8~数值分析.ppt

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1、第8章抛物型方程的差分格式模型长度为L的绝缘杆上的一维热流x=0u(0,t)=c1x=Lxu(L,t)=c2杆绝缘体热传导方程在时间t和位置x处的温度u(x,t)表示为初始温度分布为杆端点的边界值为其中k是导热率系数，σ是热量，ρ是杆的密度。8.1差分格式建立的基础一、网格将求解区域Ω分割成M∙N个小矩形，长宽分别为⊿x=h⊿t=khk0x0x1…xMxttN┆t0二、差商一阶偏导数的向前、向后、中心差商为二阶偏导数的中心差商为用(xm,tn)处的一阶向前差商近似代替微商ux，即考虑u(x,t)的Taylor展式截断误差用(xm,tn)处的一阶中心差商近似代替微商ux，

2、即截断误差为三、算子为x方向偏导数算子Tx为x方向位移算子μx为x方向平均算子x方向差分算子前差算子后差算子中心差算子（I为恒等算子）8.2显式差分格式差分方法：用差商代替微商，在网格节点上求出微分方程解的近似值的一种方法。一、一维常系数热传导方程的古典显式格式考虑一维热传导方程(2.26)在网格节点m–1,nm,nm+1,nm,n+1的差分近似取r=k/h2为步长比，得显式向前差分方程(2.29)(2.29)也称为解热传导方程(2.26)的古典显式格式。截断误差为例1用古典显式格式求解抛物型方程初始条件为边界条件为取步长⊿x=h=0.2,⊿t=k=0.02。解r=k/

3、h2=0.02/0.22=0.5,古典显式格式为…┇┇┇┇…0,00.2,00.4,01,00,0.2┇0,0.021,0.2┇1,0.02x=0.0x=0.2x=0.4x=0.6x=0.8x=1.0U=t=0.0000.64000.96000.96000.64000t=0.0200.48000.80000.80000.48000t=0.0400.40000.64000.64000.40000t=0.0600.32000.52000.52000.32000t=0.0800.26000.42000.42000.26000t=0.1000.21000.34000.34000

4、.21000t=0.1200.17000.27500.27500.17000t=0.1400.13750.22250.22250.13750t=0.1600.11130.18000.18000.11130t=0.1800.09000.14560.14560.09000t=0.2000.07280.11780.11780.07280functionu=gu_dian(f,a,b,c1,c2,m,n)%输入初值和Uh=a/(m-1);k=b/(n-1);r=k/h^2;U=zeros(n,m);%赋边界条件U(2:n,1)=c1;U(2:n,m)=c2;functiony=

5、fg(x)y=4.*(x-x.^2);%赋初始条件U(1,1:m)=fg(0:h:h*(m-1));%计算内点上u的数值解Ufori=2:nforj=2:m-1U(i,j)=(1-2*r)*U(i-1,j)+r*(U(i-1,j-1)+U(i-1,j+1));endend%gu_dianl1.m步长h=0.20,k=0.02,r=k/h2=0.5a=1;b=0.20;c1=0;c2=0;m=6;n=11;U=gu_dian('fg',a,b,c1,c2,m,n)x=0:0.2:a;y=0:0.02:b;[X,Y]=meshgrid(x,y);surf(X,Y,U)%输入

6、U后再画图若取步长h=0.2,k=0.0333,则r=k/h2=0.8333，古典显式格式为求解区域Ω={0≤x≤1,0≤t≤0.3333},由于r=0.8333<1/2，计算结果不稳定。x=0.0x=0.20x=0.40x=0.60x=0.80x=1.00U=t=0.000000.64000.96000.96000.64000t=0.033300.37340.69340.69340.37340t=0.066700.32890.42670.42670.32890t=0.100000.13640.34520.34520.13640t=0.133300.19680.1712

7、0.17120.19680t=0.166700.01150.19250.19250.01150t=0.200000.15270.04170.04170.15270t=0.23330-0.06710.13420.1342-0.06710t=0.266700.1565-0.0335-0.03350.15650t=0.30000-0.13230.12490.1249-0.13230t=0.333300.1922-0.0894-0.08940.19220%gu_dianl2.m步长h=0.20,k=0.0333,r=k/h2=0.8333a=1;