I'm asked to decide if I should solve the system dot y=(−600 400 400 −600)y, t in [t_0,t_e], y(t_0)=y_0 with either the explicit Euler method or the implicit Euler method.

Nash Frank 2022-07-21 Answered
I'm asked to decide if I should solve the system
y ˙ = ( 600 400 400 600 ) y , t [ t 0 , t e ] , y ( t 0 ) = y 0
with either the explicit Euler method or the implicit Euler method.

Using the explicit Euler method I would get the updating scheme
y n + 1 = ( 1 600 h 400 h 400 h 1 600 h ) y n
where the eigenvalues of the driving matrix is
λ 1 = 401 600 h ,
λ 2 = 399 600 h .
For the solution to be stable these need to be less than one which gives the conditions
h 4 6 = 2 3 ,
h 4 6 = 2 3 .
The last condition doesn't say anything but the first condition seems restrictive since I can't choose h as small as possible.

If I instead were to use the implicit Euler method I would get the updating scheme
y n + 1 = ( 1 600 h 400 h 400 h 1 600 h ) 1 y n .
Now I can't solve for the eigenvalues of this system but I've heard the implicit Euler is unconditionally stable so it shouldn't matter.

So is the answer that I should choose implicit Euler because it is unconditionally stable or am I missing something? The order of consistency of both is 1 so that should not matter.
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Answers (1)

Abraham Norris
Answered 2022-07-22 Author has 16 answers
Your computation of eigenvalues is wrong. Of the system matrix in y = A y there is one eigenvalue −200 with eigenvector ( 1 1 ) and one eigenvalue −1000 with eigenvector ( 1 1 ) . These translate into eigenvalues 1 200 h and 1 1000 h of the "driving matrix" for the Euler method, requiring h < 0.002 for stability.
For the implicit method there are no step size restrictions to get stability in the method, to get into the range where the error behaves like order 1 one still will need 500 h 1.
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