Error Estimation for Nordsieck Methods

The local discretization errors of general linear methods depend on the sequence of all stepsize ratios and the derivation of the exact formulas for the corresponding error estimates does not seem to be practical. In this paper we will describe an approach in which the estimates of local discretization errors are evaluated numerically as the computation proceeds from step to step.     

Nordsieck Methods with an Off-Step Point

Hybrid methods, incorporating one or more off-step points, are difficult to implement in a variable stepsize situation using the standard representation of input and output data in each step. However, instead of representing this data in terms of solution values and derivative values at a sequence of step points, it is possible to reformulate the method so that it operates on a Nordsieck vector. This has the consequence of reducing stepsize adjustments to nothing more than rescaling the components of the Nordsieck vector. This paper shows how to derive methods in both formulations and considers some implementation details. It is also possible to derive a new type of hybrid method using the Norsieck representation as the starting point and this is also discussed in the paper. The new method is found to have comparable accuracy for corresponding work expended as for standard methods.     

Nordsieck representation of DIMSIMs

A new representation for diagonally implicit multistage integration methods (DIMSIMs) is derived in which the vector of external stages directly approximates the Nordsieck vector. The methods in this formulation are zero-stable for any choice of variable mesh. They are also easy to implement since changing step-size corresponds to a simple rescaling of the vector of external approximations. The paper contains an analysis of local truncation error and of error accumulation in a variable step-size situation. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Reliable Error Estimation for Diagonally Implicit Multistage Integration Methods

A new approach to the estimation of the local discretization error for diagonally implicit multistage integration methods (DIMSIMs) is described. The error estimates that are obtained are very accurate and very reliable for both explicit and implicit methods for any stepsize pattern.This revised version was published online in October 2005 with corrections to the Cover Date.     

Experiments with a variable-order type 1 DIMSIM code

The issues related to the development of a new code for nonstiff ordinary differential equations are discussed. This code is based on the Nordsieck representation of type 1 DIMSIMs, implemented in a variable-step size variable-order mode. Numerical results demonstrate that the error estimation employed in the code is very reliable and that the step and order changing strategies are very robust. This code outperforms the Matlab ode45 code for moderate and stringent tolerances. This revised version was published online in August 2006 with corrections to the Cover Date.     

Nordsieck methods with computationally verified algebraic stability

We describe the search for algebraically stable Nordsieck methods of order p = s and stage order q = p, where s is the number of stages. This search is based on the theoretical criteria for algebraic stability proposed recently by Hill, and Hewitt and Hill, for general linear methods for ordinary differential equations. These criteria, which are expressed in terms of the non-negativity of the eigenvalues of a Hermitian matrix on the unit circle, are then verified computationally for the derived Nordsieck methods of order p ? 2.     

A Note on the Step Size Selection in Adams Multistep Methods

In this paper new algorithms for step size prediction in variable step size Adams methods are proposed. It is shown that, when large step size changes are necessary for an efficient integration, the new algorithms provide a prediction that follows more closely the local error estimation than the standard step size prediction. The new predictors can be considered as a easily computable alternative to the step size predictors given by Willé [9] in terms of differential equations.     

Efficient parallel multivalue hybrid methods for stiff differential equations

A class of efficient parallel multivalue hybrid methods for stiff differential equations are presented, which are all extremely stable at infinity,A-stable for orders 1–3 and A(α)-stable for orders 4–8. Each method of the class can be performed parallelly using two processors with each processor having almost the same computational amount per integration step as a backward differentiation formula (BDF) of the same order with the same stepsize performed in serial, whereas the former has not only much better stability properties but also a computational accuracy higher than the corresponding BDF. Theoretical analysis and numerical experiments show that the methods constructed in this paper are superior in many respects not only to BDFs but also to some other commonly used methods.     

Explicit Nordsieck methods with extended stability regions

We describe the construction of explicit Nordsieck methods of order p and stage order q = p with large regions of absolute stability. We also discuss error propagation and estimation of local discretization errors. The error estimators are derived for examples of general linear methods constructed in this paper. Some numerical experiments are presented which illustrate the effectiveness of proposed methods.     

两类变时间步长的非线性Galerkin算法的稳定性

1．引言近年来,随着计算机的飞速发展,人们越来越关心非线性发展方程解的渐进行为．为了较精确地描述解在时间t→∞时的渐进行为,人们发展了一类惯性算法,即非线性Galerkin算法．该算法是将来解空间分解为低维部分和高维部分,相应的方程可以分别投影到它们上面,它的解也相应地分解为两部分,大涡分量和小涡分量;然后核算法给出大涡分量和小涡分量之间依赖关系的一种近似,以便容易求出相应的近似解．许多研究表明,非线性Galerkin算法比通常的Galerkin算法节省可观的计算量．当数值求解微分方程时,计算机只能对已知数据进行有限位…     

两个带变时间步的两重网格算法的稳定性分析

对用于求解非线性发展方程的两个带变时间步的两重网格算法，对空间变量用有限元离散，对时间变量分别用一阶精度Euler显式和二阶精度半隐式差分格式离散，然后构造两重网格算法，通过深入的稳定性分析，得出本文的算法优于标准全离散有限元算法。     

飞机辅助动力装置引气特性计算方法

作为飞机环控系统与主发动机起动的气源,以目前广泛应用的带负载压气机结构APU(Auxiliary Power Unit)为研究对象,进行引气特性计算模型与计算方法研究。首先介绍了APU结构与引气工作特点,然后分析了建模时喘振控制阀SCV(Surge Control Valve)控制方法与APU共同工作机理,最后采用部件法建立了该类型APU引气计算数学模型。以某型APU为对象进行数值仿真并与实际试车数据比较,计算误差小于3%,表明所采用的建模方法是正确的,所建立的模型能够满足工程需求。      

Explicit General Linear Methods with Inherent Runge–Kutta Stability

A class of general linear methods is derived for application to non-stiff ordinary differential equations. A property known as inherent Runge–Kutta stability guarantees the stability regions of these methods are the same as for Runge–Kutta methods. Methods with this property have high stage order which enables asymptotically correct error estimates and high order interpolants to be computed conveniently. Some preliminary numerical experiments are given comparing these methods with some well known Runge–Kutta methods.     

Construction of General Linear Methods with Runge–Kutta Stability Properties

We describe the construction of explicit general linear methods of order p and stage order q=p with s=p+1 stages which achieve good balance between accuracy and stability properties. The conditions are imposed on the coefficients of these methods which ensure that the resulting stability matrix has only one nonzero eigenvalue. This eigenvalue depends on one real parameter which is related to the error constant of the method. Examples of methods are derived which illustrate the application of the approach presented in this paper.     

Two-Step Runge-Kutta: Theory and Practice

Local and global error for Two-Step Runge-Kutta (TSRK) methods are analyzed using the theory of B-series. Global error bounds are derived in both constant and variable stepsize environments. An embedded TSRK pair is constructed and compared with the RK5(4)6M pair of Dormand and Prince on the DETEST set of problems. Numerical results show that the TSRK performs competitively with the RK method.