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Analysis of error propagation in the RK3GL2 method

J. S. C. Prentice

TL;DR

This paper analyzes how local discretization errors propagate in the RK3GL2 method, a hybrid of an explicit RK scheme with a 2-point Gauss-Legendre quadrature. Through analytic derivations, it shows that the GL nodes introduce a quenching effect that elevates the global error order from the underlying RK3 local error to $O(h^{4})$, aided by the GL-local error being $O(h^{5})$ and RK-induced terms scaled by $h$. The key contribution is a detailed propagation and accumulation framework (including explicit expressions for $ ext{Δ}_1$, $ ext{Δ}_2$, $ ext{Δ}_3$ and their dependence on local errors and derivatives), which explains why RK3GL2 attains fourth-order global accuracy. This work provides theoretical insight into why RK-GL hybrids can achieve higher global accuracy, informing design and analysis of higher-order numerical solvers for initial value problems.

Abstract

The RK3GL2 method is a numerical method for solving initial value problems in ordinary differential equations, and is a hybrid of a third-order Runge-Kutta method and two-point Gauss-Legendre quadrature. In this paper we present an analytical study of the propagation of local errors in this method, and show that the global order of RK3GL2 is expected to be four.

Analysis of error propagation in the RK3GL2 method

TL;DR

This paper analyzes how local discretization errors propagate in the RK3GL2 method, a hybrid of an explicit RK scheme with a 2-point Gauss-Legendre quadrature. Through analytic derivations, it shows that the GL nodes introduce a quenching effect that elevates the global error order from the underlying RK3 local error to , aided by the GL-local error being and RK-induced terms scaled by . The key contribution is a detailed propagation and accumulation framework (including explicit expressions for , , and their dependence on local errors and derivatives), which explains why RK3GL2 attains fourth-order global accuracy. This work provides theoretical insight into why RK-GL hybrids can achieve higher global accuracy, informing design and analysis of higher-order numerical solvers for initial value problems.

Abstract

The RK3GL2 method is a numerical method for solving initial value problems in ordinary differential equations, and is a hybrid of a third-order Runge-Kutta method and two-point Gauss-Legendre quadrature. In this paper we present an analytical study of the propagation of local errors in this method, and show that the global order of RK3GL2 is expected to be four.
Paper Structure (12 sections, 38 equations, 1 figure)

This paper contains 12 sections, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Terminology and relevant concepts
  3. One-step methods
  4. Local and global errors
  5. Error propagation in a one-step method
  6. Gauss-Legendre quadrature
  7. The RK3GL2 algorithm
  8. Local error at the GL nodes
  9. Error propagation in the RKGL method
  10. Error propagation
  11. Error accumulation
  12. Comments

Figures (1)

  • Figure 1: RK3GL2 algorithm for the first two subintervals $H_{1}$ and $H_{2}$ on $[a,b].$