Solving the order reduction phenomenon in variable step size quasi-consistent Nordsieck methods

The phenomenon is studied of reducing the order of convergence by one in some classes of variable step size Nordsieck formulas as applied to the solution of the initial value problem for a first-order ordinary differential equation. This phenomenon is caused by the fact that the convergence of fixed step size Nordsieck methods requires weaker quasi-consistency than classical Runge-Kutta formulas, which require consistency up to a certain order. In other words, quasi-consistent Nordsieck methods on fixed step size meshes have a higher order of convergence than on variable step size ones. This fact creates certain difficulties in the automatic error control of these methods. It is shown how quasi-consistent methods can be modified so that the high order of convergence is preserved on variable step size meshes. The regular techniques proposed can be applied to any quasi-consistent Nordsieck methods. Specifically, it is shown how this technique performs for Nordsieck methods based on the multistep Adams-Moulton formulas, which are the most popular quasi-consistent methods. The theoretical conclusions of this paper are confirmed by the numerical results obtained for a test problem with a known solution.     

Scale and modify for the second and third order BDF methods

This paper investigates a ‘scale and modify’ technique used with variable stepsize BDF methods. When the stepsize is changed using the usual scaling procedure for Nordsieck methods, there can be adverse affects on the stability unless a severe restriction is placed on the allowable stepsize ratios. However, a modification to this scaling procedure may extend the range of permissible stepsize ratios. Results for the Nordsieck form of the second and third order methods indicate that this may be possible.     

Mono-implicit Runge-Kutta schemes for the parallel solution of initial value ODEs

Among the numerical techniques commonly considered for the efficient solution of stiff initial value ordinary differential equations are the implicit Runge-Kutta (IRK) schemes. The calculation of the stages of the IRK method involves the solution of a nonlinear system of equations usually employing some variant of Newton's method. Since the costs of the linear algebra associated with the implementation of Newton's method generally dominate the overall cost of the computation, many subclasses of IRK schemes, such as diagonally implicit Runge-Kutta schemes, singly implicit Runge-Kutta schemes, and mono-implicit (MIRK) schemes, have been developed to attempt to reduce these costs. In this paper we are concerned with the design of MIRK schemes that are inherently parallel in that smaller systems of equations are apportioned to concurrent processors. This work builds on that of an earlier investigation in which a special subclass of the MIRK formulas were implemented in parallel. While suitable parallelism was achieved, the formulas were limited to some extent because they all had only stage order 1. This is of some concern since in the application of a Runge-Kutta method to a system of stiff ODEs the phenomenon of order reduction can arise; the IRK method can behave as if its order were only its stage order (or its stage order plus one), regardless of its classical order. The formulas derived in the current paper represent an improvement over the previous investigation in that the full class of MIRK formulas is considered and therefore it is possible to derive efficient, parallel formulas of orders 2, 3, and 4, having stage orders 2 or 3.     

On Nordsieck's method for the numerical solution of ordinary differential equations

The connection between the class of methods suggested by Nordsieck and the class of linear multi-step methods is examined. It is shown that the starting procedure suggested by Nordsieck is specially suited to the Adams method.     

On quasi-consistent integration by Nordsieck methods

In this paper we study advantages of numerical integration by quasi-consistent Nordsieck formulas. All quasi-consistent numerical methods possess at least one important property for practical use, which has not attracted attention yet, i.e. the global error of a quasi-consistent method has the same order as its local error. This means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms. In other words, the global error control can be as cheap as the local error control in the methods under discussion.     

A Class of Multistep Method Containing Second Order Derivatives for Solving Stiff Ordinary Differential Equations

In this paper a general k-step k-order multistep method containing derivatives of second order is given. In particular, a class of k-step (k+1)th-order stiff stable multistep methods for k=3-9 is constructed. Under the same accuracy, these methods are possessed of a larger absolute stability region than those of Gear's [1] and Enright's [2]. Hence they are suitable for solving stiff initial value problems in ordinary differential equations.     

Linearly-implicit two-step methods and their implementation in Nordsieck form

We consider the Nordsieck representation of well-known linearly-implicit two-step methods in order to facilitate stepsize changes. These methods are treated as implicit schemes with one Newton step with a special predictor. A new predictor with better accuracy is discussed. Methods with 4–7 stages are tested and compared with RODAS. The numerical results show the potential of the Nordsieck implementation of the methods.     

Local error and variable order Adams codes

Error estimates needed for the selection of order in Adams codes are studied in the presence of propagated errors. A new derivation of the Nordsieck form of the Adams methods reveals that codes based on this form have changed order incorrectly and shows how to do it properly.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

Error analysis and applications of the Fourier-Galerkin Runge-Kutta schemes for high-order stiff PDEs

An integrating factor mixed with Runge-Kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. In this paper, the novel hybrid Fourier-Galerkin Runge-Kutta scheme, with the aid of an integrating factor, is proposed to solve nonlinear high-order stiff PDEs. Error analysis and properties of the scheme are provided. Application to the approximate solution of the nonlinear stiff Korteweg-de Vries (the 3rd order PDE, dispersive equation), Kuramoto-Sivashinsky (the 4th order PDE, dissipative equation) and Kawahara (the 5th order PDE) equations are presented. Comparisons are made between this proposed scheme and the competing method given by Kassam and Trefethen. It is found that for KdV, KS and Kawahara equations, the proposed method is the best.     

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.     

Variable-Order ESIRK Methods for Stiff Differential Equations

ESIRK methods (Effective order Singly-Implicit Runge–Kutta methods) have been shown to be efficient for the numerical solution of stiff differential equations. In this paper, we consider a new implementation of these methods with a variable order strategy. We show that the efficiency of the ESIRK method for stiff problems is improved by using the proposed variable order schemes.     

The numerical solution of non-linear partial differential equations by the method of lines

A new method for solving non-linear parabolic partial differential equations, based on the method of lines, is developed. Second order and fourth order finite difference approximations to the spatial derivatives are used, and in each case the band structure of the associated Jacobian is exploited. This, together with the use of Gear's fourth order stiffly stable method in Nordsieck form, leads to a method which compares favourably with the respected Sincovec and Madsen method on Burgers' equation. The method has been tested on a number of difficult problems in the literature and has proved to be most successful.     

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.     

Construction of a multirate RODAS method for stiff ODEs

Multirate time stepping is a numerical technique for efficiently solving large-scale ordinary differential equations (ODEs) with widely different time scales localized over the components. This technique enables one to use large time steps for slowly varying components, and small steps for rapidly varying ones. Multirate methods found in the literature are normally of low order, one or two. Focusing on stiff ODEs, in this paper we discuss the construction of a multirate method based on the fourth-order RODAS method. Special attention is paid to the treatment of the refinement interfaces with regard to the choice of the interpolant and the occurrence of order reduction. For stiff, linear systems containing a stiff source term, we propose modifications for the treatment of the source term which overcome order reduction originating from such terms and which we can implement in our multirate method.     

Order results for Rosenbrock type methods on classes of stiff equations

Summary In this paper we give conditions for theB-convergence of Rosenbrock type methods when applied to stiff semi-linear systems. The convergence results are extended to stiff nonlinear systems in singular perturbation form. As a special case partitioned methods are considered. A third order method is constructed.Dedicated to the memory of Professor Lothar Collatz     

Singly Diagonally Implicit Runge–Kutta Methods with an Explicit First Stage

The purpose of this paper is to construct methods for solving stiff ODEs, in particular singular perturbation problems. We consider embedded pairs of singly diagonally implicit Runge–Kutta methods with an explicit first stage (ESDIRKs). Stiffly accurate pairs of order 3/2, 4/3 and 5/4 are constructed.     

求解刚性Volterra延迟积分微分方程的隐显单支方法的稳定性与误差分析

本文主要研究用隐显单支方法求解一类刚性Volterra延迟积分微分方程初值问题时的稳定性与误差分析.我们获得并证明了结论:若隐显单支方法满足2阶相容条件,且其中的隐式单支方法是A-稳定的,则隐显单支方法是2阶收敛且关于初值扰动是稳定的.最后,由数值算例验证了相关结论.     

随机延迟微分方程平衡方法的均方收敛性与稳定性

本文讨论求解刚性随机延迟微分方程的平衡方法.证明了随机延迟微分方程平衡方法的均方收敛阶为1/2.给出了线性随机延迟微分方程平衡方法均方稳定的条件.     

A generalization of singly-implicit methods

Singly-implicit Runge-Kutta methods are characterized by a one-point spectrum property of the coefficient matrix. If a method of this type is also a collocation method, then its abscissae are proportional to the zeros of a Laguerre polynomial. The generalization introduced here is a multistep method in the style of Nordsieck and also a multistage method under the one-point spectrum constraint. It is found that much of the theory of singly-implicit methods carries over but with Laguerre polynomials replaced by their usual generalizations. Amongst the formal properties of the new method which are studied is a derivation of the similarity transformations which allow their efficient implementation. A preliminary investigation is made of the stability of the new methods.