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Associative Integrator

J. Avellar, L. G. S. Duarte, L. A. C. P. da Mota, L. O. Pereira

TL;DR

This work proposes an association of numerical integration methods (integrators) in order to optimize the performance of dynamic systems and achieves the precision of the top integrator with a processing time performance closer to the one of the lower grade integrator.

Abstract

Dynamic systems have a fundamental relevance in the description of physical phenomena. The search for more accurate and faster numerical integration methods for the resolution of such systems is, therefore, an important topic of research. The present work introduces a new approach for the numerical integration of dynamic systems. We propose an association of numerical integration methods (integrators) in order to optimize the performance. The standard we apply is the balance of the duo : precision obtained x running time. The numerical integration methods we have chosen, for this particular instance of association, were the Runge-Kutta of fourth order and seventheighth order. The algorithm was implemented in C++ language. The results showed an improvement in accuracy over the lower grade numerical integrator (actually, we have achieved, basically, the precision of the top integrator) with a processing time performance closer to the one of the lower grade integrator. Similar results can be obtained for other pairs of numerical integration methods.

Associative Integrator

TL;DR

This work proposes an association of numerical integration methods (integrators) in order to optimize the performance of dynamic systems and achieves the precision of the top integrator with a processing time performance closer to the one of the lower grade integrator.

Abstract

Dynamic systems have a fundamental relevance in the description of physical phenomena. The search for more accurate and faster numerical integration methods for the resolution of such systems is, therefore, an important topic of research. The present work introduces a new approach for the numerical integration of dynamic systems. We propose an association of numerical integration methods (integrators) in order to optimize the performance. The standard we apply is the balance of the duo : precision obtained x running time. The numerical integration methods we have chosen, for this particular instance of association, were the Runge-Kutta of fourth order and seventheighth order. The algorithm was implemented in C++ language. The results showed an improvement in accuracy over the lower grade numerical integrator (actually, we have achieved, basically, the precision of the top integrator) with a processing time performance closer to the one of the lower grade integrator. Similar results can be obtained for other pairs of numerical integration methods.

Paper Structure

This paper contains 12 sections, 1 theorem, 54 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The motivation and the jest of the approach
  3. The method: The Associative Integration Algorithm - or AI
  4. The Taylor Expansion Method
  5. The Mathematical Basis
  6. Main Result
  7. The structural basis of our AI approach
  8. The Algorithm itself
  9. Actual implementation of the Associative Integration Algorithm - AI
  10. Description of a particular implementation
  11. Performance of AI
  12. Other examples of systems calculating with AI

Key Result

Theorem 3.1

Consider the dynamical system $\dot{x_i} = f_i(\vec{x})$ with solution ${x_i}(t) = F_i(\vec{x}_0,t)$ and the mapping $M$ given by ${x_i}_{(P+\delta P)} = F_i(\vec{x}_{(P)},\delta t) = \sum_{k=0}^{\infty}{\frac{{\delta t}^{k}}{k!}}{\it X}^{k}[{x_i}]_{(P)}$, where the operator $X$ is defined by $X \eq where $r$ is a positive integer

Figures (9)

  • Figure 1: Illustration of the stitching process of the AI with $r=4$ e $p=5$
  • Figure 2: Differences between the points of the integrators in the variable $x$
  • Figure 3: Differences between the points of the integrators in the variable $y$
  • Figure 4: Differences between the points of the integrators in the variable $z$
  • Figure 5: running time performance for the integrators
  • ...and 4 more figures

Theorems & Definitions (10)

  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Definition 3.1
  • Definition 3.2
  • Remark 3.6
  • Remark 3.7