9、给出初边值问题
2
u
2
x
|
u
1
x
sin
x
0(
)1
x
(0
)0
t
0(
)1
x
u
u
2
2
t
|
u
x
|
t
0
0
u
t
1
8
|
t
0
0
用显示查分格式
U
1
n
m
2
U
n
m
U
1
n
m
2
(
Ur
n
m
1
2
U
n
m
U
n
m
1
)
求出网格节点(m,n) (m=1,2…9,n=1,2,….5)上的 n
mU 值,其中导数条件用中心差商代替,
k=h=0.1,给出解析解并与网格节点上的近似解比较。
10、考虑逼近双曲型方程
2
2
u
2
u
2
x
的隐式差分格式
t
1
1
2
n
U
m
x
2
k
试:(1)分析其稳定性
(2)编制出用于计算第九题初边值条件的计算机程序,并进行数值试验
1(1
4
h
2
U
t
2
U
x
2
U
x
1
n
m
1
4
1
2
n
m
n
m
)
2
一、算法设计
1、显示差分格式为
2
U
U
1
n
m
n
m
1
第九题
n
m
U
n
m
1
2)
U
n
m
U
1
n
m
(1)
0|
t
2
(
Ur
u
t
2
k
2、对于
0
的处理采用差分近似
U x
,
m
k U x
,
k
m
0
1
U U
m
1
m
0
在(1)中令 n=0,得
2
U U
U
2
(
r U
1
m
0
m
1
0
m
0
m
1
) 2
1
U U
m
0
m
(2)
(3)
联立(2)(3)得,消去 1
mU 得:
U
1
m
2
r
2
(
U
0
m
1
2
U U
0
m
0
m
)
U
0
m
1
于是可得:
1
n
m
1
m
1
n
m
0
m
0
m
u
U
U
U
U
U
u
x
n
m
(
U
2
(
r U
1 sin
8
2
r
2
2
(
r U
1 sin
8
2
r
2
0
0
(
U
x
U
1
m
m
1
m
0
1
2
U U
n
m
n
m
1
) 2
U U
n
m
1
n
m
0
m
1
2
U U
0
m
0
m
)
U
0
m
1
n
m
1
2
U U
n
m
n
m
1
) 2
U U
n
m
1
n
m
0
m
1
2
U U
0
m
0
m
)
U
0
m
1
2、方程的解析解为
x
sin[
(
u
1
16
3、数据:
t
)]
sin[
(
x
)]
t
精确解在给定网格格点的值
n=0
0
0
0
0
0
0
0
0
0
0
0
n=1
0
n=2
0
n=3
0
n=4
0
n=5
0
n=6
0
0.011936
0.022704
0.03125
0.036737
0.038627
0.036737
0.022704
0.043186
0.059441
0.069877
0.073473
0.069877
0.03125
0.059441
0.081814
0.096178
0.101127
0.096178
0.036737
0.069877
0.096178
0.113064
0.118882
0.113064
0.038627
0.073473
0.101127
0.118882
0.125
0.118882
0.036737
0.069877
0.096178
0.113064
0.118882
0.113064
0.03125
0.059441
0.081814
0.096178
0.101127
0.096178
0.022704
0.043186
0.059441
0.069877
0.073473
0.069877
0.011936
0.022704
0.03125
0.036737
0.038627
0.036737
2.07E-09
3.94E-09
5.42E-09
6.37E-09
6.7E-09
6.37E-09
近似解在给定网格格点的值
n=0
0
n=1
0
n=2
0
n=3
0
n=4
0
n=5
0
n=6
0
0.038627
0.036737
0.03125
0.022704
0.011936
1.04E-09
-0.01194
0.073473
0.069877
0.059441
0.043186
0.022704
1.97E-09
-0.0227
0.101127
0.096178
0.081814
0.059441
0.03125
2.71E-09
-0.03125
0.118882
0.113064
0.096178
0.069877
0.036737
3.19E-09
-0.07347
m=0
m=1
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
m=10
m=0
m=1
m=2
m=3
m=4
0.125
0.118882
0.101127
0.073473
0.038627
-0.03674
9.56E-09
0.118882
0.113064
0.096178
0.069877
-2.7E-09
0.038627
-0.03674
0.101127
0.096178
0.081814
0.022704
0.069877
-2.7E-09
-0.03125
m=5
m=6
m=7
m=8
m=9
0.073473
0.069877
0.022704
0.081814
0.022704
0
0
0.069877
0.022704
0.011936
m=10
6.7E-09
6.37E-09
0
0
0
-2E-09
-1E-09
0
-0.0227
-0.01194
0
精确解与近似解在给定网格格点的误差
n=0
0
n=1
0
n=2
0
n=3
0
n=4
0
n=5
0
n=6
0
0.038627
0.0248
0.008546
-0.00855
-0.0248
-0.03863
-0.04867
0.073473
0.047173
0.016255
-0.01625
-0.04717
-0.07347
-0.09258
0.101127
0.064928
0.022373
-0.02237
-0.06493
-0.10113
-0.12743
0.118882
0.076327
0.0263
-0.0263
-0.07633
-0.11888
-0.18654
0.125
0.080255
0.027654
-0.02765
-0.08025
-0.16174
-0.11888
0.118882
0.076327
0.0263
-0.0263
-0.11306
-0.08025
-0.1498
0.101127
0.064928
0.022373
-0.05911
-0.0263
-0.10113
-0.12743
0.073473
0.047173
-0.02048
0.022373
-0.04717
-0.07347
-0.09258
0
-0.01194
0.047173
-0.00855
-0.0248
-0.03863
-0.04867
m=0
m=1
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
m=10
6.7E-09
4.3E-09
-3.9E-09
-5.4E-09
-6.4E-09
-6.7E-09
-6.4E-09
4、代码:
clc
u=zeros(11,7);
u1=zeros(11,7);
pi=3.1415926;
h=0.1;
k=0.1;
r=k/h;
for i=1:11
end
for i=1:11
u(i,1)=(1/8)*sin(pi*(i-1)*h) ;
u(i,2)=(1/8)*sin(pi*(i-1)*h)+r*r*(1/8)*sin(pi*(i-1)*h)*(cos(pi*h)-1);
end
for j=1:7
end
for j=1:7
u(1,j)=0;
u(10,j)=0;
end
for j=2:6
for i=2:10
end
end
for i=1:11
for j=1:7
end
end
u(i,j+1)=2*(1-r*r)*u(i,j)+r*r*(u(i+1,j)+u(i-1,j))-u(i,j-1);
u1(i,j)=(1/8)*sin(pi*(i-1)*k)*sin(pi*(j-1)*h);
x=[0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1];
xlswrite('C:\Users\zhengyd\Desktop\ÎÊÌâ1.xlsx',u','sheet1');
xlswrite('C:\Users\zhengyd\Desktop\ÎÊÌâ1.xlsx',u1','sheet2');
xlswrite('C:\Users\zhengyd\Desktop\ÎÊÌâ1.xlsx',(u-u1)','sheet3');
一、稳定性分析:
第十题
1、令
u
2
2
t
u
t
2
u
2
x
uv
,
x
u
t
化为
w
w w
t
x
,
v
x
令 ( ,
v w
u
)T
A
0 1
1 0
u
t
令
n
V
m
0
uA
x
U U
k
n
m
1
n
m
,
n
W
m
n
m
1
n
m
U U
h
1
2
n
W
m
1
1
2
n
W
m
1
1
2
2
h
n
V
m
n
V
1
m
1
n
V
1
m
2
h
1
1
2
1
2
k
n
V
m
n
V
m
n
W
m
n
W
m
k
1
n
W
1
m
2
隐式差分格式等价于:
n
W
m
所以此差分格式是稳定的
二、处边值问题求解:
1
2
k
所以:
1(1
4
h
2
U
t
2
U
x
n
V
m
1
n
m
2
1
4
2
r U
1
n
1
m
(1
1
2
2
)
r U
1
n
m
1
n
m
1
2
1
4
2
U
x
n
m
1
4
2
U
x
1
n
m
)
2
r U
1
n
1
m
1
4
2
(
r U
1
n
1
m
U
1
n
1
m
)
(1
1
2
2
)
r U
1
n
m
1
2
2
(
r U
n
m
1
U
n
m
1
)
(2
2
)
r U
n
m
化为对角方程组逐层求解
m=10;
2
1
r
2
2
r
2
1
r
4
1
2
1
r
4
1
4
1
2
...
2
1
2
r
1
2
1
r
4
1
2
r
2
1
r
4
...
1
4
r
2
2
r
2
r
2
1
2
1
1
1
n
U
1
n
U
2
n
U
3
...
1
n
U
1
m
...
1
2
2
r
1
1
2
2
1
2
2
r
2
r
2
r
2
r
1
2
2
2
r
...
2
1
r
2
...
1
2
r
2
n
U
1
n
U
2
n
U
3
...
n
U
1
m
2
...
2
r
1
2
2
n
r U
0
1
2
n
r U
0
0
0
...
1
2
n
r U
m
1
2
2
n
r U
m
2
r
)
1
2
2
r
(1
1
4
三、数据
1
4
(1
2
r
1
2
2
r
1
4
2
r
)
1
4
(1
2
r
1
2
...
1
n
U
1
1
n
U
2
1
n
U
3
...
1
n
U
1
m
)
1
4
1
4
1
2
n
r U
0
1
2
n
r U
m
1
4
1
4
...
1
2
(1
2
r
2
r
)
2
r
1
4
...
1
4
2
r
近似解在给定网格格点的值
n=0
0
n=1
0
n=2
0
n=3
0
n=4
0
n=5
0
n=6
0
0.038627
0.036737
0.031336
0.022941
0.012354
0.000587
-0.01124
0.073473
0.069877
0.059604
0.043637
0.023499
0.001117
-0.02137
0.101127
0.096178
0.082038
0.060061
0.032344
0.001537
-0.02942
0.118882
0.113064
0.096442
0.070605
0.038023
0.001807
-0.03458
0.125
0.118882
0.101405
0.074239
0.039979
0.0019
-0.03636
0.118882
0.113064
0.096442
0.070605
0.038023
0.001807
-0.03458
0.101127
0.096178
0.082038
0.060061
0.032344
0.001537
-0.02942
0.073473
0.069877
0.059604
0.043637
0.023499
0.001117
-0.02137
0.038627
0.036737
0.031336
0.022941
0.012354
0.000587
-0.01124
m=0
m=1
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
m=10
0
0
0
0
0
0
0
精确解在给定网格格点的值
n=0
0
0
0
0
0
m=0
m=1
m=2
m=3
m=4
n=1
0
n=2
0
n=3
0
n=4
0
n=5
0
n=6
0
0.011936
0.022704
0.03125
0.036737
0.038627
0.036737
0.022704
0.043186
0.059441
0.069877
0.073473
0.069877
0.03125
0.059441
0.081814
0.096178
0.101127
0.096178
0.036737
0.069877
0.096178
0.113064
0.118882
0.113064
m=5
m=6
m=7
m=8
m=9
m=10
0
0
0
0
0
0
0.038627
0.073473
0.101127
0.118882
0.125
0.118882
0.036737
0.069877
0.096178
0.113064
0.118882
0.113064
0.03125
0.059441
0.081814
0.096178
0.101127
0.096178
0.022704
0.043186
0.059441
0.069877
0.073473
0.069877
0.011936
0.022704
0.03125
0.036737
0.038627
0.036737
2.07E-09
3.94E-09
5.42E-09
6.37E-09
6.7E-09
6.37E-09
精确解与近似解在给定网格格点的误差
n=0
n=1
n=2
n=3
n=4
n=5
n=6
0
0
0
0
0
0
0
0.038627
0.0248
0.008631
-0.00831
-0.02438
-0.03804
-0.04797
0.073473
0.047173
0.016418
-0.0158
-0.04638
-0.07236
-0.09125
0.101127
0.064928
0.022597
-0.02175
-0.06383
-0.09959
-0.12559
0.118882
0.076327
0.026565
-0.02557
-0.07504
-0.11708
-0.14765
0.125
0.080255
0.027932
-0.02689
-0.0789
-0.1231
-0.15524
0.118882
0.076327
0.026565
-0.02557
-0.07504
-0.11708
-0.14765
0.101127
0.064928
0.022597
-0.02175
-0.06383
-0.09959
-0.12559
0.073473
0.047173
0.016418
-0.0158
-0.04638
-0.07236
-0.09125
0.038627
0.0248
0.008631
-0.00831
-0.02438
-0.03804
-0.04797
m=0
m=1
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
0
-2.1E-09
-3.9E-09
-5.4E-09
-6.4E-09
-6.7E-09
-6.4E-09
m=10
四、代码:
clc
u=zeros(9,7);
u1=zeros(11,7);
a=zeros(9,9);
b=zeros(9,9);
c=zeros(9,9);
pi=3.1415926;
h=0.1;
k=0.1;
r=k/h;
for i=2:10
u(i-1,1)=(1/8)*sin(pi*(i-1)*h) ;
end
for i=2:10
u(i-1,2)=(1/8)*sin(pi*(i-1)*h)+r*r*(1/8)*sin(pi*(i-1)*h)*(cos(pi*h)-1)
;
end
for i=1:8
a(i,i)=1+r*r/2;
b(i,i)=2-r*r;
c(i,i)=(1+r*r/2)*(-1);
a(i,i+1)=(-1)/4*r*r;
a(i+1,i)=(-1)/4*r*r;
b(i,i+1)=1/2*r*r;
b(i+1,i)=1/2*r*r;
c(i,i+1)=1/4*r*r;
c(i+1,i)=1/4*r*r;
end
a(9,9)=1+r*r/2;
b(9,9)=2-r*r;
c(9,9)=(1+r*r/2)*(-1);
for j=3:7
u(:,j)=inv(a)*b*u(:,j-1)+inv(a)*c*u(:,j-2);
j=j+1;
end
u1(i,j)=(1/8)*sin(pi*(i-1)*k)*sin(pi*(j-1)*h);
w(i,:)=u(i-1,:)
end
for j=1:7
for i=1:11
for j=1:7
end
end
w=zeros(11,7);
for i=2:10
u(1,j)=0;
end
for j=1:7
u(11,j)=0;
end
% xlswrite('C:\Users\zhengyd\Desktop\ÎÊÌâ1.xlsx',u','sheet4');
% xlswrite('C:\Users\zhengyd\Desktop\ÎÊÌâ1.xlsx',u1','sheet2');
xlswrite('C:\Users\zhengyd\Desktop\ÎÊÌâ1.xlsx',(w-u1)','sheet5');