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A new family of fourth-order energy-preserving integrators

Yuto Miyatake

TL;DR

It is demonstrated that by carefully choosing these free parameters a simplified Newton iteration applied to the integrators of order four can be parallelizable, resulting in faster and more efficient integrators compared with existing fourth-order energy-preserving integrators.

Abstract

For Hamiltonian systems with non-canonical structure matrices, a new family of fourth-order energy-preserving integrators is presented. The integrators take a form of a combination of Runge--Kutta methods and continuous-stage Runge--Kutta methods and feature a set of free parameters that offer greater flexibility and efficiency. Specifically, we demonstrate that by carefully choosing these free parameters a simplified Newton iteration applied to the integrators of order four can be parallelizable. This results in faster and more efficient integrators compared with existing fourth-order energy-preserving integrators.

A new family of fourth-order energy-preserving integrators

TL;DR

It is demonstrated that by carefully choosing these free parameters a simplified Newton iteration applied to the integrators of order four can be parallelizable, resulting in faster and more efficient integrators compared with existing fourth-order energy-preserving integrators.

Abstract

For Hamiltonian systems with non-canonical structure matrices, a new family of fourth-order energy-preserving integrators is presented. The integrators take a form of a combination of Runge--Kutta methods and continuous-stage Runge--Kutta methods and feature a set of free parameters that offer greater flexibility and efficiency. Specifically, we demonstrate that by carefully choosing these free parameters a simplified Newton iteration applied to the integrators of order four can be parallelizable. This results in faster and more efficient integrators compared with existing fourth-order energy-preserving integrators.
Paper Structure (12 sections, 5 theorems, 55 equations, 1 figure, 10 tables)

This paper contains 12 sections, 5 theorems, 55 equations, 1 figure, 10 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hamiltonian systems and CSRK methods
  4. Poisson systems and PCSRK methods
  5. A family of fourth-order energy-preserving integrators
  6. Parameter selection
  7. Solving the nonlinear equations system
  8. Parallelizable integrators
  9. Discussion for the accuracy
  10. Numerical verification
  11. Concluding remarks
  12. Coefficients of the elementary differentials

Key Result

theorem 1

When applied to eq:hamilton, a CSRK method is energy-preserving if $M$ is symmetric.

Figures (1)

  • Figure 1: Error at $t=1$ of numerical solutions for the Lotka--Volterra system. The proposed method with the parameters \ref{['eq:opt_param']} and $\tilde{\alpha} = -234$ is compared with the 2nd and 4th order methods proposed in ch11. Dashed and dotted lines show the slope for the 2nd and 4th-order convergence, respectively.

Theorems & Definitions (13)

  • definition thmcounterdefinition: CSRK methods
  • theorem 1: mi14mb16, see also ta14
  • definition thmcounterdefinition: PCSRK methods
  • theorem 2: mi15
  • theorem 3: cf. ch11
  • remark thmcounterremark
  • proposition thmcounterproposition
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • ...and 3 more