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 recent work on global Richardson extrapolation, 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_654_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> times. Then, we prove that the accelerated sequence has convergence order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_654_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p+\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>+</mo> <mi>ℓ</mi> </mrow> </math></EquationSource> </InlineEquation>. The version we present here is global RE (GRE, also known as passive RE), since the terms of the linear combinations are calculated independently. Thus, the resulting higher-order LMM-RGRE methods can be implemented in a parallel fashion and existing LMM codes can directly be used without any modification. We also investigate how the linear stability properties of the base method (e.g., A- or <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_654_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$A(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-stability) are preserved by the LMM-RGRE methods.

Linear multistep methods with repeated global Richardson extrapolation

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2025年5月25日

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I. Fekete,
L. Lóczi

摘要

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 recent work on global Richardson extrapolation, 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 $\ell $ $ℓ$ times. Then, we prove that the accelerated sequence has convergence order $p+\ell $ $p + ℓ$ . 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 $A(\alpha )$ $A (α)$ -stability) are preserved by the LMM-RGRE methods.