共查询到20条相似文献,搜索用时 10 毫秒
1.
Automatic Control and Adaptive Time-Stepping 总被引:1,自引:0,他引:1
Gustaf Söderlind 《Numerical Algorithms》2002,31(1-4):281-310
Adaptive time-stepping is central to the efficient solution of initial value problems in ODEs and DAEs. The error committed in the discretization method primarily depends on the time-step size h, which is varied along the solution in order to minimize the computational effort subject to a prescribed accuracy requirement. This paper reviews the recent advances in developing local adaptivity algorithms based on well established techniques from linear feedback control theory, which is introduced in a numerical context. Replacing earlier heuristics, this systematic approach results in a more consistent and robust performance. The dynamic behaviour of the discretization method together with the controller is analyzed. We also review some basic techniques for the coordination of nonlinear equation solvers with the primary stepsize controller in implicit time-stepping methods. 相似文献
2.
Separation of variables is a well‐known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude–shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well‐established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi‐implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude–shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
3.
H. Lamba 《BIT Numerical Mathematics》2000,40(2):314-335
Initial value problems for ODEs are often solved numerically using adaptive timestepping algorithms. These algorithms are controlled by a user-defined tolerance which bounds from above the estimated error committed at each step. We formulate a large class of such algorithms as discrete dynamical systems which are discontinuous and of higher dimension than the underlying ODE. By assuming sufficiently strong finite-time convergence results on some neighbourhood of an attractor of the ODE we prove existence and upper semicontinuity results for a nearby numerical attractor as the tolerance tends to zero.This assumption of sufficiently strong finite-time convergence results is then examined for adaptive algorithms that use a pair of explicit Runge-Kutta methods of different order to estimate the one-step error. For arbitrary Runge-Kutta pairs the necessary finite-time convergence results fail to hold on a set of points in the phase space that includes all the equilibria of the ODE. Therefore, in general, the asymptotic convergence results cannot be applied to attractors containing equilibria. However, for a particular class of Runge-Kutta pairs, the finite-time convergence results can be strengthened to include neighbourhoods of equilibrium points for which the Jacobian is invertible. 相似文献
4.
In biochemical systems some of the chemical species are present with only small numbers of molecules. In this situation discrete and stochasticsimulation approaches are more relevant than continuous and deterministic ones. The fundamental Gillespie’s stochastic simulation algorithm (SSA) accounts for every reaction event, which occurs with a probability determined by the configuration of the system. This approach requires a considerable computational effort for models with many reaction channels and chemical species.In order to improve efficiency, tau-leaping methods represent multiple firings of each reaction during a simulation step by Poisson random variables. For stiff systems the mean of this variable is treated implicitly in order to ensure numerical stability. This paper develops fully implicit tau-leaping-like algorithms that treat implicitly both the mean and the variance of the Poisson variables. The construction is based on adapting weakly convergent discretizations of stochastic differential equations to stochastic chemical kinetic systems. Theoretical analyses of accuracy and stability of the new methods are performed on a standard test problem. Numerical results demonstrate the performance of the proposed tau-leaping methods. 相似文献
5.
Ladislav Adamec 《Applications of Mathematics》1997,42(4):293-309
The aim of the paper is to give some preliminary information concerning a class of nonlinear differential equations often used in physical chemistry and biology. Such systems are often very large and it is well known that where studying properties of such systems difficulties rapidly increase with their dimension. One way how to get over the difficulties is to use special forms of such systems. 相似文献
6.
Ladislav Adamec 《Applications of Mathematics》1998,43(2):111-117
The paper gives the answer to the question of the number and qualitative character of stationary points of an autonomous detailed balanced kinetical system. 相似文献
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This article is concerned with
the asymptotic accuracy of the
Computational Singular Perturbation (CSP)
method developed by Lam and Goussis [The CSP method for simplifying kinetics,
Int. J. Chem. Kin. 26 (1994) 461–486]
to reduce the dimensionality of a
system of chemical kinetics equations.
The method,
which is generally applicable to
multiple-time scale problems
arising in a broad array of scientific disciplines,
exploits the presence of
disparate time scales to model
the dynamics by an evolution equation
on a lower-dimensional slow manifold.
In this article it is shown that
the successive applications
of the CSP algorithm generate,
order by order,
the asymptotic expansion
of a slow manifold.
The results are illustrated
on the Michaelis–Menten–Henri
equations of enzyme kinetics. 相似文献
9.
Urszula Foryś Natalia Jankowska Katarzyna Cytlak Monika Joanna Piotrowska 《Mathematical Methods in the Applied Sciences》2020,43(18):10593-10618
In this paper, a delayed model of interactions between two actors in the context of their internal optimism and pessimism is studied. Considered model is based on the model proposed earlier in the context of romantic relationships. With the use of the system of nonlinear delay differential equations, we describe the change of emotions of two actors. Delays in the inertial component and in the influence function are introduced, and their influence on the system dynamics is investigated focusing on most beneficial meetings for actors. Finally, the modified systems are compared with the nondelayed case, and results are illustrated by numerical solutions for particular investigated scenarios. 相似文献
10.
Hossein Aminikhah Jafar Biazar 《Numerical Methods for Partial Differential Equations》2010,26(2):480-489
In this paper, we introduce a new version of the homotopy perturbation method (NHPM) that efficiently solves linear and non‐linear ordinary differential equations. Several examples, including Euler‐Lagrange, Bernoulli and Ricatti differential equations, are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 相似文献
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12.
Carsten Elsner 《Proceedings of the American Mathematical Society》1999,127(1):139-143
It is shown that the S-chains solving Rubel's universal fourth-order differential equation also satisfy a third-order functional equation.
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14.
研究了红树林自然保护区自然环境和人类社会活动对于生态系统的影响,考虑了生物之间的相互关系,将生物量、生物生长的面积等作为主要指标,建立了常微分方程组模型,对生态系统的变化情况进行了描述,借助稳定性分析对方程进行了研究,并进行了数值模拟。根据理论分析和数值模拟的结果,对保护区的林木恢复工作提出了合理的建议。 相似文献
15.
Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):509-518
The paper describes the general form of an ordinary differential equation of the order n + 1 (n 1) which allows a nontrivial global transformation consisting of the change of the independent variable. A result given by J. Aczél is generalized. A functional equation of the form
where
are given functions,
is solved on
. 相似文献
16.
We establish some uniqueness and existence results for first-order ordinary differential equations with constant-signed discontinuous nonlinear parts. Several examples are given to illustrate the applicability of our work. 相似文献
17.
Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(2):265-278
For linear differential and functional-differential equations of the n-th order criteria of equivalence with respect to the pointwise transformation are derived. 相似文献
18.
Nicolas Champagnat 《偏微分方程通讯》2013,38(5):786-816
We study existence and uniqueness for the classical dynamics of a particle in a force field in the phase space. Through an explicit control on the regularity of the trajectories, we show that this is well posed if the force belongs to the Sobolev space H 3/4. 相似文献
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Ladislav Adamec 《Czechoslovak Mathematical Journal》2001,51(1):121-126
The paper describes asymptotic properties of a strongly nonlinear system
. The existence of an n/2 parametric family of solutions tending to zero is proved. Conditions posed on the system try to be independent of its linear approximation. 相似文献