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1.
Avoiding Order Reduction of Runge–Kutta Discretizations for Linear Time-Dependent Parabolic Problems
A technique is developed in this paper to avoid order reduction when discretizing linear parabolic problems with time dependent
operator using Runge–Kutta methods in time and standard schemes in space. In an abstract framework, the boundaries of the
stages of the Runge–Kutta method which would completely avoid the order reduction are given. Then, the possible practical
implementations for the calculus of those boundaries from the given data are studied, and the full discretization is completely
analyzed. Some numerical experiments are included.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
2.
We construct A‐stable and L‐stable diagonally implicit Runge–Kutta methods of which the diagonal vector in the Butcher matrix
has a minimal maximum norm. If the implicit Runge–Kutta relations are iteratively solved by means of the approximately factorized
Newton process, then such iterated Runge–Kutta methods are suitable methods for integrating shallow water problems in the
sense that the stability boundary is relatively large and that the usually quite fine vertical resolution of the discretized
spatial domain is not involved in the stability condition.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
3.
Summary. This paper studies the convergence properties of general Runge–Kutta methods when applied to the numerical solution of a
special class of stiff non linear initial value problems. It is proved that under weaker assumptions on the coefficients of
a Runge–Kutta method than in the standard theory of B-convergence, it is possible to ensure the convergence of the method
for stiff non linear systems belonging to the above mentioned class. Thus, it is shown that some methods which are not algebraically
stable, like the Lobatto IIIA or A-stable SIRK methods, are convergent for the class of stiff problems under consideration.
Finally, some results on the existence and uniqueness of the Runge–Kutta solution are also presented.
Received November 18, 1996 / Revised version received October 6, 1997 相似文献
4.
Georgios E. Zouraris 《Numerische Mathematik》1997,77(1):123-142
Summary. We analyze a class of algebraically stable Runge–Kutta/standard Galerkin methods for inhomogeneous linear parabolic equations,
with time–dependent coefficients, under Neumann boundary conditions, and derive an error bound of provided is bounded.
Received June 25, 1994 / Revised version received February 26, 1996 相似文献
5.
We investigate conservative properties of Runge–Kutta methods for Hamiltonian partial differential equations. It is shown
that multi-symplecitic Runge–Kutta methods preserve precisely the norm square conservation law. Based on the study of accuracy
of Runge–Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy
of conservation laws of energy and momentum for Hamiltonian PDEs under Runge–Kutta discretizations.
J. Hong, S. Jiang and C. Li are supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS,
the NNSFC (No. 19971089, No. 10371128, No. 60771054) and the Special Funds for Major State Basic Research Projects of China
2005CB321701. 相似文献
6.
J. G. Verwer 《Numerische Mathematik》2009,112(3):485-507
We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous
wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated
in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer
efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special
attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.
相似文献
7.
Panagiotis Chatzipantelidis 《Numerische Mathematik》1999,82(3):409-432
We introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in . The method is based on nonuniform triangulations with piecewise linear nonconforming spaces. We prove optimal order error
estimates in the –norm and a mesh dependent –norm.
Received September 10, 1997 / Revised version received March 18, 1998 相似文献
8.
Solving high-order or mixed-order boundary value problems by general purpose software often requires the system to be first
converted to a larger equivalent first-order system. The cost of solving such problems is generally O(m
3), where m is the dimension of the equivalent first-order system. In this paper, we show how to reduce this cost by exploiting the special
structure the “equivalent” first-order system inherits from the original associated mixed-order system. This technique applies
to a broad class of boundary value methods. We illustrate the potential benefits by considering in detail a general purpose
Runge–Kutta method and a multiple shooting method.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
9.
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
10.
Summary. We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations
and extend thus recent results concerning the discretization of nonlinear parabolic equations. The resulting schemes are linearly
implicit and include as particular cases implicit–explicit multistep schemes as well as the combination of implicit Runge–Kutta
schemes and extrapolation. We establish optimal order error estimates. The abstract results are applied to a third–order evolution
equation arising in the modelling of flow in a fluidized bed. We discretize this equation in space by a Petrov–Galerkin method.
The resulting fully discrete schemes require solving some linear systems to advance in time with coefficient matrices the
same for all time levels.
Received October 22, 2001 / Revised version received April 22, 2002 /
Published online December 13, 2002
Mathematics Subject Classification (1991): Primary 65M60, 65M12; Secondary 65L06
Correspondence to: G. Akrivis 相似文献
11.
In this paper we present an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes
equation, governing American and European option pricing. The method is based on a fitted finite volume spatial discretization
and an implicit time stepping technique. The analysis is performed within the framework of the vertical method of lines, where
the spatial discretization is formulated as a Petrov–Galerkin finite element method with each basis function of the trial
space being determined by a set of two-point boundary value problems. We establish the stability and an error bound for the
solutions of the fully discretized system. Numerical results are presented to validate the theoretical results. 相似文献
12.
Chengming Huang 《Numerische Mathematik》2009,111(3):377-387
This paper is concerned with the study of the delay-dependent stability of Runge–Kutta methods for delay differential equations.
First, a new sufficient and necessary condition is given for the asymptotic stability of analytical solution. Then, based
on this condition, we establish a relationship between τ(0)-stability and the boundary locus of the stability region of numerical methods for ordinary differential equations. Consequently,
a class of high order Runge–Kutta methods are proved to be τ(0)-stable. In particular, the τ(0)-stability of the Radau IIA methods is proved. 相似文献
13.
In this article, we study positivity properties of exponential Runge–Kutta methods for abstract evolution equations. Our problem
class includes linear ordinary differential equations with a time-dependent inhomogeneity. We show that the order of a positive
exponential Runge–Kutta method cannot exceed two. On the other hand there exist second-order methods that preserve positivity
for linear problems. We give some examples for the latter. 相似文献
14.
An error analysis of Runge–Kutta convolution quadrature is presented for a class of non-sectorial operators whose Laplace
transform satisfies, besides the standard assumptions of analyticity in a half-plane Re s > σ
0 and a polynomial bound
\operatornameO(|s|m1){\operatorname{O}(|s|^{\mu_1})} there, the stronger polynomial bound
\operatornameO(sm2){\operatorname{O}(s^{\mu_2})} in convex sectors of the form
|\operatorname*arg s| £ p/2-q{|\operatorname*{arg} s| \leq \pi/2-\theta} for θ > 0. The order of convergence of the Runge–Kutta convolution quadrature is determined by μ
2 and the underlying Runge–Kutta method, but is independent of μ
1. Time domain boundary integral operators for wave propagation problems have Laplace transforms that satisfy bounds of the
above type. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour
of Runge–Kutta convolution quadrature for this class of applications. Our results show in particular that the full classical
order of the Runge–Kutta method is attained away from the scattering boundary. 相似文献
15.
An error analysis is given for convolution quadratures based on strongly A-stable Runge–Kutta methods, for the non-sectorial
case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation
depends on the classical order and stage order of the Runge–Kutta method and on the growth exponent of the Laplace transform.
Numerical experiments with convolution quadratures based on the Radau IIA methods are given on an example of a time-domain
boundary integral operator. 相似文献
16.
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
17.
Summary. Our task in this paper is to present a new family of methods of the Runge–Kutta type for the numerical integration of perturbed
oscillators. The key property is that those algorithms are able to integrate exactly, without truncation error, harmonic oscillators,
and that, for perturbed problems the local error contains the perturbation parameter as a factor. Some numerical examples
show the excellent behaviour when they compete with Runge–Kutta–Nystr?m type methods.
Received June 12, 1997 / Revised version received July 9, 1998 相似文献
18.
Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non
necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous
one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward
Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity,
of the exact solution. Numerical illustrations are provided. 相似文献
19.
J.C. Butcher 《Numerical Algorithms》1998,17(3-4):193-221
Almost Runge–Kutta methods (or “ARK methods”) have many of the advantages of Runge–Kutta methods but, for many problems, are
capable of greater accuracy. In this paper a complete classification of fourth order ARK methods with 4 stages is presented.
The paper also analyzes fifth order methods with 5 or with 6 stages. Some limited numerical experiments show that the new
methods are capable of excellent performance, comparable to that of known highly efficient Runge–Kutta methods.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
20.
The representation of order conditions for general linear methods formulated using an algebraic theory by Butcher, and the
alternative using B-series by Hairer and Wanner for treating vector initial value problems in ordinary differential equations
are well-known. Each relies on a recursion over rooted trees; yet tractable forms—for example, those which may be solved to
yield particular methods—often are obtained only after extensive computation. In contrast, for Runge–Kutta methods, tractable
forms have been used by various authors for obtaining methods. Here, the corresponding recursion formula for two-step Runge–Kutta
methods is revised to yield tractable forms which may be exploited to derive such methods and to motivate the selection of
efficient algorithms in an obvious way. The new recursion formula is utilized in a MAPLE code. 相似文献