%0 Journal Article
%T 求解二维Allen-Cahn方程的保正隐式差分格式<br>Positive-Preserving Implicit Difference Scheme for Solving Two-Dimensional Allen-Cahn Equation
%A 赵紫琳
%J Advances in Applied Mathematics
%P 328-338
%@ 2324-8009
%D 2025
%I Hans Publishing
%R 10.12677/aam.2025.145262
%X 本文研究求解二维Allen-Cahn方程的保正隐式差分格式。通过证明得到当网格比满足<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mn>1</mn><mo>+</mo><mi>&#x03C4;</mi><mo>&#x2212;</mo><mn>2</mn><msub>
   <mi>R</mi>
   <mi>x</mi>
  </msub>
  <mo>&#x2212;</mo><mn>2</mn><msub>
   <mi>R</mi>
   <mi>y</mi>
  </msub>
  <mo>&#x2265;</mo><mn>0</mn></mrow>
</math>

时&#65292;差分解具有保正性&#65292;且在无穷范数意义下有<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>O</mi><mrow><mo>(</mo>
   <mrow>
    <mi>&#x03C4;</mi><mo>+</mo><msubsup>
     <mi>h</mi>
     <mi>x</mi>
     <mn>2</mn>
    </msubsup>
    <mo>+</mo><msubsup>
     <mi>h</mi>
     <mi>y</mi>
     <mn>2</mn>
    </msubsup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

的收敛阶&#65292;最后数值实验表明数值结果与理论结果相吻合。<br>In this paper, we study the positive-preserving implicit difference scheme for solving two-dimensional Allen-Cahn equation. It is proved that when the grid ratio satisfies <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mn>1</mn><mo>+</mo><mi>&#x03C4;</mi><mo>&#x2212;</mo><mn>2</mn><msub>
   <mi>R</mi>
   <mi>x</mi>
  </msub>
  <mo>&#x2212;</mo><mn>2</mn><msub>
   <mi>R</mi>
   <mi>y</mi>
  </msub>
  <mo>&#x2265;</mo><mn>0</mn></mrow>
</math>

, the difference solution is positive-preserving and has the convergence order of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>O</mi><mrow><mo>(</mo>
   <mrow>
    <mi>&#x03C4;</mi><mo>+</mo><msubsup>
     <mi>h</mi>
     <mi>x</mi>
     <mn>2</mn>
    </msubsup>
    <mo>+</mo><msubsup>
     <mi>h</mi>
     <mi>y</mi>
     <mn>2</mn>
    </msubsup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

 in the sense of infinite norm. Finally, numerical experiments show that the numerical results are consistent with the theoretical results.
%K Allen-Cahn方程&#65292
%K 保正隐式差分格式&#65292
%K 保正性&#65292
%K 收敛性<br>Allen-Cahn Equation
%K Positive-Preserving Implicit Difference Scheme
%K Positive-Preserving
%K Convergence
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=115571