[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-326249-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-326249-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","chapter-1-finite-difference-approximations","Chapter 1 - Finite Difference Approximations","","Finite difference approximations replace derivatives in differential equations with finite difference formulas built from function values at discrete grid points. The chapter motivates this approach by first deriving one-sided and centered difference estimates for the derivative u0(x) from known u(x) values, then comparing their truncation errors. It introduces higher-order formulas (including a third-order accurate scheme) and demonstrates how error magnitude scales with powers of the step size h. It also explains log-log error plots for extracting the order of accuracy.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/chapter-1-finite-difference-approximations/326249/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/chapter-1-finite-difference-approximations/326249.png","ImageObject",300,407,{"name":42,"@type":43},"Jiven","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-03","2026-09-21",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",11,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"Why do finite difference methods replace derivatives with finite difference approximations?","Question",{"text":63,"@type":64},"Finite difference methods replace derivatives in differential equations with discrete approximations using function values at grid points. This turns the differential equation into a solvable finite algebraic system on a computer.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How are one-sided, centered, and higher-order derivative approximations related?",{"text":68,"@type":64},"The chapter derives one-sided forward and backward differences and a centered difference that averages them. It then presents a third-order accurate approximation (with error proportional to h^3) to show improved accuracy.",{"name":70,"@type":61,"acceptedAnswer":71},"How does truncation error determine the order of accuracy?",{"text":72,"@type":64},"By expanding function values in Taylor series around a grid point, the chapter shows that different finite difference formulas have leading error terms proportional to different powers of h. The power of h directly indicates whether the method is first, second, or third order accurate.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},326249,1790478653,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":97,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":112,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":116,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145,"preview_page_count":4},1099513958607,"https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399","Downloaded 06/09/ 16 to [205.155.65.226. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php](205.155.65.226. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php)  \nChapter 1  \nFinite Difference Approximations  \nOur goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to this function) that satisfies a given relationship between various of its derivatives on some given region of space and/or time, along with some boundary conditions along the edges of this domain. In general this is a difficult problem, and only rarely can an analytic formula be found for the solution. A finite difference method proceeds by replacing the derivatives in the differential equations with finite difference approximations. This gives a large but finite algebraic system of equations to be solved in place of the differential equation, something that can be done on a computer.  \nBefore tackling this problem, we first consider the more basic question of how we can approximate the derivatives of a known function by finite difference formulas based only on values of the function itself at discrete points. Besides providing a basis for the later development of finite difference methods for solving differential equations, this allows us to investigate several key concepts such as the order of accuracy of an approximation in the simplest possible setting.  \nLet u.x/ represent a function of one variable that, unless otherwise stated, will always be assumed to be smooth, meaning that we can differentiate the function several times and each derivative is a well-defined bounded function over an interval containing a particular onsntyuleeesse.aatn tofinapproite numbeouf0pibtsynaeaitedOerenceobviouasppchrooion bauld bseet  \nDC u.Nx/ 􀀑 u.~~N~~x~~ ~~C~~ ~~h/h~~ ~~􀀀~~ ~~u.~~N~~x/~~ ~~ (1.1)  \nfor some small value of h. This is motivated by the standard definition of the derivative astieteimpoTilgnetithp(sit)eaaiodne-shhed(!seapep0Fro. iNgxith1 1ioan)ttDoCu0 usi./ceisuthiseeoalpueatofedtenlliynaet values of x 􀀕 Nx . Another one-sided approximation would be  \nD 􀀀 u.Nx/ 􀀑 u.~~N~~x/~~ ~~􀀀~~ ~~uh.~~N~~x~~ ~~􀀀~~ ~~h/ : (1.2)  \nDownloaded 06/09/ 16 to [205.155.65.226. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php](205.155.65.226. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php)  \n4 Chapter 1 . Finite Difference Approximations  \nslope DCu.Nx/  \nslope u0 .Nx/  \nslope D 􀀀 u.Nx/  \nslope D0 u.Nx/  \nNx 􀀀 h Nx Nx C h u.x/  \nFigure 1.1. Various approximations to u0 .Nx/ interpreted as the slope of secant lines.  \nEach ofthe sizehesf thee ormurror iasroguivgehyapsptoorrtiraaltuhraittslafpproximation to u0 .Nx/, meaning that Another possibility is to use the centered approximation  \nD0 u.Nx/ 􀀑 u.~~N~~x~~ ~~C~~ ~~h/u.~~N~~x~~ ~~􀀀~~ ~~h/ D 12 .DC u.Nx/ C D 􀀀 u.Nx//: (1.3)  \nTohfitshte so olonpe-odthed alpine interpproximatiooigefiated bvhe . andFrCFh aigu i1s1siimply thet shouldverage cleaer thatapprwoxwoumatiesxpectIn faDct0uth.i/gtoivegsivaesa betteecondapprdeoxaicmatiocuratenhan epproxiieaoo—the otheneer-sr proportional to h2 and hence is much smaller than the error in a first order approximation when h is small.  \nOther approximations are also possible, for example,  \nD3 u.Nx/ 􀀑 61h Œ2u.Nx C h/ C 3u.Nx/ 􀀀 6u.Nx 􀀀 h/ C u.Nx 􀀀 2h/􀂍: (1.4)  \nIt may not be clear where this came from or why it should approximate u0 at all, but in fact it turns out to be a third order accurate approximation—the error is proportional to h3 when h is small.  \nOur first goal is to develop systematic ways to derive such formulas and to analyze their accuracy and relative worth. First we will look at a typical example of how the errors in these formulas compare.  \n0 Example 1.1. Let u.x/ D sin.x/ and Nx D 1; thus we are trying to approximate uof.1h/fDorceoac.1/of Dthe0f:5o3u0l2a3sa.","cbCait0NYdmeaKgg","https://ap.wps.com/l/cbCait0NYdmeaKgg","pdf",135020,"English","# Chapter 1 Finite Difference Approximations\n## Truncation errors","[{\"question\":\"Why do finite difference methods replace derivatives with finite difference approximations?\",\"answer\":\"Finite difference methods replace derivatives in differential equations with discrete approximations using function values at grid points. This turns the differential equation into a solvable finite algebraic system on a computer.\"},{\"question\":\"How are one-sided, centered, and higher-order derivative approximations related?\",\"answer\":\"The chapter derives one-sided forward and backward differences and a centered difference that averages them. It then presents a third-order accurate approximation (with error proportional to h^3) to show improved accuracy.\"},{\"question\":\"How does truncation error determine the order of accuracy?\",\"answer\":\"By expanding function values in Taylor series around a grid point, the chapter shows that different finite difference formulas have leading error terms proportional to different powers of h. The power of h directly indicates whether the method is first, second, or third order accurate.\"}]","Chapter 1 - Finite Difference Approximations | PDF",1789973752,23]