Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order

AbstractThe eight main contributions of the author to the field of approximate solutions of ordinary differential equations described herein are all application-oriented, with the purposes of simplification and the increase in efficiency and effectiveness of the Runge-Kutta processes generated. They range from the determination of an initial trial step-size to be adopted to expedite the approximation process through embedded Runge-Kutta algorithms to a more recent procedure for upgrading the order of Runge-Kutta processes. These contributions encompass all classes of differential equations of all orders, such as explicit, implicit, single or systems, and their treatment by Runge-Kutta processes of scalar or vector type (with the related equivalence conditions), of discrete or continuous kind, including the computer derivations of nonlinear algebraic equations associated with the Runge-Kutta processes. Specifically, the author developed the first fifth order Runge-Kutta formulae with fourth order embedded and the first C1 approximate solution through interpolation and Runge-Kutta formulae, which he improved by developing C1 embeddings with Runge-Kutta formulae without the use of interpolative techniques

Research Achievements Review Series no. 20 - Mathematics and computation research

Computational mathematics, perturbed orbit three-body problem, and periodic trajectories solutions through computer method

Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems

Low order Runge-Kutta formulas with step control for heat transfer problem

A Sixth-Order Extension to the MATLAB Package bvp4c of J. Kierzenka and L. Shampine

A new two-point boundary value problem algorithm based upon the MATLAB bvp4c package of Kierzenka and Shampine is described. The algorithm, implemented in a new package bvp6c, uses the residual control framework of bvp4c (suitably modified for a more accurate finite difference approximation) to maintain a user specified accuracy. The new package is demonstrated to be as robust as the existing software, but more efficient for most problems, requiring fewer internal mesh points and evaluations to achieve the required accuracy

Splitting and composition methods in the numerical integration of differential equations

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table

Implicit Runge-Kutta formulae for the numerical integration of ODEs

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Fourth-order 2N-storage Runge-Kutta schemes

A family of five-stage fourth-order Runge-Kutta schemes is derived; these schemes required only two storage locations. A particular scheme is identified that has desirable efficiency characteristics for hyperbolic and parabolic initial (boundary) value problems. This scheme is competitive with the classical fourth-order method (high-storage) and is considerably more efficient and accurate than existing third-order low-storage schemes

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances