In this paper, we investigate a family of implicit-explicit (IMEX) schemes of high order and stage order that are well-suited to deal with the systems of differential equations whose right hand side is naturally split into non-stiff (or mildly stiff) and stiff parts. For the proposed IMEX schemes, the diagonally implicit multistage integration methods (DIMSIMs) and the second derivative diagonally implicit multistage integration methods (SDIMSIMs) respectively serve as explicit and implicit solvers, where the implicit solver is A- or L-stable and possesses the Runge–Kutta stability property. The order and stage order conditions for the proposed IMEX schemes of order p and stage order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3202_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$q=p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> are derived. Then, by analyzing the linear stability properties of these methods, examples of the methods of order and stage order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3202_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$p=q=5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3202_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$p=q=6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> with desirable stability properties are constructed. Finally, we furnish some numerical experiments to demonstrate the efficiency of the constructed IMEX methods and confirm their robust performance by applying them to some well-known real-life problems.

Construction of high order implicit-explicit SDIMSIMs for ODEs

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发表日期：

2025年4月24日

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Mohammad Sharifi,
Ali Abdi,
Gholamreza Hojjati

摘要

In this paper, we investigate a family of implicit-explicit (IMEX) schemes of high order and stage order that are well-suited to deal with the systems of differential equations whose right hand side is naturally split into non-stiff (or mildly stiff) and stiff parts. For the proposed IMEX schemes, the diagonally implicit multistage integration methods (DIMSIMs) and the second derivative diagonally implicit multistage integration methods (SDIMSIMs) respectively serve as explicit and implicit solvers, where the implicit solver is A- or L-stable and possesses the Runge–Kutta stability property. The order and stage order conditions for the proposed IMEX schemes of order p and stage order $q=p$ $q = p$ are derived. Then, by analyzing the linear stability properties of these methods, examples of the methods of order and stage order $p=q=5$ $p = q = 5$ and $p=q=6$ $p = q = 6$ with desirable stability properties are constructed. Finally, we furnish some numerical experiments to demonstrate the efficiency of the constructed IMEX methods and confirm their robust performance by applying them to some well-known real-life problems.