Math 104C Homework #4
1.考虑初始条件下单位为[0,1]⇥[0,1]的热方程

ut = D（uxx + uyy）
u（0，x，y）= sin（⇡x）sin（⇡y）（1）

（b）准确性检查。验证方案的准确性并作为可能的检查

v（k）= u + c2k2 + c3k3 +···，（2）
u是确切值。保持空间结果固定（hx = hy = h，均匀

v（k / 2）= u +
1个
4
c2k2 +
1个
8
c3k3 +···（3）

R（k）= v（k）v（k / 2）
v（k / 2）v（k / 4）（4）
（c）计算D = 1的解，并绘制三个不同的时间。
course所有课程资料（课堂讲课和讨论，讲义，家庭作业，考试，网络资料）以及课程本身的知识内容均受美国联邦版权法，

1个

2. Consider the one-way wave equation ut + ux = 0 on the interval [1, 3] and for t 0 with
the following two sets of initial conditions:
u(x, 0) = (
1 |x| if |x|  1,
0 otherwise, (5)
and
u(x, 0) = e5x2
. (6)
(a) Use the forward-time forward-space scheme:
un+1
j un
j
k +
un
j+1 un
j
h = 0,
with right-point boundary condition un+1
M = un+1
M1 where xM = 3 to compute an approximation to the solution at several (up to 40) time steps. Use h = 0.02 and = k/h = 0.8.
Demonstrate numerically (plot the solution) the instability of the scheme and show that
the instability appears sooner with the less smooth initial data.
(b) Comment on the localization of the onset of instability for initial data (5) and give an
estimate of the expected growth rate of the instability per time step.
(c) Using the left boundary condition u(1) = 0, write a stable scheme and compute the
corresponding approximation for data sets (5) and (6). Plot the approximations at
representative time steps. Use again h = 0.02 and = k/h = 0.8.
2