偏微分方程数值解实验报告内容参考.docx

偏微分方程数值解实验报告内容参考.docx

《偏微分方程数值解实验报告内容参考.docx》由会员分享，可在线阅读，更多相关《偏微分方程数值解实验报告内容参考.docx（14页珍藏版）》请在冰豆网上搜索。

偏微分方程数值解实验报告内容参考

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

0.98230.98510.98770.99010.99210.99400.99560.99690.99810.99890.9995

Columns100through101

0.99991.0000

ans=

Columns1through10

00.01570.03140.04710.06280.07850.09410.10970.12530.1409

Columns11through20

0.15640.17190.18740.20280.21810.23350.24870.26390.27900.2940

Columns21through30

0.30900.32390.33870.35350.36810.38270.39720.41150.42580.4400

Columns31through40

0.45400.46790.48180.49550.50910.52250.53580.54900.56210.5750

Columns41through50

0.58780.60040.61290.62530.63740.64950.66130.67300.68460.6959

Columns51through60

0.70710.71810.72900.73970.75010.76040.77050.78050.79020.7997

Columns61through70

0.80900.81820.82710.83580.84440.85270.86080.86870.87630.8838

Columns71through80

0.89100.89810.90490.91140.91780.92390.92980.93550.94090.9461

Columns81through90

0.95110.95580.96030.96460.96860.97240.97590.97930.98230.9851

Columns91through100

0.98770.99010.99210.99400.99560.99690.99810.99890.99950.9999

Column101

1.0000

2、function[u]=Q_2（P）

formatlong

ifnargin<1

P=16;

end

f=2;beta