Two-step Runge-Kutta Methods Research Articles

Order conditions for Two-Step Runge-Kutta methods

Order conditions for Two-Step Runge-Kutta methods are derived using the algebraic criteria for order.

A class of explicit parallel two-step Runge-Kutta methods

The aim of the present paper is to construct a class of two-step Runge-Kutta methods of arbitrarily high order for application to parallel computers. Starting with ans-stage implicit two-step Runge-Kutta method of orderp withk=p/2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P(EC) m E mode. In this way, we obtain an explicit two-step Runge-Kutta method that has orderp for allm and that requiresk(m+1) right-hand side evaluations per step of which eachk evaluation can be computed in parallel. By a number of numerical experiments we show the superiority of the parallel predictor-corrector methods proposed here over parallel method available in the literature.

Implementation of two-step Runge-Kutta methods for ordinary differential equations

We investigate the potential for efficient implementation of two-step Runge-Kutta methods (TSRK), a new class of methods introduced recently by Jackiewicz and Tracogna for numerical integration of ordinary differential equations. The implementation issues addressed are the local error estimation, changing stepsize using Nordsieck technique and construction of interpolants. The numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment.

Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations

A general class of variable stepsize continuous two-step Runge-Kutta methods is investigated. These methods depend on stage values at two consecutive steps. The general convergence and order criteria are derived and examples of methods of orderp and stage orderq=p orq=p−1 are given forp≤5. Numerical examples are presented which demonstrate that high order and high stage order are preserved on nonuniform meshes with large variations in ratios between consecutive stepsizes.

Collocation-based two-step Runge-Kutta methods

In this paper we propose a new class of numerical methods of Runge-Kutta type for integrating initial-value problems for first-order ODEs, that is two-step Runge-Kutta (TRK) methods. The advantage of TRK methods is that for a given order of accuracyp we can construct methods of orderp with any desired number of implicit relations. In the very first investigation, the TRK methods are shown to be efficient numerical integration methods.

A General Class of Two-Step Runge–Kutta Methods for Ordinary Differential Equations

A general class of two-step Runge–Kutta methods that depend on stage values at two consecutive steps is studied. These methods are special cases of general linear methods introduced by Butcher and are quite efficient with respect to the number of function evaluations required for a given order. General order conditions are derived using the approach proposed recently by Albrecht, and examples of methods are given up to the order 5. These methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments.

Explicit two-step Runge-Kutta methods

The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order $p\le 5$ the minimal number of stages for explicit TSRK method of order $p$ is equal to the minimal number of stages for explicit Runge-Kutta method of order $p-1$. Numerical results are presented which demonstrate that constant step size TSRK can be both effectively and efficiently used in an Euler equation solver. Furthermore, a comparison with a variable step size formulation shows that in these solvers the variable step size formulation offers no advantages compared to the constant step size implementation.

Variable-Stepsize Explicit Two-Step Runge-Kutta Methods

Variable-step explicit two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are studied. Order conditions are derived and the results about the minimal number of stages required to attain a given order are established up to order five. The existence of embedded pairs of continuous Runge-Kutta methods and two-step Runge-Kutta methods of order $p - 1$ and p is proved. This makes it possible to estimate local discretization error of continuous Runge-Kutta methods without any extra evaluations of the right-hand side of the differential equation. An algorithm to construct such embedded pairs is described, and examples of (3, 4) and (4, 5) pairs are presented. Numerical experiments illustrate that local error estimation of continuous Runge-Kutta methods based on two-step Runge-Kutta methods appears to be almost as reliable as error estimation by Richardson extrapolation, at the same time being much more efficient.

Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods

In this paper it is shown that the local discretization error ofs-stage singly-implicit methods of orderp can be estimated by embedding these methods intos-stage two-step Runge-Kutta methods of orderp+1, wherep=s orp=s+1. These error estimates do not require any extra evaluations of the right hand side of the differential equations. This is in contrast with the error estimation schemes based on embedded pairs of two singly-implicit methods proposed by Burrage.

Variable-stepsize explicit two-step Runge-Kutta methods

Variable-step explicit two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are studied. Order conditions are derived and the results about the minimal number of stages required to attain a given order are established up to order five. The existence of embedded pairs of continuous Runge-Kutta methods and two-step Runge-Kutta methods of order p − 1 p - 1 and p is proved. This makes it possible to estimate local discretization error of continuous Runge-Kutta methods without any extra evaluations of the right-hand side of the differential equation. An algorithm to construct such embedded pairs is described, and examples of (3, 4) and (4, 5) pairs are presented. Numerical experiments illustrate that local error estimation of continuous Runge-Kutta methods based on two-step Runge-Kutta methods appears to be almost as reliable as error estimation by Richardson extrapolation, at the same time being much more efficient.

Two-Step Runge–Kutta Methods

Implicit two-step Runge–Kutta methods are studied. It will be shown that these methods require fewer stages to achieve the same order as one-step Runge–Kutta methods, which means the two-step methods are potentially more efficient than one-step methods. Order conditions are derived and examples of two-step one-stage methods of order 2 and two-step two-stage methods of order 4 are presented. Stability properties of these methods with respect to $y' = ay$ are studied and A-stable two-step methods of order 2 are characterized. Two-step two-stage methods of order 4 which are A-stable are found by an extensive computer search. Semi-implicit two-stage methods of order 4 were also constructed. This is in contrast to the situation encountered in the Runge–Kutta theory where the unique two-stage method of order 4 is not semi-implicit.

Two-Step Runge-Kutta Methods and Hyperbolic Partial Differential Equations

The purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation. Jameson recently introduced the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. Improvements in efficiency up to 80

Two-step Runge-Kutta methods and hyperbolic partial differential equations

The purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation. Jameson recently introduced the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. Improvements in efficiency up to 80

Factored two-step Runge-Kutta methods

A class of two-step fourth order implicit Runge-Kutta Methods designed for stiff systems of ODEs has recently been investigated by the author and others. In this paper certain computational aspects of this class of methods are discussed. Similar algorithms which, in particular, exploit any sparseness properties inherent in the Jacobian matrix are considered along with an efficient way of estimating the local truncation error using embedded methods.