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Numerical integrators that contract volume

Robert I McLachlan, G R W Quispel

Abstract

We study numerical integrators that contract phase space volume even when the ODE does so at an arbitrarily small rate. This is done by a splitting into two-dimensional contractive systems. We prove a sufficient condition for Runge-Kutta methods to have the appropriate contraction property for these two-dimensional systems; the midpoint rule is an example.

Numerical integrators that contract volume

Abstract

We study numerical integrators that contract phase space volume even when the ODE does so at an arbitrarily small rate. This is done by a splitting into two-dimensional contractive systems. We prove a sufficient condition for Runge-Kutta methods to have the appropriate contraction property for these two-dimensional systems; the midpoint rule is an example.

Paper Structure

This paper contains 4 sections, 7 theorems, 30 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Dissipative schemes in two dimensions
  3. 3. More than two dimensions
  4. Acknowledgements

Key Result

Proposition 3

The midpoint rule, $x_{n+1} = x_n + h f(\bar{x})$, $\bar{x} = (x_n + x_{n+1})/2$, is contractive in two dimensions.

Theorems & Definitions (16)

  • Definition 1
  • Example 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • ...and 6 more