这个作业是完成线性问题相关的数学问题

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
中点规则(1.12)。您无需证明theta方法(1.13)的收敛性。
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,
我们证明了
| en,h | ≤c
λ
(exp(t
?λ)− 1)h,
其中λ是与f(y)相关的Lipschitz常数,而c是与

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)。
找到λ的值,因此求出| en,h |形式的误差范数的界。 ≤αh
其中α∈R
+。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

一种
2小时
证明此错误界限。当a = −2和t
? = 100,将此误差与之进行比较
在(b)中找到。
提示:表明
| en,h | = |(1 +啊)
n-exp(anh)|。
然后将其用于−1 x≤0且n = 0、1、2 …
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,
哪里b
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
). What is p? Explain your answer.
(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
solution? Explain your answer.
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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