差分作业张国宇终稿

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1、华中科技大学研究生课程考试答题本考生姓名考生学号M201570061系、年级数学与统计学院、2015级类别非定向考试科目有限元法与有限差分法的应用题号得分题号得分总分：评卷人：注：1、无评卷人签名试卷无效。2、必须用钢笔或圆珠笔阅卷，使用红色。用铅笔阅卷无效。题号回答内容得分Pleasededucethesecond-orderforwarddifferential,second-orderbackwarddifferentialandsecond-ordercentraldifferentialofy=f（x）,thendedu

2、cetheprecisionofy=f（x）second-ordercentraldifferential.（10points）Answer：（1）Firstofall,threedifferentialformulaaregivenasfollows:Thesecond-orderforwarddifferentialformula：A〉二A（4y）二A［/（x+心）-/（尢）］Ax2Ay22_纣0+心）一纣（兀）A?［/（兀+2心）一/（兀+心）］一［/（兀+心）一/（兀）］ZP_f（x+2Ar）一2f{x+Ax）+f（x）A

3、x2Thesecond・orderbackwarddifferentialformula：A2yA（A^）A［/（x）一f（x一心）］—二3AjtArAr=纣（兀）-纣（无-心）二［/（X）—/（兀一心）］一［/（兀一心）一/（兀一2心）］Ax2/⑴一2/（无一心）+/（兀一2心）Ax2Thesecond・ordercentraldifferentialformula：A2y_△（△》）_△［•/'（尤+£心）_./'（乂_£心）］Ay7-Ar2ZPA/（x+£心）-A/（x-~心）Ax5=［/（X+Ay）—/（兀）］一［/（兀）

4、一/（兀一Ax）］Ax2=/（兀+心）一2/（兀）+f（x一心）Ax2（2）Thefunctionf（x+Ar）andf（x-areexpandedatthepointofxtillthe3rdorderwiththeremainderomitted,/(x+Ax)_/(x)+/(x)Ax+2,(Ax)+3j(Ax)+0((3)/(X一Ax)=f(x)-fx}x+心)2—(心)3+0((心T)ThenbringtheTaylorexpansionof/(x+Ax)and/(x-Ax)tothesecond-ordercentr

5、aldifferentialformula,afterrearrangingwegetthatA2y_/(x+Ax)-2f(x)+f(x-Ax)Ax2Ax2=m)+o((3)Nowwecansaythattheprecisionofthesecond・ordercentraldifferentialformulais2.题号回答内容得分PleasewritetheFTCSformatoftheconvectionequation洋+a埜=0

6、recision.(15points)Answer：(1)Usingdifferencequotienttoreplacethederivative,weget:xi=x()+iAx,Z=0,1，2,•…(哲丫〜鈔-JIdt丿~"些〜:n(dx12Axtn=r0+nAt,〃=0,1,2，…So,atthepointof(石丿打)，theconvectionequationcanbeapproximatedbyj+iyd+Q―=0

7、onvectionequation.Afterrearrangement,wecangettheFTCSformatofthegivenconvectionequation：Vb-Q半(算-列)<2Ar席=%)(2)Forageneralfunctionf(x),weuseTaylor'sexpansionatthepointofxtoget:.伦+3=加+f(站+今(3+匕严3+0((3)/(x-Ax)=/(x)-/(x)Ax+(Ax)2-(Ax)3+0((Ar)3)⑴一⑵By,wehave:⑴⑵Similarly,•心+2)—

8、/⑴-f(x)+O(Ax)Ax/(7)-/(7)=f(x)+0((3)Sotheerrorofthe1-orderforwarddifferentialformulais0(Ar),theerrorofthe1-ordercentraldiffer