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Linear multistep methods with repeated global Richardson extrapolation

Imre Fekete, Lajos Lóczi

TL;DR

This paper addresses accelerating convergence of sequences from linear multistep methods for initial-value problems using repeated global Richardson extrapolation (RGRE). It shows that if the base LMM has order $p$, applying RGRE $\\ell$ times yields order $p+\\ell$, under mild stability and starting-value closeness assumptions, and frames the analysis within the GLM/hairer–Lubich theory. This yields AB$k$-$\\ell$GRE, AM$k$-$\\ell$GRE, and BDF$k$-$\\ell$GRE methods with improved convergence while preserving $A(\\alpha)$-stability in relevant cases. Numerical experiments on the Dahlquist, Lotka–Volterra, and van der Pol problems demonstrate the predicted convergence orders and practical efficiency gains, including notable speedups for mildly stiff systems.

Abstract

In this work, we further investigate the application of the well-known Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for numerically solving initial-value problems of systems of ordinary differential equations. By extending the ideas of our previous paper, we now utilize some advanced versions of RE in the form of repeated RE (RRE). Assume that the underlying LMM -- the base method -- has order $p$ and RE is applied $l$ times. Then we prove that the accelerated sequence has convergence order $p+l$. The version we present here is global RE (GRE, also known as passive RE), since the terms of the linear combinations are calculated independently. Thus, the resulting higher-order LMM-RGRE methods can be implemented in a parallel fashion and existing LMM codes can directly be used without any modification. We also investigate how the linear stability properties of the base method (e.g. $A$- or $A(α)$-stability) are preserved by the LMM-RGRE methods.

Linear multistep methods with repeated global Richardson extrapolation

TL;DR

This paper addresses accelerating convergence of sequences from linear multistep methods for initial-value problems using repeated global Richardson extrapolation (RGRE). It shows that if the base LMM has order , applying RGRE times yields order , under mild stability and starting-value closeness assumptions, and frames the analysis within the GLM/hairer–Lubich theory. This yields AB-GRE, AM-GRE, and BDF-GRE methods with improved convergence while preserving -stability in relevant cases. Numerical experiments on the Dahlquist, Lotka–Volterra, and van der Pol problems demonstrate the predicted convergence orders and practical efficiency gains, including notable speedups for mildly stiff systems.

Abstract

In this work, we further investigate the application of the well-known Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for numerically solving initial-value problems of systems of ordinary differential equations. By extending the ideas of our previous paper, we now utilize some advanced versions of RE in the form of repeated RE (RRE). Assume that the underlying LMM -- the base method -- has order and RE is applied times. Then we prove that the accelerated sequence has convergence order . The version we present here is global RE (GRE, also known as passive RE), since the terms of the linear combinations are calculated independently. Thus, the resulting higher-order LMM-RGRE methods can be implemented in a parallel fashion and existing LMM codes can directly be used without any modification. We also investigate how the linear stability properties of the base method (e.g. - or -stability) are preserved by the LMM-RGRE methods.
Paper Structure (6 sections, 8 theorems, 24 equations, 1 figure, 4 tables)

This paper contains 6 sections, 8 theorems, 24 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Convergence analysis
  4. Linear stability analysis
  5. Numerical tests
  6. Conclusions

Key Result

Theorem 2.1

Assume that a given GLM is consistent of order $p\ge 1$ and satisfies assumptions A1--A3: Then the global discretization error has an asymptotic expansion of the form where $\mathbf{E}$ is bounded uniformly in $h\in[0, h_0]$ and for $t^*$ in compact intervals not containing $t_0$.

Figures (1)

  • Figure 1: Number of grid points versus the global error in maximum norm for the van der Pol equation \ref{['eq:vanderPol1']}-\ref{['eq:vanderPol2']} for $4^\text{th}$-order and $5^\text{th}$-order LMM-RGREs

Theorems & Definitions (17)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1: Hairer & Lubich, 1984
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • ...and 7 more