二維波動(dòng)方程的有限差分法

1、學(xué)生實(shí)驗(yàn)報(bào)告實(shí)驗(yàn)課程名稱偏微分方程數(shù)值解開課實(shí)驗(yàn)室數(shù)統(tǒng)學(xué)院學(xué)院數(shù)統(tǒng)年級(jí)20院專業(yè)班信計(jì)02班學(xué)生姓名學(xué)號(hào)開課時(shí)間2015至2016學(xué)年第2學(xué)期總成績(jī)教師簽名數(shù)學(xué)與統(tǒng)計(jì)學(xué)院制開課學(xué)院、實(shí)驗(yàn)室：數(shù)統(tǒng)學(xué)院實(shí)驗(yàn)時(shí)間：2016年6月20日實(shí)驗(yàn)項(xiàng)目名稱二維波動(dòng)方程的有限差分法實(shí)驗(yàn)項(xiàng)目類型驗(yàn)證演示綜合設(shè)計(jì)其他指導(dǎo)教師日力成績(jī)是一.實(shí)驗(yàn)?zāi)康耐ㄟ^該實(shí)驗(yàn)，要求學(xué)生掌握求解二維波動(dòng)方程的有限差分法，并能通過計(jì)算機(jī)語言編程實(shí)現(xiàn)。.實(shí)驗(yàn)內(nèi)容考慮如下的初值問題:-2_2_23=羊+察(4丫產(chǎn)Q=(0,1),te(0,1.4):t二x:y227(x,y,0產(chǎn)sinwxsinyy,一u(x,y,0)=0,(x,y產(chǎn)(0,1)

2、x,y,t)=0,x,y；：1t10,1.412.3.4.在第三部分寫出問題（1）三層顯格式。根據(jù)你寫出的差分格式，編寫有限差分法程序。將所寫程序放到第四部分。取h=0.14=0.1h,分別將t=0.5,1.0,1.4時(shí)刻的數(shù)值解畫圖顯示。該問題的解析解為U（x,y,t）=cosJ2：tsinnxsinny,將四個(gè)時(shí)刻的數(shù)值解的誤差畫圖顯示，對(duì)數(shù)值結(jié)果進(jìn)行簡(jiǎn)單的討論。.實(shí)驗(yàn)原理、方法（算法）、步驟網(wǎng)格劃分h=0.1,7=0.1h，故N=1.4htk=kT,k=0,1,111,140。在內(nèi)網(wǎng)點(diǎn)(K,yj,tk)=10,M=140,Xi=ih,yj=jh,i,j=0,1川，10,利用二階中心差商，

3、對(duì)（1）建立差分格式:k1八kUi,j一2。2k-1kkkkkk'Ui,j=Ui1,j-2ui,j.Ui4,j.Ui,j1-2ui,j,Ui,jh2h2(2)整理得到:k12kkUi,j=rUi1,j'Ui4,jUikj1Uikj42-4r2qkj-UiT(3)其中，i,j=1,2,|119k=1,2川|,139,網(wǎng)比r=2=0.1,局部截?cái)嗾`差為ol+h2）。h考慮邊界條件ux,y,t=0,x,y）m.,t.0,1.41,差分格式為：_k_k_k_ku0,0=u0,N=uN,0=uN,N=0,k=0,1JH,140(4)考慮初始條件u（x,y,0）=sinnxsinny,差分

4、格式為:0ui,j=sin二xsinr：yj=sin二ihsin二jh,i,j=0,1,l|l,10(5)2考慮初始條件ut（x,y,0）=0,（x,y產(chǎn)（0,1）,利用二階差冏近似:11Uj-uij,j,j=0,i,j=0,1,111,102.設(shè)k=0時(shí)刻的點(diǎn)為內(nèi)點(diǎn)，則滿足差分格式（2）,代入上式得到:u；j=r2（u'j+u：+u：j書+5+）十（24r2從u：(6)將（6）得到的結(jié)果u：j=5：代入（7）中，整理得到:1ui,j綜上（2）、(4)、k1uui,j12000020三-2r31,jujui,j15,j1-2rui,j(5)、(8)得到三層顯格式的差分格式為：2kkkk

5、2kkrui1,juy,jui,j1u,j2-4ru,j-u5(8)0uui,jui1ji,j=1,2,|l|,9,k=1,2川,139u0,0=u0,N=uN,0=uN,N-0,k-0,1,|I,140二sin-xisin.：yjisin二ihsin二jh,i,j=0,1,|l,10=22(50書+u0j書+u：j)+(12r2以/，j=0,1,|,10(9)其中=三=0.1,局部截?cái)嗾`差為o（T2+h2）。h四.實(shí)驗(yàn)環(huán)境（所用軟件、硬件等）及實(shí)驗(yàn)數(shù)據(jù)文件Matlab%二維波動(dòng)方程數(shù)值計(jì)算（關(guān)鍵：怎么運(yùn)用i,j,k三個(gè)指標(biāo)建立循環(huán)）clc;%可以將代碼換成函數(shù)m文件h=0.1;tau=0.1

6、*h;%定義步長(zhǎng)r=tau/h;%網(wǎng)比空間網(wǎng)格剖分x,y,t=meshgrid(0:h:1,0:h:1,0:tau:1.4);%uu=cos(sqrt(2)*pi*t).*sin(pi*x).*sin(pi*y);%精確解計(jì)算%第一層網(wǎng)點(diǎn)計(jì)算u=sin(pi*x).*sin(pi*y);%初始條件u1=u(:,:,1);%因?yàn)榇藭r(shí)得到的u為11x11x141,故只取第一層%第二層網(wǎng)點(diǎn)計(jì)算fori=2:10forj=2:10u(i,j,2)=0.5*rA2*(u(i+1,j,1)+u(i-1,j,1)+u(i,j+1,1)+u(i,j-1,1)+(1-2*rA2)*u(i,j,1)；u(11,:

7、,2)=0;u(:,11,2)=0;endendu2=u(:,:,2);%S3-141層網(wǎng)點(diǎn)計(jì)算fork=2:140fori=2:10forj=2:10u(i,j,k+1)=rA2*(u(i+1,j,k)+u(i-1,j,k)+u(i,j+1,k)+u(i,j-1,k)+(2-4*rA2)*u(i,j,k)-u(i,j,k-1);u(11,:,k+1)=0;u(:,11,k+1)=0;endend%end%wucha=abs(u-uu);%求絕對(duì)誤差矩陣11x11x141wucha1=wucha(:,:,11);%計(jì)算t=0.1時(shí)刻的絕對(duì)誤差矩陣11x11wucha2=wucha(:,:,51

8、);%計(jì)算t=0.5時(shí)刻的絕對(duì)誤差矩陣11x11wucha3=wucha(:,:,101);%計(jì)算t=1.0時(shí)刻的絕對(duì)誤差矩陣11x11wucha4=wucha(:.:.141);%計(jì)算t=1.4時(shí)亥1I的絕對(duì)誤差矢口陣11x11x0=0:h:1;y0=0:h:1;%作t=0.1時(shí)刻的絕對(duì)誤差圖subplot（2,2,1）;mesh（x0,y0,wucha1）;title（'t=0.1時(shí)刻的絕對(duì)誤差'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'絕對(duì)誤差值'）；%作t=0.5時(shí)刻的絕對(duì)誤差圖subpl

9、ot（2,2,2）;mesh（x0,y0,wucha2）;title（'t=0.5時(shí)刻的絕對(duì)誤差'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'絕對(duì)誤差值'）；%乍t=1.0時(shí)刻的絕對(duì)誤差圖subplot（2,2,3）;mesh（x0,y0,wucha3）;title（'t=1.0時(shí)刻的絕對(duì)誤差'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'絕對(duì)誤差值'）；%作t=1.4時(shí)刻的絕對(duì)誤差圖subplot（2,2

10、,4）;mesh（x0,y0,wucha4）;title（'t=1.4時(shí)亥1J的絕對(duì)誤差'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'絕對(duì)誤差值'）；%1%1解%t=0.1、0.5時(shí)刻的數(shù)值解與精確解subplot（2,2,1）;mesh（x0,y0,u（:,:,11）;%作t=0.1時(shí)刻的數(shù)值解title（'t=0.1時(shí)刻的數(shù)值解'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'u值'）；subplot（2

11、,2,2）;mesh（x0,y0,uu（:,:,11）;%作t=0.1時(shí)刻的精確解title（'t=0.1時(shí)刻的精確解）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'u值'）；%t=0.5時(shí)刻的數(shù)值解與精確解subplot（2,2,3）;mesh（x0,y0,u（:,:,51）;%作t=0.5時(shí)刻的數(shù)值解title（'t=0.5時(shí)刻的數(shù)值解'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'u值'）；subplot（2,2

12、,4）;mesh（x0,y0,uu（:,:,51）;%作t=0.5時(shí)刻的精確解title（'t=0.5時(shí)刻的精確解'）;%xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'u值'）；%«%（%（%t=1.0、1.4時(shí)刻的數(shù)值解與精確解subplot（2,2,1）;mesh（x0,y0,u（:,:,101）;%作t=1.0時(shí)刻的數(shù)值解title（'t=1.0時(shí)刻的數(shù)值解'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'

13、u值'）；subplot（2,2,2）;mesh（x0,y0,uu（:,:,101）;%作t=1.0時(shí)刻的精確解title（'t=1.0時(shí)刻的精確解'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'u值'）；%t=1.4時(shí)刻的數(shù)值解與精確解subplot（2,2,3）;mesh（x0,y0,u（:,:,141）;%作t=1.4時(shí)刻的數(shù)值解title（'t=1.4時(shí)刻的數(shù)值解'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（

14、'u值'）；subplot（2,2,4）;mesh（x0,y0,uu（:,:,141）;%作t=1.4時(shí)刻的精確解title（'t=1.4時(shí)刻的精確解'）;xlabel（'x變量'）;ylabel（'y變量'）;zlabel（'u值'）；五.實(shí)驗(yàn)結(jié)果及實(shí)例分析1、t=0.1、051.R1.4時(shí)刻的數(shù)值解與精確解圖圖1t=0.1、0.5時(shí)刻的數(shù)值解、精確解圖2t=1.0、1.4時(shí)刻的數(shù)值解、精確解注：上兩圖為四個(gè)時(shí)刻的數(shù)值解與精確解，r='=0.1<-L（p代表維數(shù)）,本文p=2h、.P,三層顯格式達(dá)二

15、階收斂，不難看出，收斂效果很好，符合理論。下圖是四個(gè)時(shí)刻的絕對(duì)誤差圖像，從圖中看出，絕對(duì)誤差較小，且經(jīng)過計(jì)算得到，收斂階近似于2,正好符合理論值2、t=0.1、051.01.4時(shí)刻的絕對(duì)誤差圖t=1。劃的拾"一蝴弱的施好總XMUJM的推時(shí)-一mof”屁星燈盤.無0.02作。1序.0.1?10006.0.-3,1口圖3四個(gè)時(shí)刻的絕對(duì)誤差3、四個(gè)時(shí)刻（t=0.1、0.5、1.0、1.4）的絕對(duì)誤差表t=0.1時(shí)刻的絕對(duì)誤差0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00010.0001

16、0.00020.00020.00020.00020.00020.00010.00010.00000.00000.00010.00030.00040.00040.00050.00040.00040.00030.00010.00000.00000.00020.00040.00050.00060.00060.00060.00050.00040.00020.00000.00000.00020.00040.00060.00070.00070.00070.00060.00040.00020.00000.00000.00020.00050.00060.00070.00080.00070.00060.0005

17、0.00020.00000.00000.00020.00040.00060.00070.00070.00070.00060.00040.00020.00000.00000.00020.00040.00050.00060.00060.00060.00050.00040.00020.00000.00000.00010.00030.00040.00040.00050.00040.00040.00030.00010.00000.00000.00010.00010.00020.00020.00020.00020.00020.00010.00010.00000.00000.00000.00000.0000

18、0.00000.00000.00000.00000.00000.00000.0000t=0.5時(shí)刻的絕對(duì)誤差0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00070.00130.00180.00210.00220.00210.00180.00130.00070.00000.00000.00130.00250.00340.00400.00420.00400.00340.00250.00130.00000.00000.00180.00340.00470.00550.00580.00550.0047

19、0.00340.00180.00000.00000.00210.00400.00550.00650.00680.00650.00550.00400.00210.00000.00000.00220.00420.00580.00680.00710.00680.00580.00420.00220.00000.00000.00210.00400.00550.00650.00680.00650.00550.00400.00210.00000.00000.00180.00340.00470.00550.00580.00550.00470.00340.00180.00000.00000.00130.0025

20、0.00340.00400.00420.00400.00340.00250.00130.00000.00000.00070.00130.00180.00210.00220.00210.00180.00130.00070.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000t=1.0時(shí)刻的絕對(duì)誤差0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00160.00310.00430.00510.00530.0051

21、0.00430.00310.00160.00000.00000.00310.00590.00820.00960.01010.00960.00820.00590.00310.00000.00000.00430.00820.01130.01320.01390.01320.01130.00820.00430.00000.00000.00510.00960.01320.01560.01640.01560.01320.00960.00510.00000.00000.00530.01010.01390.01640.01720.01640.01390.01010.00530.00000.00000.0051

22、0.00960.01320.01560.01640.01560.01320.00960.00510.00000.00000.00430.00820.01130.01320.01390.01320.01130.00820.00430.00000.00000.00310.00590.00820.00960.01010.00960.00820.00590.00310.00000.00000.00160.00310.00430.00510.00530.00510.00430.00310.00160.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000t=1.4時(shí)刻的絕