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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.
Andreas Rößler 《BIT Numerical Mathematics》2007,47(3):657-680
The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m–dimensional Wiener process is studied. Therefore, a new class of stochastic Runge–Kutta methods is introduced. As the main
novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due
to the author is applied to determine order conditions for the new stochastic Runge–Kutta methods assuring convergence with
order two in the weak sense. Further, some coefficients for second order stochastic Runge–Kutta schemes are calculated explicitly.
AMS subject classification (2000) 65C30, 65L06, 60H35, 60H10 相似文献
3.
For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods
exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal
order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned
Runge–Kutta method with infinitely many stages. 相似文献
4.
In our previous paper [3], the performance of a variable step‐size implementation of Parallel Iterated Methods based on Multistep
Runge–Kutta methods (PIMRK) is far from satisfactory. This is due to the fact that the underlying parameters of the Multistep
Runge–Kutta (MRK) method, and the splitting matrices W that are needed to solve the nonlinear system are designed on a fixed step‐size basis. Similar unsatisfactory results based
on this method were also noted by Schneider [12], who showed that the method is only suitable when the step‐size does not
vary too often. In this paper, we design the Variable step‐size Multistep Runge–Kutta (VMRK) method as the underlying formula
for Parallel Iterated methods. The numerical results show that Parallel Iterated Variable step‐size MRK (PIVMRK) methods improve
substantially on the PIMRK methods and are usually competitive with Parallel Iterated Runge–Kutta methods (PIRKs).
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
5.
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
6.
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
7.
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
8.
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
9.
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
10.
We consider the construction of a special family of Runge–Kutta(RK) collocation methods based on intra-step nodal points ofChebyshev–Gauss–Lobatto type, with A-stability andstiffly accurate characteristics. This feature with its inherentimplicitness makes them suitable for solving stiff initial-valueproblems. In fact, the two simplest cases consist in the well-knowntrapezoidal rule and the fourth-order Runge–Kutta–LobattoIIIA method. We will present here the coefficients up to eighthorder, but we provide the formulas to obtain methods of higherorder. When the number of stages is odd, we have considereda new strategy for changing the step size based on the use ofa pair of methods: the given RK method and a linear multistepone. Some numerical experiments are considered in order to checkthe behaviour of the methods when applied to a variety of initial-valueproblems. 相似文献
11.
Runge-Kutta methods without order reduction for linear initial boundary value problems 总被引:1,自引:0,他引:1
Isaías Alonso-Mallo 《Numerische Mathematik》2002,91(4):577-603
Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization
of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values
in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time
is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta
methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element
techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved.
Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001 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.
General Linear Methods (GLMs) were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep
methods. An extension of GLMs, so-called SGLMs (GLM with second derivative), was introduced to the case in which second derivatives,
as well as first derivatives, can be calculated. In this paper, we introduce the definitions of consistency, stability and
convergence for an SGLM. It will be shown that in SGLMs, stability and consistency together are equivalent to convergence.
Also, by introducing a subclass of SGLMs, we construct methods of this subclass up to the maximal order which possess Runge–Kutta
stability property and A-stability for implicit ones. 相似文献
14.
Optimal explicit Runge–Kutta methods consider more stages in order to include a particular spectrum in their stability domain
and thus reduce time-step restrictions. This idea, so far used mostly for real-line spectra, is generalized to more general
spectra in the form of a thin region. In thin regions the eigenvalues may extend away from the real axis into the imaginary
plane. We give a direct characterization of optimal stability polynomials containing a maximal thin region and calculate these
polynomials for various cases. Semi-discretizations of hyperbolic–parabolic equations are a relevant application which exhibit
a thin region spectrum. As a model, linear, scalar advection–diffusion is investigated. The second-order-stabilized explicit
Runge–Kutta methods derived from the stability polynomials are applied to advection–diffusion and compressible, viscous fluid
dynamics in numerical experiments. Due to the stabilization the time step can be controlled solely from the hyperbolic CFL
condition even in the presence of viscous fluxes. 相似文献
15.
In this paper we consider ordinary differential equations with a known Lyapunov function. We study the use of Runge–Kutta
methods provided with a dense output and a projection technique to preserve any given Lyapunov function. This approach extends
previous work of Grimm and Quispel (BIT 45, 2005), allowing the use of Runge–Kutta methods for which the associated quadrature
formula does not need to have positive or zero coefficients. Some numerical experiments show the good performance of the proposed
technique. 相似文献
16.
Tree and forest spaces, which are at the heart of the theory of Runge–Kutta methods, are formulated recursively, and it is
shown that the forest space is an algebra. To obtain order conditions in a systematic manner, Banach algebras are introduced
to generate both the elementary weights for a general Runge–Kutta method and the corresponding quantities based on the Picard
integral. To connect these two concepts, the Picard integral is written as the limiting case of an s-stage Runge–Kutta method, equivalent to s steps of the Euler method, as s tends to infinity. This approach makes it possible to make direct use of the tree space without going over to the dual space.
By choosing linear combinations of trees, appropriate to a particular application, it is shown how to obtain alternative ways
of writing the order conditions. This leads to a simpler and more direct derivation of particular methods. 相似文献
17.
The De Donder–Weyl (DW) Hamilton–Jacobi equation is investigated in this paper, and the connection between the DW Hamilton–Jacobi
equation and multi-symplectic Hamiltonian system is established. Based on the DW Hamilton–Jacobi theory, generating functions
for multi-symplectic Runge–Kutta (RK) methods and partitioned Runge–Kutta (PRK) methods are presented.
The work is supported by the Foundation of ICMSEC, LSEC, AMSS and CAS, the NNSFC (No.10501050, 19971089 and 10371128) and
the Special Funds for Major State Basic Research Projects of China (2005CB321701). 相似文献
18.
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
19.
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
20.
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