Ferreiro 《Journal of Mathematical Analysis and Applications》2011,380(1):124-136
This paper is devoted to the study of differential equations with piecewise constant arguments coupled with nonlinear boundary value conditions. Under suitable assumptions on the data of the equation, by means of the method of (weakly coupled) lower and upper solutions, we derive the existence of extremal solutions and extremal quasi-solutions. Moreover some results are given concerning the uniqueness of solutions. Furthermore, we deduce some maximum principles related to the linear equation which allow us to develop the monotone iterative method. Some illustrative examples are also presented. 相似文献
8.
Construction of highly stable parallel two-step Runge–Kutta methods for delay differential equations
Z. Bartoszewski Z. Jackiewicz 《Journal of Computational and Applied Mathematics》2008,220(1-2):257-270
It is shown that any A-stable two-step Runge–Kutta method of order and stage order for ordinary differential equations can be extended to the P-stable method of uniform order for delay differential equations. 相似文献
9.
This article is concerned with the stability analysis of the analytic and numerical solutions of a partial differential equation with piecewise constant arguments of mixed type. First, by means of the similar technique in Wiener and Debnath [Int J Math Math Sci 15 (1992), 781–788], the sufficient conditions under which the analytic solutions asymptotically stable are obtained. Then, the θ‐methods are used to solve the above‐mentioned equation, the sufficient conditions for the asymptotic stability of numerical methods are derived. Finally, some numerical experiments are given to demonstrate the conclusions.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1‐16, 2014 相似文献
10.
In this paper we study numerical methods for addressing hybrid fuzzy differential equations by an application of the Runge–Kutta method for fuzzy differential equations using the Seikkala derivative. We state a convergence result and give a numerical example to illustrate the theory. 相似文献
11.
Trygve K. Nilssen Gunnar A. Staff Kent‐Andre Mardal 《Numerical Methods for Partial Differential Equations》2011,27(5):1290-1312
The partial differential equation part of the bidomain equations is discretized in time with fully implicit Runge–Kutta methods, and the resulting block systems are preconditioned with a block diagonal preconditioner. By studying the time‐stepping operator in the proper Sobolev spaces, we show that the preconditioned systems have bounded condition numbers given that the Runge–Kutta scheme is A‐stable and irreducible with an invertible coefficient matrix. A new proof of order optimality of the preconditioners for the one‐leg discretization in time of the bidomain equations is also presented. The theoretical results are verified by numerical experiments. Additionally, the concept of weakly positive‐definite matrices is introduced and analyzed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq Eq 27: 1290–1312, 2011 相似文献
12.
We study scalar advanced and delayed differential equations with piecewise constant generalized arguments, in short DEPCAG of mixed type, that is, the arguments are general step functions. It is shown that the argument deviation generates, under certain conditions, oscillations of the solutions, which is an impossible phenomenon for the corresponding equation without the argument deviations. Criteria for existence of periodic solutions of such equations are discussed. New criteria extend and improve related results reported in the literature. The efficiency of our criteria is illustrated via several numerical examples and simulations. 相似文献
13.
Green's function for second-order periodic boundary value problems with piecewise constant arguments
Juan J. Nieto 《Journal of Mathematical Analysis and Applications》2005,304(1):33-57
We obtain, under suitable conditions, the Green's function to express the unique solution for a second-order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function. This expression is given in terms of the solutions for certain associated problems. The sign of the solution is determined taking into account the sign of that Green's function. 相似文献
14.
In this work, we present some existence theorems of weighted pseudo almost periodic solutions for N-th order neutral differential equations with piecewise constant argument by means of weighted pseudo almost periodic solutions of relevant difference equations. 相似文献
15.
PIAO Daxiong 《中国科学A辑(英文版)》2001,44(9)
In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences. 相似文献
16.
Alberto Cabada Juan B Ferreiro 《Journal of Mathematical Analysis and Applications》2004,291(2):690-697
In this paper we obtain the expression of the Green's function related with a first order periodic differential equation with piecewise constant argument. We derive comparison results for the treated linear operator by studying the sign of the obtained Green's function. 相似文献
17.
Numerical solution of evolutionary integral equations with completely monotonic kernel by Runge–Kutta convolution quadrature 下载免费PDF全文
Da Xu 《Numerical Methods for Partial Differential Equations》2015,31(1):105-142
We study the numerical solutions of the initial boundary value problems for the Volterra‐type evolutionary integal equations, in which the integral operator is a convolution product of a completely monotonic kernel and a positive definite operator, such as an elliptic partial‐differential operator. The equation is discretized in time by the Runge–Kutta convolution quadrature. Error estimates are derived and numerical experiments reported. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 105–142, 2015 相似文献
18.
In this paper we give a brief overview of the application of delay differential equations with piecewise constant arguments
(EPCAs) for obtaining numerical approximation of delay differential equations, and we show that this method can be used for
numerical approximation in a class of age-dependent population models. We also formulate an open problem for stability and
oscillation of a class of linear delay equations with continuous and piecewise constant arguments.
This research was partially supported by Hungarian NFSR Grant No. T046929. 相似文献
19.
Daxiong Piao 《中国科学A辑(英文版)》2001,44(9):1156-1161
In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost
periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the
notion of pseudo almost periodic vector sequences 相似文献
20.
The improved linear multistep methods for differential equations with piecewise continuous arguments
This paper deals with the convergence of the linear multistep methods for the equation x′(t) = ax(t) + a0x([t]). Numerical experiments demonstrate that the 2-step Adams-Bashforth method is only of order p = 0 when applied to the given equation. An improved linear multistep methods is constructed. It is proved that these methods preserve their original convergence order for ordinary differential equations (ODEs) and some numerical experiments are given. 相似文献