[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-326359-105":59,"doc-detail-326359-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","chapter-15-finite-difference-approximation-of-derivatives","Chapter 15 - Finite Difference Approximation of Derivatives","","Explains how derivatives defined in continuous calculus are approximated on discrete computational grids using finite-difference formulas. Introduces forward Euler, backward Euler, and centered difference schemes, emphasizing that computers cannot evaluate the limit as Δx→0. Uses Taylor series expansion to derive truncation errors, showing that both forward and backward approximations have errors of order O(Δx) for sufficiently smooth functions and small grid spacing.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-15-finite-difference-approximation-of-derivatives/326359/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/chapter-15-finite-difference-approximation-of-derivatives/326359.png","ImageObject",300,407,{"name":92,"@type":93},"River Wang","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-25","2026-09-21",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"Why are finite-difference approximations needed instead of the standard derivative definition?","Question",{"text":113,"@type":114},"Computers cannot compute the limit as Δx approaches 0, so derivatives must be approximated using values from a finite, discrete set of grid points.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What is the forward Euler approximation for u'(xi)?",{"text":118,"@type":114},"It approximates the derivative at xi using values at xi and xi+1, expressed as (ui+1 − ui)/Δx, corresponding to truncating the Taylor series after the first relevant term.",{"name":120,"@type":111,"acceptedAnswer":121},"How is the truncation error for forward and backward differences characterized?",{"text":122,"@type":114},"By Taylor series analysis, both forward and backward schemes have truncation errors that behave linearly with the grid spacing, described as O(Δx) for sufficiently smooth functions and small Δx.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},326359,1790437763,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},1099514067438,"https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542","Chapter 15  \nFinite Di↵erence Approximation of Derivatives  \n15.1 Introduction  \nThe standard definition of derivative in elementary calculus is the following  \nu (x + ∆x) − u(x)  \nu0 (x) = lim (15.1)  \n∆x!0 ∆x  \nComputers however cannot deal with the limit of ∆x ! 0, and hence a discrete analogue of the continuous case need to be adopted. In a discretization step, theset of points on which the function is defined is finite, and the function value is available on a discrete set of points. Approximations to the derivative will have to come from this discrete table of the function.  \nFigure 15 . 1 shows the discrete set of points xi where the function is known. We will use the notation ui = u(xi ) to denote the value of the function at the i-th node of the computational grid. The nodes divide the axis into a set of intervals of width ∆xi = xi+1 − xi. When the grid spacing is fixed, i.e. all intervals are of equal size, we will refer to the grid spacing as ∆x. There are definite advantages to a constant grid spacing as we will see later.  \n15.2 Finite Di↵erence Approximation  \nThe definition of the derivative in the continuum can be used to approximate the derivative in the discrete case:  \nu0 (xi) ⇡ u~~ ~~(xi~~ ~~+~~ ~~x)~~ ~~−~~ ~~u(xi) = ui+−x~~ ~~ui~~ ~~ (15.2)  \nwhere now ∆x is finite and small but not necessarily infinitesimally small, i.e   This is known as a forward Euler approximation since it uses forward di↵erencing.  \n114CHAPTER 15 . FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES  \nxi−1 xi xi+1  \nFigure 15.1: Computational grid and example of backward, forward, and central approximation to the derivative at point xi. The dash-dot line shows the centered parabolic interpolation, while the dashed line show the backward (blue), forward (red) and centered (magenta) linear interpolation to the function.  \nIntuitively, the approximation will improve, i.e. the error will be smaller, as ∆x is made smaller. The above is not the only approximation possible, two equally valid approximations are:  \nbackward Euler:  \nu0 (xi) ⇡ u~~ ~~(xi)~~ ~~−~~ ~~(xxi~~ ~~−~~ ~~∆x) = ui~~ ~~uxi−1~~ ~~ (15.3)  \nCentered Di↵erence  \nu0 (xi) ⇡ u~~ ~~(xi~~ ~~+~~ ~~∆x)2ux(xi~~ ~~−~~ ~~∆x) = ui+12uxi−1~~ ~~ (15.4)  \nAll these definitions are equivalent in the continuum but lead to di↵erent approximations in the discrete case. The question becomes which one is better, and is there a way to quantify the error committed. The answer lies in the application of Taylor series analysis. We briefly describe Taylor series in the next section, before applying them to investigate the approximation errors of finite di↵erence formulae.  \n15.2.1 Taylor series and finite di↵erences  \nTaylor series have been widely used to study the behavior of numerical approximation to di↵erential equations. Let us investigate the forward Euler with Taylor series. To do so, we expand the function u at xi+1 about the point xi:  \nu (xi + ∆xi ) = u(xi )+ ∆xi @u !!! + ∆x2i @2 u !!! + ∆x3i @3 u !!! + ... (15 .5)  \n@x !! xi 2! @x2 !! xi 3! @x3 !! xi  \n15.2. FINITE DIFFERENCE APPROXIMATION 115  \nThe Taylor series can be rearranged to read as follows: u~~ ~~(xi~~ ~~+~~ ~~i)xi~~ ~~−~~ ~~u(xi) − @@ux !! xi = x!~~ ~~i @@2~~ ~~ux2 ! xi + x!~~ ~~2~~i~~ @@3~~ ~~ux3 !!!!! xi + ..  .  \n}  \n{z  \nTruncation Error  \n(15.6)  \nwhere it is now clear that the forward Euler formula (15.2) corresponds to truncating the Taylor series after the second term. The right hand side of equation (15.6) is the error committed in terminating the series and is referred to as the truncation error. The tuncation error can be defined as the di↵erence between the partial derivative and its finite di↵erence representation. For sufficiently smooth functions, i.e. ones that possess continuous higher order derivatives, and sufficiently small ∆xi , the first term in the series can be used to characterize the order of magnitude of the error. The first term in the truncation error is the product of the second derivative evaluated at xi and the grid s","cbCaiaNRoffffCAS","https://ap.wps.com/l/cbCaiaNRoffffCAS","pdf",617683,14,"English","# 15.1 Introduction\n# 15.2 Finite Difference Approximation\n## 15.2.1 Taylor series and finite differences","[{\"question\":\"Why are finite-difference approximations needed instead of the standard derivative definition?\",\"answer\":\"Computers cannot compute the limit as Δx approaches 0, so derivatives must be approximated using values from a finite, discrete set of grid points.\"},{\"question\":\"What is the forward Euler approximation for u'(xi)?\",\"answer\":\"It approximates the derivative at xi using values at xi and xi+1, expressed as (ui+1 − ui)/Δx, corresponding to truncating the Taylor series after the first relevant term.\"},{\"question\":\"How is the truncation error for forward and backward differences characterized?\",\"answer\":\"By Taylor series analysis, both forward and backward schemes have truncation errors that behave linearly with the grid spacing, described as O(Δx) for sufficiently smooth functions and small Δx.\"}]","Chapter 15 - Finite Difference Approximation of Derivatives | PDF",1789973952,35]