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<p>This paper presents a linearized Backward Euler (BE) scheme for coupled Burgers’ equations. By adopting the temporal-spatial error splitting technique, we rigorously establish the boundedness of numerical solutions in the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norm. By employing the special high-accuracy results of the linear triangular element and the interpolated post-processing approach, we derive unconditional supercloseness and global superconvergence results with order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(O(h^2+\tau )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norm on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(h^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-parallelogram meshes, where <i>h</i> denotes the mesh size and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> the time step. In addition, the theoretical outcomes and good performance of the proposed scheme are confirmed by numerical experiments.</p>

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Unconditional Superconvergence Analysis of an Efficient Backward Euler Scheme for Coupled Burgers’ Equations on \(h^2\)-Parallelogram Meshes

  • 期刊论文

  • 发表日期：

    2026年2月13日

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  • Weijun Zhu,
  • Yingxia Chen,
  • Dongyang Shi

摘要

This paper presents a linearized Backward Euler (BE) scheme for coupled Burgers’ equations. By adopting the temporal-spatial error splitting technique, we rigorously establish the boundedness of numerical solutions in the \(H^1\) H 1 -norm. By employing the special high-accuracy results of the linear triangular element and the interpolated post-processing approach, we derive unconditional supercloseness and global superconvergence results with order \(O(h^2+\tau )\) O ( h 2 + τ ) in \(H^1\) H 1 -norm on \(h^2\) h 2 -parallelogram meshes, where h denotes the mesh size and \(\tau \) τ the time step. In addition, the theoretical outcomes and good performance of the proposed scheme are confirmed by numerical experiments.