偏微分方程数值解实验报告.docx
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偏微分方程数值解实验报告
一、题目：
1、用有限元方法求下列边值问题的数值解：
其中其精确解为，取h=0.1
要求：
（1）将精确解与用有限元得到的数值解画在同一图中
（2）、、
2、用线性元求解下列问题的数值解：
精确到小数点后第六位，并画出解曲面。
3、用Crank-Nicolson差分法求解Burger方程
其中取要求画出解曲面。
迭代格式如下：
二、代码：
1、
%RitzGalerkin方法求解方程
functionu1=Ritz（x）
%定义步长
h=1/100;
x=0:
h:
1;
n=1/h;
a=zeros（n-1,1）;
b=zeros（n,1）;
c=zeros（n-1,1）;
d=zeros（n,1）;
%求解Ritz方法中内点系数矩阵
fori=1:
1:
n-1
b（i）=（1/h+h*pi*pi/12）*2;
d（i）=h*pi*pi/2*sin（pi/2*（x（i）+h））/2+h*pi*pi/2*sin（pi/2*x（i+1））/2;
end
%右侧导数条件边界点的计算
b（n）=（1/h+h*pi*pi/12）;
d（n）=h*pi*pi/2*sin（pi/2*（x（i）+h））/2;
fori=1:
1:
n-1
a（i）=-1/h+h*pi*pi/24;
c（i）=-1/h+h*pi*pi/24;
end
%调用追赶法
u=yy（a,b,c,d）
%得到数值解向量
u1=[0,u]
%对分段区间做图
plot（x,u1）
%得到解析解
y1=sin（pi/2*x）;
holdon
plot（x,y1,'o'）
legend（'数值解','解析解'）
functionx=yy（a,b,c,d）
n=length（b）;
q=zeros（n,1）;
p=zeros（n,1）;
q
（1）=b
（1）;
p
（1）=d
（1）;
fori=2:
1:
n
q（i）=b（i）-a（i-1）*c（i-1）/q（i-1）;
p（i）=d（i）-p（i-1）*c（i-1）/q（i-1）;
end
x（n）=p（n）/q（n）;
forj=n-1:
-1:
1
x（j）=（p（j）-a（j）*x（j+1））/q（j）;
end
x
x=
Columns1through11
0.01570.03140.04710.06280.07850.09410.10970.12530.14090.15640.1719
Columns12through22
0.18740.20280.21810.23350.24870.26390.27900.29400.30900.32390.3387
Columns23through33
0.35350.36810.38270.39720.41150.42580.44000.45400.46790.48180.4955
Columns34through44
0.50910.52250.53580.54900.56210.57500.58780.60040.61290.62530.6374
Columns45through55
0.64950.66130.67300.68460.69590.70710.71810.72900.73970.75010.7604
Columns56through66
0.77050.78050.79020.79970.80900.81820.82710.83580.84440.85270.8608
Columns67through77
0.86870.87630.88380.89100.89810.90490.91140.91780.92390.92980.9355
Columns78through88
0.94090.94610.95110.95580.96030.96460.96860.97240.97590.97930.9823
Columns89through99
0.98510.98770.99010.99210.99400.99560.99690.99810.99890.99950.9999
Column100
1.0000
u=
Columns1through11
0.01570.03140.04710.06280.07850.09410.10970.12530.14090.15640.1719
Columns12through22
0.18740.20280.21810.23350.24870.26390.27900.29400.30900.32390.3387
Columns23through33
0.35350.36810.38270.39720.41150.42580.44000.45400.46790.48180.4955
Columns34through44
0.50910.52250.53580.54900.56210.57500.58780.60040.61290.62530.6374
Columns45through55
0.64950.66130.67300.68460.69590.70710.71810.72900.73970.75010.7604
Columns56through66
0.77050.78050.79020.79970.80900.81820.82710.83580.84440.85270.8608
Columns67through77
0.86870.87630.88380.89100.89810.90490.91140.91780.92390.92980.9355
Columns78through88
0.94090.94610.95110.95580.96030.96460.96860.97240.97590.97930.9823
Columns89through99
0.98510.98770.99010.99210.99400.99560.99690.99810.99890.99950.9999
Column100
1.0000
u1=
Columns1through11
00.01570.03140.04710.06280.07850.09410.10970.12530.14090.1564
Columns12through22
0.17190.18740.20280.21810.23350.24870.26390.27900.29400.30900.3239
Columns23through33
0.33870.35350.36810.38270.39720.41150.42580.44000.45400.46790.4818
Columns34through44
0.49550.50910.52250.53580.54900.56210.57500.58780.60040.61290.6253
Columns45through55
0.63740.64950.66130.67300.68460.69590.70710.71810.72900.73970.7501
Columns56through66
0.76040.77050.78050.79020.79970.80900.81820.82710.83580.84440.8527
Columns67through77
0.86080.86870.87630.88380.89100.89810.90490.91140.91780.92390.9298
Columns78through88
0.93550.94090.94610.95110.95580.96030.96460.96860.97240.97590.9793
Columns89through99
0.98230.98510.98770.99010.99210.99400.99560.99690.99810.99890.9
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