AMATH 342 Fall 2020: Assignment 1

1.（5分）练习Arieh Iserles撰写的本书1.1。只证明隐式收敛

2.考虑标量线性问题y
0 = f（y），y（0）= 1，t∈[0，t？
]，其中f（y）= ay与
a <0。
（a）（1分）写下此问题的欧拉方法。
（b）（3分）现在取a = −2和t
？ =100。为证明Euler方法是收敛的，

| en，h | ≤c
λ
（exp（t
？λ）− 1）h，

2
y（tn + 1）的泰勒级数展开中的）项（请参见定理1.1的证明）
Arieh Iserles的书）。合理的选择是c = 2（请参阅Arieh Iserles著作的第7页）。

+。在实践中这有什么用吗？
（c）（4分）通过以下方法可以改善误差范围：
| en，h | ≤1
2
Ť

2小时

？ = 100，将此误差与之进行比较

| en，h | = |（1 +啊）
n-exp（anh）|。

exp（nx）-
1个
2
nx2
exp（（n-1）x）≤（1 + x）
n≤exp（nx）。
2
3.我们得到以下ODE：
ÿ
00 + by0 + cy = 0，t∈[0，1]，y（0）= 1，y0
（0）= 0，

2 = 4分。
（a）（1分）找到上述问题的确切解决方案。
（b）（1分）将上面的二阶ODE编写为一阶ODE的系统。
(c) (5 points) Implement Euler’s method to solve the system of first order ODEs. Take
b = 10. In 3 separate figures plot both the numerical solution and the exact solution. In
Figure 1 use h = 0.5, in Figure 2 use h = 0.05 and in Figure 3 use h = 0.005.
(d) (2 points) Let the error on a grid with time step h be defined as:
Eh = max
0≤nh≤1
|yn,h − y(nh)|
For h = 0.5, h = 0.05 and h = 0.005 compute Eh. You will see that the error satisfies
Eh = O(h
p
(e) (5 point) Write down the theta-method for the system of first order ODEs of question
(b). Repeat questions (c) and (d) but instead of the Euler method, use the θ-method
with θ = 0.5. Which of the two methods discussed here converges faster to the exact
4. The Lorenz equations are a simplified model of convection in the earths atmosphere, and is
given by
dx
dt =σ(y − x)
dy
dt =x(ρ − z) − y
dz
dt =xy − βz
where σ, ρ and β are system parameters.
(a) (5 points) Take σ = 10, ρ = 28 and β = 8/3. Implement Euler’s method to solve
the Lorenz equations. Let t ∈ [0, 50] and take h = 0.002. As initial condition take
(x0, y0, z0) = (1, 0, 0). Plot the solution where the horizontal axis is x and the vertical
axis is z.
(b) (1 point) Repeat question (a) but with ρ = 14.
(c) (2 points) Search online or in a textbook for the definitions of ’chaos’, ’strange attractor’, ’Lorentz attractor’, etc. to explain the plots found in questions (a) and (b).

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