Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation

Recently, Kulikov presented the idea of double quasi-consistency, which facilitates global error estimation and control, considerably. More precisely, a local error control implemented in such methods plays a part of global error control at the same time. However, Kulikov studied only Nordsieck formulas and proved that there exists no doubly quasi-consistent scheme among those methods.Here, we prove that the class of doubly quasi-consistent formulas is not empty and present the first example of such sort. This scheme belongs to the family of superconvergent explicit two-step peer methods constructed by Weiner, Schmitt, Podhaisky and Jebens. We present a sample of s-stage doubly quasi-consistent parallel explicit peer methods of order s−1 when s=3. The notion of embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi-consistent peer method and, hence, its global error at the same time. Numerical examples of this paper confirm clearly that the usual local error control implemented in doubly quasi-consistent numerical integration techniques is capable of producing numerical solutions for user-supplied accuracy conditions in automatic mode.     

Methods for parallel integration of stiff systems of odes

This paper presents a class of parallel numerical integration methods for stiff systems of ordinary differential equations which can be partitioned into loosely coupled sub-systems. The formulas are called decoupled backward differentiation formulas, and they are derived from the classical formulas by restricting the implicit part to the diagnonal sub-system. With one or several sub-systems allocated to each processor, information only has to be exchanged after completion of a step but not during the solution of the nonlinear algebraic equations.The main emphasis is on the formula of order 1, the decoupled implicit Euler formula. It is proved that this formula even for a wide range of multirate formulations has an asymptotic global error expansion permitting extrapolation. Besides, sufficient conditions for absolute stability are presented.     

On implicit Runge-Kutta methods with high stage order

It is well known that high stage order is a desirable property for implicit Runge-Kutta methods. In this paper it is shown that it is always possible to construct ans-stage IRK method with a given stability function and stage orders−1 if the stability function is an approximation to the exponential function of at least orders. It is further indicated how to construct such methods as well as in which cases the constructed methods will be stiffly accurate.     

On symmetrizers for Gauss methods

Summary In this paper the maximum attainable order of a special class of symmetrizers for Gauss methods is studied. In particular, it is shown that a symmetrizer of this type for thes-stage Gauss method can attain order 2s-1 only for 1 s 3, and that these symmetrizers areL-stable. A classification of the maximum attainable order of symmetrizers for some higher stages is presented. AnL-stable symmetrizer is also shown to exist for each of the methods studied.     

Superconvergent explicit two-step peer methods

We consider explicit two-step peer methods for the solution of nonstiff differential systems. By an additional condition a subclass of optimally zero-stable methods is identified that is superconvergent of order p=s+1$p=s+1$, where s$s$ is the number of stages. The new condition allows us to reduce the number of coefficients in a numerical search for good methods. We present methods with 4–7 stages which are tested in FORTRAN90 and compared with DOPRI5 and DOP853. The results confirm the high potential of the new class of methods.     

Relationships among some classes of implicit Runge-Kutta methods and their stability functions

In this paper we apply the theory for implicit Runge-Kutta methods presented by Stetter to a number of subclasses of methods that have recently been discussed in the literature. We first show how each of these classes can be expressed within this theoretical framework and from this we are able to establish a number of relationships among these classes. In addition to improving the current state of understanding of these methods, their expression within this theoretical framework makes it possible for us to obtain results giving general forms for their stability functions.This work was supported by the Natural Sciences and Engineering Research Council of Canada.     

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.Work supported by the Ministero della Ricerca Scientifica, 40% project, and C.N.R. (contract of research # 92.00535.01).     

Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.     

A note on pseudo-symplectic Runge-Kutta methods

Aubry and Chartier introduced (1998) the concept of pseudo-symplecticness in order to construct explicit Runge-Kutta methods, which mimic symplectic ones. Of particular interest are methods of order (p, 2p), i.e., of orderp and pseudo-symplecticness order 2p, for which the growth of the global error remains linear. The aim of this note is to show that the lower bound for the minimal number of stages can be achieved forp=4 andp=5.     

Derivation of continuous explicit two-step Runge–Kutta methods of order three

We describe a construction of continuous extensions to a new representation of two-step Runge–Kutta methods for ordinary differential equations. This representation makes possible the accurate and reliable estimation of local discretization error, facilitates the efficient implementation of these methods in variable stepsize environment, and adapts readily to the numerical solution of a class of delay differential equations. A number of numerical tests carried out on the obtained methods of order 3 with quadratic interpolants show their efficiency and robust performance which allow them to compete with the state-of-the-art dde23 code from Matlab.     

Structure preservation of exponentially fitted Runge–Kutta methods

The preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge–Kutta (EFRK) methods is considered. A complete characterisation of EFRK methods that preserve linear or quadratic invariants is given and, following the approach of Bochev and Scovel [On quadratic invariants and symplectic structure, BIT 34 (1994) 337–345], the sufficient conditions on symplecticity of EFRK methods derived by Van de Vyver [A fourth-order symplectic exponentially fitted integrator, Comput. Phys. Comm. 174 (2006) 255–262] are obtained. Further, a family of symplectic EFRK two-stage methods with order four has been derived. It includes the symplectic EFRK method proposed by Van de Vyver as well as a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. Finally, the results of some numerical experiments are presented to compare the relative merits of several fitted and nonfitted fourth-order methods in the integration of oscillatory systems.     

A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article, we develop an explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter. The parameter is used for the optimization of the method in order to solve efficiently the Schrödinger equation and related oscillatory problems. We evaluate the local truncation error and the interval of periodicity as functions of the parameter. We reveal a direct relationship between the periodicity interval and the local truncation error. We also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature. The analysis and the numerical results help us to determine the optimal values of the parameter, which render the new method highly efficient.     

On some explicit Adams multistep methods for fractional differential equations

In this paper we present a family of explicit formulas for the numerical solution of differential equations of fractional order. The proposed methods are obtained by modifying, in a suitable way, Fractional-Adams–Moulton methods and they represent a way for extending classical Adams–Bashforth multistep methods to the fractional case. The attention is hence focused on the investigation of stability properties. Intervals of stability for k$k$-step methods, k=1,…,5$k=1,\dots ,5$, are computed and plots of stability regions in the complex plane are presented.     

Parallel frontal methods for ODE's

This paper concerns parallel frontal predictor-corrector methods. Order and stability of these methods are investigated, when the corrector is solved both by the fixed point iteration method and by the Newton method.This work has been partially supported by the Italian C.N.R. within the Finalized Project Sistemi Informatici e Calcolo Parallelo.     

A fourth-order Runge–Kutta method based on BDF-type Chebyshev approximations

In this paper we consider a new fourth-order method of BDF-type for solving stiff initial-value problems, based on the interval approximation of the true solution by truncated Chebyshev series. It is shown that the method may be formulated in an equivalent way as a Runge–Kutta method having stage order four. The method thus obtained have good properties relatives to stability including an unbounded stability domain and large α$\alpha$-value concerning A(α)$A(\alpha )$-stability. A strategy for changing the step size, based on a pair of methods in a similar way to the embedding pair in the Runge–Kutta schemes, is presented. The numerical examples reveals that this method is very promising when it is used for solving stiff initial-value problems.     

A stability result for general linear methods with characteristic function having real poles only

It is shown that there exist A-stable multistep formulae, with a characteristic function havings poles, all of which are real, with orderp satisfyingp>s+1. This contradicts the widely held belief thatp=s+1 is the maximum possible order of such a method.     

A criterion forP-stability properties of Runge-Kutta methods

P-stability is an analogous stability property toA-stability with respect to delay differential equations. It is defined by using a scalar test equation similar to the usual test equation ofA-stability. EveryP-stable method isA-stable, but anA-stable method is not necessarilyP-stable. We considerP-stability of Runge-Kutta (RK) methods and its variation which was originally introduced for multistep methods by Bickart, and derive a sufficient condition for an RK method to have the stability properties on the basis of an algebraic characterization ofA-stable RK methods recently obtained by Schere and Müller. By making use of the condition we clarify stability properties of some SIRK and SDIRK methods, which are easier to implement than fully implicit methods, applied to delay differential equations.     

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.     

NP-stability of Runge-Kutta methods based on classical quadrature

Recently Bellen, Jackiewicz and Zennaro have studied stability properties of Runge-Kutta (RK) methods for neutral delay differential equations using a scalar test equation. In particular, they have shown that everyA-stable collocation method isNP-stable, i.e., the method has an analogous stability property toA-stability with respect to the test equation. Consequently, the Gauss, Radau IIA and Lobatto IIIA methods areNP-stable. In this paper, we examine the stability of RK methods based on classical quadrature by a slightly different approach from theirs. As a result, we prove that the Radau IA and Lobatto IIIC methods equipped with suitable continuous extensions are alsoNP-stable by virtue of fundamental notions related to those methods such as simplifying conditions, algebraic stability, and theW-transformation.