Exercise 15.2. Show that the discretized energy defined in (15.5) decreases as n in the discretized system (15.6).
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please solve 15.2, thanks.
![Exercise 15.2. Show that the discretized energy defined in (15.5) decreases as n in the
discretized system (15.6).
Reference
Exercise 15.1. Show that the discretized energy
1
en = ;v+
(15.5)
2
increases as n in the discretized system (15.4).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1317e820-ee32-43c7-854c-25ce4b67ab45%2F51d44b44-ed1a-4e6a-ab63-3cba4d41a5b0%2F7zypsos_processed.jpeg&w=3840&q=75)
![69
implicit method
In order to deal with the deficit of the Euler method in Exercise 15.1, we consider an
'implicit' method which approximates the right hand side of (15.3) by the value of new
time step:
In+1 - xn
= Yn+1,
At
(15.6)
Yn+1 - Yn
-w?In+1.
At
mass-spring system
The simplest example (cf. Example 9.1) is an equation describing the motion of a
mass attached to a spring:
ma" = -kx, x(0) = a, x'(0)
(15.1)
= 6
In this system, the total energy
k
m
E =
(15.2)
= me
is preserved. The equation (15.1) is reformulated into a system of ODES of the first order:
()-(-)-(G)) --
d
1
k
(15.3)
dt
-w2 0
m
The first order Euler method (14.4) or (14.5) approximates the system (15.3) by
In+1 - In
= Yn,
At
(15.4)
Yn+1
Yn
At
where an
and
Yn
denote values of x and y att = nAt, respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1317e820-ee32-43c7-854c-25ce4b67ab45%2F51d44b44-ed1a-4e6a-ab63-3cba4d41a5b0%2Fgvos1p_processed.jpeg&w=3840&q=75)
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