{"title":"结合非标准与紧致有限差分格式求解FitzHugh-Nagumo方程的高阶解","authors":"Zhi-Chen Li , Yang-Wen Yu , Xiao-Yu Zhang , Qing Fang","doi":"10.1016/j.apnum.2025.07.004","DOIUrl":null,"url":null,"abstract":"<div><div>This study presents a novel numerical method for solving the FitzHugh-Nagumo equation by combining nonstandard finite difference (NSFD) and high-order compact finite difference schemes. Through rigorous mathematical analysis, we demonstrate the stability and convergence of our approach, revealing that instability arises only under extremely rare conditions. To verify the efficiency of our scheme, we calculated the <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>l</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> errors as well as the convergence rate by comparing the numerical results with the exact solution. Experiments show that our combined scheme not only ensures stability, but also possesses the lowest error while maintaining high order convergence.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 436-450"},"PeriodicalIF":2.4000,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A higher-order solver for the FitzHugh-Nagumo equation by combining nonstandard and compact finite difference scheme\",\"authors\":\"Zhi-Chen Li , Yang-Wen Yu , Xiao-Yu Zhang , Qing Fang\",\"doi\":\"10.1016/j.apnum.2025.07.004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This study presents a novel numerical method for solving the FitzHugh-Nagumo equation by combining nonstandard finite difference (NSFD) and high-order compact finite difference schemes. Through rigorous mathematical analysis, we demonstrate the stability and convergence of our approach, revealing that instability arises only under extremely rare conditions. To verify the efficiency of our scheme, we calculated the <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>l</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> errors as well as the convergence rate by comparing the numerical results with the exact solution. Experiments show that our combined scheme not only ensures stability, but also possesses the lowest error while maintaining high order convergence.</div></div>\",\"PeriodicalId\":8199,\"journal\":{\"name\":\"Applied Numerical Mathematics\",\"volume\":\"217 \",\"pages\":\"Pages 436-450\"},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2025-07-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Numerical Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S016892742500145X\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S016892742500145X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
A higher-order solver for the FitzHugh-Nagumo equation by combining nonstandard and compact finite difference scheme
This study presents a novel numerical method for solving the FitzHugh-Nagumo equation by combining nonstandard finite difference (NSFD) and high-order compact finite difference schemes. Through rigorous mathematical analysis, we demonstrate the stability and convergence of our approach, revealing that instability arises only under extremely rare conditions. To verify the efficiency of our scheme, we calculated the and errors as well as the convergence rate by comparing the numerical results with the exact solution. Experiments show that our combined scheme not only ensures stability, but also possesses the lowest error while maintaining high order convergence.
期刊介绍:
The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are:
(i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments.
(ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers.
(iii) Short notes, which present specific new results and techniques in a brief communication.