用Crank-Nicolson差分格式计算抛物型方程

1、实验二（习题2.2）用Crank-Nicolson差分格式计算抛物型方程2=52t:t:满足初始条件u|t=sin二x0_x_1和边界条件ulxJ=u1x4=0tB0在t=0.1,0.2处的解，.：t=k=0.1,.：x=h=0.12、程序#include#includeconstdoublepi=3.1415926;constintN=11;constintM=11;constdoublet=0.1;constdoubleh=0.1;constdoublee=2.71828;doubleUt(doublex);初始时刻值doubleUx1(doubletime);/左边值doubleUx2(

2、doubletime);/右边值doubleFUN(doublex,doubletime);voidmain()inti,k;doubleU1111,d9;doublea,b,T1,Tn,r;doubleg9,w9;coutab;coutT1Tn;r=t/(h*h);for(k=0;k11;k+)Uk0=Ux1(T1+t*k);Uk10=Ux2(T1+t*k);for(i=0;i11;i+)U0i=Ut(a+h*i);for(k=1;k11;k+)计算方程常数项d0=(1-r)*Uk-10+0.5*r*Uk-11;d8=(1-r)*Uk-19+0.5*r*Uk-18;for(i=1;i8;i+

3、)di=(1-r)*Uk-1i+0.5*r*(Uk-1i+1+Uk-1i-1);g0=d0/(1+r);w0=0.5*r/(1+r);for(i=1;i0;i-)Uki=gi-1+wi-1*Uki+1;cout?古典显格式计算结果(t=0.1,h=0.1):n;for(i=0;i11;i+)coutU1i;coutn;cout?精确计算结果(t=0.1,h=0.1):n;for(i=0;iN;i+)coutFUN(a+i*h,0.1);coutn;cout?古典显格式计算结果(t=0.2h=0.1):n;for(i=0;i11;i+)coutU2i;coutn;cout?精确计算结果(t=0.

4、2,h=0.1):n;for(i=0;iN;i+)coutFUN(a+i*h,0.2);coutn;doubleUt(doublex)初始时刻值if(x-int(x)=O)x=0;return(sin(pi*x);doubleUx1(doubletime)/左边值return0;doubleUx2(doubletime)/右边值return0;doubleFUN(doublex,doubletime)doubles1,s2;if(x-int(x)=0)x=0;s1=-pow(pi,2)*time;s2=pow(e,s1)*sin(pi*x);returns2;3、运行结果节点t=0.1t=0.

5、2数值解真解数值解真解000000.10.2647750.1151730.128410.0429260.20.2734880.2190720.01772720.081650.30.3053450.3015270.1136970.1123820.40.3382490.3544660.1545650.1321130.50.3561930.3727080.1642280.1389110.60.3482590.3544660.1540490.1321130.70.3078640.3015270.1293180.1123820.80.2319270.2190720.09325980.081650.90.1197650.1151730.04982320.042926100004、结果分析Crank-Nicolson格式（六点对称格式）在各个网格节点处对t,x的差商方法分别为：:u