One step of Forward Euler method: u j + 1 = u j + h f j Find the region of absolute statibility for the Forward Euler method?I'm struggling to solve this problem, could I please get a hint?

Question

One step of Forward Euler method:       u          j      +      1        =      u    j    +  h      f    j  Find the region of absolute statibility for the Forward Euler method?I&#039;m struggling to solve this problem, could I please get a hint?

Salma Baird · Accepted Answer

Let&#039;s consider the so-called linear test equation            d      u              d      t        =  f  (  u  )  =  λ  u  ,where   λ  ∈      C   is a system parameter which mimics the eigenvalues of linear systems of differential equations.The Forward Euler discretization will get you      u          j      +      1        =      u    j    +  h      f    j    =      u    j    +  h  λ      u    j    =  (  1  +  h  λ  )      u    j  Note that                              u          1                =        (        1        +        h        λ        )                  u          0                                              u          2                =        (        1        +        h        λ        )                  u          1                =        (        1        +        h        λ                  )          2                          u          0                                              u          3                =        (        1        +        h        λ        )                  u          2                =        (        1        +        h        λ                  )          2                          u          1                =        (        1        +        h        λ                  )          3                          u          0                    and so on.For absolute stability, we require  |      1    +    z    |    ≤    1  where   z  =  h  λ  ∈      C  . It follows that z must lie inside a disk of radius 1 centblack at z = −1 in the complex plane. We call that region the absolute stability of the Forward Euler’s method.Facit: For stable ODEs with a fast decaying solution (Real(  λ) &amp;lt;&amp;lt; −1) or highly oscillatory modes (Im(  λ) &amp;gt;&amp;gt; 1) the explicit Euler method demands small step sizes. This makes the method inefficient for these so-called stiff systems.

One step of Forward Euler method: u_j+1=u_j+hf_j Find the region of absolute statibility for the Forward Euler method?

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