Two-step Runge-Kutta Methods Research Articles

Extended Runge–Kutta-like formulae

In this paper we present a new family of extended Runge–Kutta formulae in which, just like in Enright's methods, it is assumed that the user will evaluate both f and f ′ readily when solving the autonomous system y ′ = f ( y ) numerically. This means that we introduce some new parameters in the extended Runge–Kutta-like formulae in order to enhance the order of accuracy of the solutions using evaluations of both f and f ′ , instead of the evaluations of f only. Moreover, if f ′ is approximated by a difference quotient of past and current evaluations of f, the order of convergence can be retained. The resulting two-step Runge–Kutta method can be regarded as replacing the function evaluations of f ′ with approximations of f ′ . Specifically, the proposed formulae with f ′ are more efficient for cases where f ′ is not more expensive to evaluate than f and the proposed ‘derivative-free’ formulae are more attractive for use when past values of f are available. Furthermore error estimates and step-choose strategies are considered for the ‘derivative-free’ extended Runge–Kutta methods.

Improved starting methods for two-step Runge–Kutta methods of stage-order [formula omitted

In [Japan JIAM 19 (2002) 227], Jackiewicz and Verner derived formulas for, and tested the implementation of two-step Runge–Kutta (TSRK) pairs. For pairs of orders 3 and 4, the error estimator accurately tracked the exact local truncation error on several nonlinear test problems. However, for pairs designed to achieve order 8, the results appeared to be only of order 6. This deficiency was identified in [SIAM J. Numer. Anal. 34 (1997) 2087 [2]] by Hairer and Wanner who used B-series to formulate a complete set of order conditions for TSRK methods, and showed that if the order of a TSRK method is at least two greater than its stage-order, special starting values are necessary for the first step. In forthcoming paper [Starting methods for two-step Runge–Kutta methods of stage-order 3 and order 6, J. Comput. Appl. Math.], Verner showed that such starting values have to be perturbed from their asymptotically correct values to include errors of precisely the form which the selected TSRK formula is designed to propagate from step to step. For TSRK methods of order 6, it was shown that a complementary set of Runge–Kutta methods could be utilized to obtain suitably perturbed starting values, and that each method of the set could be derived by solving appropriately modified order conditions directly. The design used there required solving an intricate polynomial equation. Here, the design is improved, and new starting methods are simpler to derive, and perhaps may lead to starting methods for TSRK methods of order 8.

Starting methods for two-step Runge–Kutta methods of stage-order 3 and order 6

Jackiewicz and Tracogna [SIAM J. Numer. Anal. 32 (1995) 1390–1427] proposed a general formulation of two step Runge–Kutta (TSRK) methods. Using formulas for two-step pairs of TSRK methods constructed in [Japan JIAM 19 (2002) 227–248], Jackiewicz and Verner obtain results for order 8 pairs that fail to show this designated order. Hairer and Wanner [SIAM J. Numer. Anal. 34 (1997) 2087–2089] identify the problem by using B-series to formulate a complete set of order conditions for TSRK methods, and emphasize that special starting methods are necessary for the first step of implementation. They observe that for methods with stage order at least p - 1 , and design order p , starting methods of order at least p are sufficient. In this paper, the more general challenge to provide correct starting values for methods of low stage-order is met by showing how perturbed starting values should be selected for methods of order 6 and stage-order 3. The approach is sufficiently general that it may (and later will) be provided for such methods of higher orders. Evidence of the accompanying improvement in the implementation of TSRK methods illustrates that carefully designed starting methods are essential for efficient production codes based on methods of low stage-order.

Nordsieck representation of two-step Runge–Kutta methods for ordinary differential equations

We describe a new representation of explicit two-step Runge–Kutta methods for ordinary differential equations. This representation makes it possible for the accurate and reliable estimation of local discretization error and facilitates the efficient implementation of these methods in a variable stepsize environment.

The wall-jetting effect in Mach reflection: NavierStokes simulations

The wall-jetting effect in Mach reflections in viscous pseudo-steady flows (as obtained in shock tubes) is investigated numerically. The W-modification of Godunov's scheme has been modified to solve the Navier-Stokes equations using a splitting into physical processes. The viscous terms are approximated using an explicit scheme with central differences in space and a two-step Runge-Kutta method in time. Two analytical models are considered. The first is a self-similar viscous flow model in which we consider a flow field with characteristic size L, and assume that as the characteristic size grows from 0 to L, the viscosity of the gas ahead of the shock wave varies from 0 to μ 0 . Consequently, the flow can be made self-similar by using the parameter Re = ρ 0 a 0 L/μ 0 . The second is a real non-stationary viscous flow, in which the molecular viscosity during the growth of a characteristic size from 0 to L remains constant and is equal μ 0 . As a result the viscous effects are only partially accounted for in the self-similar viscous flow model in comparison to a real non-stationary viscous flow model, since they are smaller in the former case

Exponentially‐fitted algorithms: fixed or frequency dependent knot points?

AbstractExponentially‐fitted algorithms are constructed for the derivation of Gauss formulae and implicit Runge‐Kutta methods of collocation type making them tuned for oscillatory (or exponential) functions. The weights and the abscissas of these formulae can depend naturally on the frequency ω by the very construction. For twopoints Gauss formulae and two‐step Runge‐Kutta methods a detailed study of the obtained results is made. In particular the difference in the numerical application of these algorithms with fixed points and/or frequency dependent nodes is analysed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Two-step Runge–Kutta methods of order over five with reduced function evaluations

Two-step Runge–Kutta formulae of order five with reduced function evaluations are discussed and both of the order conditions for two-step Runge–Kutta formulae of order five and six are offered. Numerical experiments show these new formulae are feasible and effective.

Construction of two-step Runge–Kutta methods with large regions of absolute stability

We describe the construction of explicit two-step Runge–Kutta methods of order p and stage order q= p−1 or q= p with large regions of absolute stability. This process is illustrated for the methods of order p=2, and 3 and leads to new methods which are competitive with explicit Runge–Kutta methods of the same order.

Parallel predictor-corrector iteration of pseudo two-step RK methods for nonstiff IVPs

A parallel predictor-corrector (PC) iteration scheme for a general class of pseudo two-step Runge-Kutta methods (PTRK methods) of arbitrarily high order is analyzed for solving first-order nonstiff initial-value problems (IVPs) on parallel computers. Starting with ans-stage pseudo two-step RK method of orderp* withw implicit stages, we apply the highly parallel PC iteration process in P(EC)mE mode. The resulting parallel-iterated pseudo two-step RK method (PIPTRK method) uses an optimal number of processors equal tow. By a number of numerical experiments, we show the superiority of the PIPTRK methods proposed in this paper over both sequential and parallel methods available in the literature.

Stability analysis of two-step Runge-Kutta methods for delay differential equations

We investigate stability properties of two-step Runge-Kutta methods with respect to the linear test equation y′( t) = ay( t) + by( t − τ), t ≥ 0, y( t) = g( t), t ϵ [−τ,0], where a and b are complex parameters. It is known that the solution y( t) to this equation tends to zero as t → ∞ if | b| < - Re( a). We will show that under some conditions this property is inherited by any A-stable two-step Runge-Kutta method applied on a constrained mesh to delay differential equations, i.e., that the corresponding method is P-stable.

Derivation and implementation of Two-Step Runge-Kutta pairs

Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two-step Runge-Kutta methods strive to improve the efficiency by utilizing approximations to the solution and its derivatives from the previous step. This article suggests a strategy for computing embeddedpairs of such two-step methods using a smaller number of function evaluations than that required for traditional Runge-Kutta methods of the same order. This leads to the efficient and reliable estimation of local discretization error and a robust step control strategy. The change of stepsize is achieved by a suitable interpolation of stage values from the previous step and does not require any additional function evaluations. Two examples illustrate the features of these pairs.

A General Family of Pseudo Two-step Runge-Kutta Methods

The aim of this paper is to design a new family of numerical methods of arbitrarily high order for systems of first-order differential equations which are to be termed pseudo two-step Runge-Kutta methods. By using collocation techniques, we can obtain an arbitrarily high-order stable pseudo two-step Runge-Kutta method with any desired number of implicit stages in retaining the two-step nature. In very first investigations, the pseudo two-step Runge-Kutta methods are shown to be promising numerical integration methods.

A General Family of Pseudo Two-step Runge-Kutta Methods

A General Family of Pseudo Two-step Runge-Kutta Methods

GP-Stability of two-step implicit runge-kutta methods for delay differential equations

The GP-stability of two-step implicit Runge-Kutta (TIRK) methods for the numerical solution of systems of delay differential equations (DDEs) is considered. We focus on the stability behaviour of TIRK methods in the solution of the linear constant-coefficient systems of DDE with a single delay. We present a sufficient condition of GP-stability of TIRK methods.

High Order Explicit Two-Step Runge-Kutta Methods for Parallel Computers

In this paper we study a class of explicit pseudo two-step Runge-Kutta methods (EPTRK methods) with additional weights v. These methods are especially designed for parallel computers. We study s-stage methods with local stage order s and local step order s + 2 and derive a sufficient condition for global convergence order s + 2 for fixed step sizes. Numerical experiments with 4- and 5-stage methods show the influence of this superconvergence condition. However, in general it is not possible to employ the new introduced weights to improve the stability of high order methods. We show, for any given s-stage method with extended weights which fulfills the simplifying conditions B(s) and C(s - 1), the existence of a reduced method with a simple weight vector which has the same linear stability behaviour and the same order.

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.

Continuous variable stepsize explicit pseudo two-step RK methods

The aim of this paper is to apply a class of constant stepsize explicit pseudo two-step Runge-Kutta methods of arbitrarily high order to nonstiff problems for systems of first-order differential equations with variable stepsize strategy. Embedded formulas are provided for giving a cheap error estimate used in stepsize control. Continuous approximation formulas are also considered for use in an eventual implementation of the methods with dense output. By a few widely used test problems, we compare the efficiency of two pseudo two-step Runge-Kutta methods of orders 5 and 8 with the codes DOPRI5, DOP853 and PIRK8. This comparison shows that in terms of ƒ-evaluations on a parallel computer, these two pseudo two-step Runge-Kutta methods are a factor ranging from 3 to 8 cheaper than DOPRI5, DOP853 and PIRK8. Even in a sequential implementation mode, fifth-order new method beats DOPRI5 by a factor more than 1.5 with stringent error tolerances.

Explicit pseudo two-step runge-kutta methods for parallel computers*

The aim of this paper is to investigate a class of explicit pseudo two-step Runge-Kutta methods of arbitrarily high order for nonstiff problems for systems of first-order differential equations. By using collocation techniques we can obtain for any given order of accuracy p, a stable pth-order explicit pseudo two-step Runge-Kutta method requiring only one effective sequential right-hand side evaluation per step on multiprocessor computers. By a few widely-used test problems, we show the superiority of the methods considered in this paper over both sequential and parallel methods available in the literature.

Numerical experiments with some explicit pseudo two-step RK methods on a shared memory computer

This paper investigates the performance of two explicit pseudo two-step Runge-Kutta methods of order 5 and 8 for first-order nonstiff ODEs on a parallel shared memory computer. For expensive right-hand sides the parallel implementation gives a speed-up of 3–4 with respect to the sequential one. Furthermore, we compare the codes with the two efficient nonstiff codes DOPRI5 and DOP853. For problems where the stepsize is determined by accuracy rather than by stability our codes are shown to be more efficient.

Construction of two-step Runge-Kutta methods of high order for ordinary differential equations

The construction of two-step Runge-Kutta methods of order p and stage order q=p with stability polynomial given in advance is described. This polynomial is chosen to have a large interval of absolute stability for explicit methods and to be A-stable and L-stable for implicit methods. After satisfying the order and stage order conditions the remaining free parameters are computed by minimizing the sum of squares of the difference between the stability function of the method and a given polynomial at a sufficiently large number of points in the complex plane.