System theory for numerical analysis has recently become a focus of research. In this paper we regard dynamics of Newton’s method as a nonlinear feedback system and derive convergence conditions, based on the internal model principle and systems of Lur’e type. We then focus our attention on the analysis of the region of absolute stability of Runge-Kutta type methods. We derive a linear matrix i...

In the context of solving nonlinear partial differential equations, Shu and Osher introduced representations of explicit Runge-Kutta methods, which lead to stepsize conditions under which the numerical process is totalvariation-diminishing (TVD). Much attention has been paid to these representations in the literature. In general, a Shu-Osher representation of a given Runge-Kutta method is not u...

This paper continues earlier work by the same author concerning the stability and B-convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable, Bor optimally B-convergent ...

Abstract—In this paper zero-dissipative explicit Runge-Kutta method is derived for solving second-order ordinary differential equations with periodical solutions. The phase-lag and dissipation properties for Runge-Kutta (RK) method are also discussed. The new method has algebraic order three with dissipation of order infinity. The numerical results for the new method are compared with existing ...

New software based on explicit Runge-Kutta formulas have been developed to replace well-established, widely-used codes written by the authors (RKF45 and its successors in the SLATEC Library and the NAG Fortran 77 Library Runge-Kutta codes). The new software has greater functionality than its predecessors. Also, it is more eecient, more robust and better documented.

A new selection is made of the most practical of the many explicit Runge-Kutta formulas of order 4 which have been proposed. A new formula is considered, formulas are modified to improve their quality and efficiency in agreement with improved understanding of the issues, and formulas are derived which permit interpolation. It is possible to do a lot better than the pair of Fehlberg currently re...

Additive Runge Kutta (ARK) methods are investigated for application to the spatially discretized one dimensional convection diffusion reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge Kutta methods are grouped into a single composite method. Then, implicit explicit, N = 2, additive Runge Kutta (ARK2) m...

This paper presents a family of Runge{Kutta type integration schemes of arbitrarily high order for diierential equations evolving on manifolds. We prove that any classical Runge{Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement.

We prove that to every rational function R(z) satisfying R(−z)R(z) = 1, there exists a symplectic Runge-Kutta method with R(z) as stability function. Moreover, we give a surprising relation between the poles of R(z) and the weights of the quadrature formula associated with a symplectic Runge-Kutta method.