Traditional Runge-Kutta Methods Research Articles

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.

Explicit mixed finite order Runge–Kutta methods

We investigate the asymptotic behavior of systems of nonlinear differential equations and introduce a family of mixed methods from combinations of explicit Runge–Kutta methods. These methods have better stability behavior than traditional Runge–Kutta methods and generally extend the range of validity of the calculated solutions. These methods also give a way of determining if the numerical solutions are real or spurious. Emphasis is put on examples coming from mathematical models in ecology.

An efficient method for solving implicit and explicit stiff differential equations

When the s-stage fully implicit Runge–Kutta (RK) method is used to solve a system of n ordinary differential equations (ODE) the resulting algebraic system has a dimension ns. Its solution by Gauss elimination is expensive and requires 2s3n3/3 operations. In this paper we present an efficient algorithm, which differs from the traditional RK method. The formal procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. Its solution is s2/2 times faster than the original, nondiagonalized system, for s even, and s3/(s−1) for s odd in terms of number of multiplications, as well as s2 times in terms of number of additions/multiplications. In particular, for s=3 the method has the same precision and stability properties as the well-known RK-based RadauIIA quadrature of Ehle, implemented by Hairer and Wanner in RADAU5 algorithm. Unlike RADAU5, however, the method is applicable with any s and not only to the explicit ODEs My′=f(x, y), where M=const., but also to the general implicit ODEs of the form f(x, y, y′)=0. The block-diagonal form of the algebraic system allows parallel processing. The algorithm formally differs from the implicit RK methods in that the solution for y is assumed to have a form of the algebraic polynomial whose coefficients are found by enforcing y to satisfy the differential equation at the collocation points. Locations of those points are found from the derived stability function such as to guarantee either A- or L-stability properties as well as a superior precision of the algorithm. If constructed such as to be L-stable the method is a good candidate for solving differential-algebraic equations (DAEs). Although not limited to any specific field, the application of the method is illustrated by its implementation in the multibody dynamics described by both ODEs and DAEs. Copyright © 2000 John Wiley & Sons, Ltd.

Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames

We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E 1, ... , E n on the manifold. The numerical approximation is obtained by composing flows of certain vector fields in the linear span of E 1, ... , E n that are tangent to the differential system at various points. The methods reduce to traditional Runge-Kutta methods if the frame vector fields are chosen as the standard basis of euclidean ℝ n . A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results.