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<p>The Exponential Time Difference Runge–Kutta method (ETDRK) is an important approach for simulating gradient flow models, and whether it can maintain energy stability is an important research topic. New adaptive ETDRK method (ETDRK32) has been designed in [<CitationRef CitationID="CR4">4</CitationRef>] and proven to possess the property of long time unconditional energy stability for one specific parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2840_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha =\frac{2}{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we adopt a new method of semi-positive definite matrix decomposition to prove the long-term energy stability of the schemes when the parameter <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2840_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is within a certain range. Furthermore, we provide numerical simulations to verify the theoretical analysis and demonstrate the convergence and energy stability of the new schemes.</p>

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Energy Stability of Adaptive Exponential Time Difference Runge-Kutta Method (ETDRK32)

  • 期刊论文

  • 发表日期：

    2025年3月12日

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  • Hengli Yang,
  • Weichen Cao,
  • Wenbin Chen

摘要

The Exponential Time Difference Runge–Kutta method (ETDRK) is an important approach for simulating gradient flow models, and whether it can maintain energy stability is an important research topic. New adaptive ETDRK method (ETDRK32) has been designed in [4] and proven to possess the property of long time unconditional energy stability for one specific parameter \(\alpha =\frac{2}{3}\) α = 2 3 . In this paper, we adopt a new method of semi-positive definite matrix decomposition to prove the long-term energy stability of the schemes when the parameter \(\alpha \) α is within a certain range. Furthermore, we provide numerical simulations to verify the theoretical analysis and demonstrate the convergence and energy stability of the new schemes.