{"title":"用于求解二维 EFK 方程的四阶紧凑差分方案","authors":"Kai Qu, Shuguang Li, Longjie Lv, Xin Liu","doi":"10.1016/j.rinam.2024.100441","DOIUrl":null,"url":null,"abstract":"<div><p>In this article, a fourth order compact difference scheme for solving the two-dimensional extended Fisher–Kolmogorov (2D EFK) equation is proposed and analyzed. This scheme is three-level implicit, based on a novel time discretization idea of <span><math><mrow><mi>u</mi><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>≈</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mrow><mo>(</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>. The discrete energy functional method is used to obtain prior estimates of numerical solutions in the maximum norm. Furthermore, the convergence of the difference solutions in the maximum norm is analyzed, and the convergence rate is obtained as <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, which without any restriction on the grid ratio with time step <span><math><mi>τ</mi></math></span> and mesh size <span><math><mi>h</mi></math></span>. Finally, numerical examples are given to support the theoretical analysis.</p></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"22 ","pages":"Article 100441"},"PeriodicalIF":1.4000,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2590037424000116/pdfft?md5=086d043a63b5fa6d73ba13132c8b642d&pid=1-s2.0-S2590037424000116-main.pdf","citationCount":"0","resultStr":"{\"title\":\"A fourth-order compact difference scheme for solving 2D EFK equation\",\"authors\":\"Kai Qu, Shuguang Li, Longjie Lv, Xin Liu\",\"doi\":\"10.1016/j.rinam.2024.100441\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this article, a fourth order compact difference scheme for solving the two-dimensional extended Fisher–Kolmogorov (2D EFK) equation is proposed and analyzed. This scheme is three-level implicit, based on a novel time discretization idea of <span><math><mrow><mi>u</mi><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>≈</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mrow><mo>(</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>+</mo><mn>2</mn><msubsup><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>. The discrete energy functional method is used to obtain prior estimates of numerical solutions in the maximum norm. Furthermore, the convergence of the difference solutions in the maximum norm is analyzed, and the convergence rate is obtained as <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, which without any restriction on the grid ratio with time step <span><math><mi>τ</mi></math></span> and mesh size <span><math><mi>h</mi></math></span>. Finally, numerical examples are given to support the theoretical analysis.</p></div>\",\"PeriodicalId\":36918,\"journal\":{\"name\":\"Results in Applied Mathematics\",\"volume\":\"22 \",\"pages\":\"Article 100441\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-02-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S2590037424000116/pdfft?md5=086d043a63b5fa6d73ba13132c8b642d&pid=1-s2.0-S2590037424000116-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Results in Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2590037424000116\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Results in Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2590037424000116","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
本文提出并分析了求解二维扩展费希尔-科尔莫戈罗夫（2D EFK）方程的四阶紧凑差分方案。该方案是三级隐式的，基于 u(xi,yj,tn)≈14(Ui,jn+1+2Ui,jn+Ui,jn-1) 的新颖时间离散化思想。离散能量函数法用于获得最大规范数值解的先验估计值。此外，还分析了最大规范差分解的收敛性，得到收敛率为 O(τ2+h4)，这对时间步长 τ 和网格大小 h 的网格比没有任何限制。
A fourth-order compact difference scheme for solving 2D EFK equation
In this article, a fourth order compact difference scheme for solving the two-dimensional extended Fisher–Kolmogorov (2D EFK) equation is proposed and analyzed. This scheme is three-level implicit, based on a novel time discretization idea of . The discrete energy functional method is used to obtain prior estimates of numerical solutions in the maximum norm. Furthermore, the convergence of the difference solutions in the maximum norm is analyzed, and the convergence rate is obtained as , which without any restriction on the grid ratio with time step and mesh size . Finally, numerical examples are given to support the theoretical analysis.
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