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1.
Many iterative processes can be interpreted as discrete dynamical systems and, in certain cases, they correspond to a time
discretization of differential systems. In this paper, we propose to derive iterative schemes for solving linear systems of
equations by modeling the problem to solve as a stable state of a proper differential system; the solution of the original
linear problem is then computed numerically by applying a time marching scheme. We discuss some aspects of this approach,
which allows to recover some known methods but also to introduce new ones. We give convergence results and numerical illustrations.
AMS subject classification 65F10, 65F35, 65L05, 65L12, 65L20, 65N06 相似文献
2.
Functionally-fitted methods are generalizations of collocation techniques to integrate an equation exactly if its solution is a linear combination of a chosen set of basis functions. When these basis functions are chosen as the power functions, we recover classical algebraic collocation methods. This paper shows that functionally-fitted methods can be derived with less restrictive conditions than previously stated in the literature, and that other related results can be derived in a much more elegant way. The novelty in our approach is to fully retain the collocation framework without reverting back into derivations based on cumbersome Taylor series expansions. AMS subject classification (2000) 65L05, 65L06, 65L20, 65L60 相似文献
3.
We consider ordinary differential equations (ODEs) with a known Lyapunov function V. To ensure that a numerical integrator reflects the correct dynamical behaviour of the system, the numerical integrator should
have V as a discrete Lyapunov function. Only second-order geometric integrators of this type are known for arbitrary Lyapunov functions.
In this paper we describe projection-based methods of arbitrary order that preserve any given Lyapunov function.
AMS subject classification (2000) 65L05, 65L06, 65L20, 65P40 相似文献
4.
The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied
to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs).
Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately
smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in
this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential
equations with small noise.
AMS subject classification (2000) 65L06, 65L80, 65L50, 65L05 相似文献
5.
The numerical solution of a linear singularly-perturbed reaction–diffusion two-point boundary value problem is considered.
The method used is adaptive movement of a fixed number of mesh points by monitor-function equidistribution. A partly heuristic
argument based on truncation error analysis leads to several suitable monitor functions, but also shows that the standard
arc-length monitor function is unsuitable for this problem. Numerical results are provided to demonstrate the effectiveness
of our preferred monitor function.
AMS subject classification Primary: 65L50; secondary: 65L10, 65L12, 65L70
Research supported by the Boole Centre for Research in Informatics, National University of Ireland, Cork, Ireland.
Natalia Kopteva: This paper was written while the first author was visiting the Department of Mathematics, National University
of Ireland, Cork, Ireland. 相似文献
6.
High even order generalizations of the traditional upwind method are introduced to solve second order ODE-BVPs without recasting
the problem as a first order system. Both theoretical analysis and numerical comparison with central difference schemes of
the same order show that these new methods may avoid typical oscillations and achieve high accuracy. Singular perturbation
problems are taken into account to emphasize the main features of the proposed methods.
AMS subject classification (2000) 65L10, 65L12, 65L50 相似文献
7.
T. Linss 《BIT Numerical Mathematics》2007,47(2):379-391
A non-monotone FEM discretization of a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits
strong layers is considered. The method is shown to be maximum-norm stable although it is not inverse monotone. Both a priori
and a posteriori error bounds in the maximum norm are derived. The a priori result can be used to deduce uniform convergence
of various layer-adapted meshes proposed in the literature. Numerical experiments complement the theoretical results.
AMS subject classification (2000) 65L10, 65L50, 65L60 相似文献
8.
In this paper, the asymptotical stability of the analytic solution and the numerical methods with constant stepsize for pantograph
equations is investigated using the Razumikhin technique. In particular, the linear pantograph equations with constant coefficients
and variable coefficients are considered. The stability conditions of the analytic solutions of those equations and the numerical
solutions of the θ-methods with constant stepsize are obtained. As a result Z. Jackiewicz’s conjecture is partially proved.
Finally, some experiments are given.
AMS subject classification (2000) 65L02, 65L05, 65L20 相似文献
9.
J. C. Butcher 《BIT Numerical Mathematics》2006,46(3):479-489
The 1976 paper of G. Dahlquist, [13], has had a wide-ranging impact on our understanding of numerical methods for the solution of stiff differential equation systems. The present paper surveys some of the work of Dahlquist in this area. It also shows how this has led to contributions by other authors. In particular, the paper contains a review of non-linear stability for Runge–Kutta and general linear methods. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65L05, 65L06, 65L20 相似文献
10.
Tobin A. Driscoll Folkmar Bornemann Lloyd N. Trefethen 《BIT Numerical Mathematics》2008,48(4):701-723
In Matlab, it would be good to be able to solve a linear differential equation by typing u = L\f, where f, u, and L are representations
of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to
be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described
in which such calculations are indeed possible, at least in one space dimension, based on the previously developed chebfun
system in object-oriented Matlab. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution.
AMS subject classification (2000) 65L10, 65M70, 65N35 相似文献
11.
In recent papers the technique for a local and global error estimation and the local-global step size control were presented
to solve both ordinary differential equations and semi-explicit index 1 differential-algebraic systems by multistep methods
with any reasonable accuracy attained automatically. Now those results are extended to the concept of multistep extrapolation,
and the paper demonstrates with numerical examples how such methods work in practice. Especially, we develop an efficient
technique to calculate higher derivatives of a numerical solution with Hermite interpolating polynomials. The necessary theory
is also provided.
AMS subject classification (2000) 65L06, 65L70, 65L80 相似文献
12.
J. F. B. M. Kraaijevanger 《BIT Numerical Mathematics》1985,25(4):652-666
We present upper bounds for the global discretization error of the implicit midpoint rule and the trapezoidal rule for the case of arbitrary variable stepsizes. Specializing our results for the case of constant stepsizes they easily prove second order optimal B-convergence for both methods.1980 AMS Subject Classification: 65L05, 65L20. 相似文献
13.
We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization.
AMS subject classification 65L10, 65L12, 65L15 相似文献
14.
Classical collocation RK methods are polynomially fitted in the sense that they integrate an ODE problem exactly if its solution
is an algebraic polynomial up to some degree. Functionally fitted RK (FRK) methods are collocation techniques that generalize
this principle to solve an ODE problem exactly if its solution is a linear combination of a chosen set of arbitrary basis
functions. Given for example a periodic or oscillatory ODE problem with a known frequency, it might be advantageous to tune
a trigonometric FRK method targeted at such a problem. However, FRK methods lead to variable coefficients that depend on the
parameters of the problem, the time, the stepsize, and the basis functions in a non-trivial manner that inhibits any in-depth
analysis of the behavior of the methods in general. We present the class of so-called separable basis functions and show how
to characterize the stability function of the methods in this particular class. We illustrate this explicitly with an example
and we provide further insight for separable methods with symmetric collocation points.
AMS subject classification (2000) 65L05, 65L06, 65L20, 65L60 相似文献
15.
To solve ODE systems with different time scales which are localized over the components, multirate time stepping is examined.
In this paper we introduce a self-adjusting multirate time stepping strategy, in which the step size for a particular component
is determined by its own local temporal variation, instead of using a single step size for the whole system. We primarily
consider implicit time stepping methods, suitable for stiff or mildly stiff ODEs. Numerical results with our multirate strategy
are presented for several test problems. Comparisons with the corresponding single-rate schemes show that substantial gains
in computational work and CPU times can be obtained.
AMS subject classification (2000) 65L05, 65L06, 65L50 相似文献
16.
In this paper, we consider a class of explicit exponential integrators that includes as special cases the explicit exponential
Runge–Kutta and exponential Adams–Bashforth methods. The additional freedom in the choice of the numerical schemes allows,
in an easy manner, the construction of methods of arbitrarily high order with good stability properties. We provide a convergence
analysis for abstract evolution equations in Banach spaces including semilinear parabolic initial-boundary value problems
and spatial discretizations thereof. From this analysis, we deduce order conditions which in turn form the basis for the construction
of new schemes. Our convergence results are illustrated by numerical examples.
AMS subject classification (2000) 65L05, 65L06, 65M12, 65J10 相似文献
17.
Summary. We study the numerical solution of singularly perturbed Schr?-dinger equations with time-dependent Hamiltonian. Based on
a reformulation of the equations, we derive time-reversible numerical integrators which can be used with step sizes that are
substantially larger than with traditional integration schemes. A complete error analysis is given for the adiabatic case.
To deal with avoided crossings of energy levels, which lead to non-adiabatic behaviour, we propose an adaptive extension of
the methods which resolves the sharp transients that appear in non-adiabatic state transitions.
Received November 12, 2001 / Revised version received May 8, 2002 / Published online October 29, 2002
Mathematics Subject Classification (1991): 65L05, 65M15, 65M20, 65L70. 相似文献
18.
K. Wright 《BIT Numerical Mathematics》2007,47(1):197-212
Various adaptive methods for the solution of ordinary differential boundary value problems using piecewise polynomial collocation
are considered. Five different criteria are compared using both interval subdivision and mesh redistribution. The methods
are all based on choosing sub-intervals so that the criterion values have (approximately) equal values in each sub-interval.
In addition to the main comparison it is shown by example that at least when accuracy is low then equidistribution may not
give a unique solution.
The main results that using interval size times maximum residual as criterion gives very much better results than using maximum
residual itself. It is also shown that a criterion based on a global error estimate while giving very good results in some
cases, is unsatisfactory in other cases. The other criteria considered are that given by De Boor and the last Chebyshev series
coefficient.
AMS subject classification (2000) 65L10, 65L50, 65L60 相似文献
19.
We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in
the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation
and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic
one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic
mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In
particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods,
the Milne–Simpson method and the BDF method.
AMS subject classification (2000) 60H35, 65C30, 65L06, 65L20 相似文献
20.
Summary. We prove that the numerical solution of partitioned Runge-Kutta methods applied to constrained Hamiltonian systems (e.g., the Rattle algorithm or the Lobatto IIIA–IIIB pair) is formally equal to the exact solution of a constrained Hamiltonian system with a globally defined modified Hamiltonian. This property is essential for a better understanding of their longtime behaviour. As an illustration, the equations of motion of an unsymmetric top are solved using a parameterization with Euler parameters.
Mathematics Subject Classification (2000):65L06, 65L80, 65P10 相似文献