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.

Question

I&#039;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&#039;t say anything but the first condition seems restrictive since I can&#039;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&#039;t solve for the eigenvalues of this system but I&#039;ve heard the implicit Euler is unconditionally stable so it shouldn&#039;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.

Abraham Norris · Accepted Answer

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 &quot;driving matrix&quot; for the Euler method, requiring   h  &amp;lt;  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.

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.

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