I am having trouble finding the region of absolute stability for modified Euler method: w i + 1 ∗ = w i + h f ( t i , w i ) w i + 1 = w i + h 2 [ f ( t i , w i ) + f ( t i + 1 , w i + 1 ∗ ) ] . DEFINITION: We define a region R of absolute stability for a one-step method as the region in the complex plane satisfying: R = { h κ ∈ C : | Q ( h κ ) | &lt; 1 } .I don't fully understand the above definition of region of absolute stability and how to apply it. Clear and step by step help would be much appreciated. Thank you

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I am having trouble finding the region of absolute stability for modified Euler method:                              w                      i            +            1                    ∗                                    =                  w          i                +        h        f        (                  t          i                ,                  w          i                )                                      w                      i            +            1                                              =                  w          i                +                  h          2                          [          f          (                      t            i                    ,                      w            i                    )          +          f          (                      t                          i              +              1                                ,                      w                          i              +              1                        ∗                    )          ]                .            DEFINITION: We define a region R of absolute stability for a one-step method as the region in the complex plane satisfying:  R  =  {    h      κ    ∈  C    :        |    Q  (  h      κ    )      |    &amp;lt;  1    }  .I don&#039;t fully understand the above definition of region of absolute stability and how to apply it. Clear and step by step help would be much appreciated. Thank you

cdtortosadn · Accepted Answer

The function   Q, which more often than not will be a polynomial or rational function, is the factor that approximates the exponential       e          h              κ             in the numerical solution of       w    ′    (  t  )  =      κ    w  (  t  ). In the exact solution   w  (      t          i      +      1        )  =      e          h              κ              w  (      t    i    ), in the numerical solution       w          i      +      1        =  Q  (  h      κ    )      w    i  . In your scheme                              w                      i            +            1                    ∗                                    =                  w          i                +        h        ·        (                  κ                          w          i                )        =        (        1        +        h                  κ                )                  w          i                                              w                      i            +            1                                              =                  w          i                +                  h          2                (        (                  κ                          w          i                )        +                  κ                (        1        +        h                  κ                )                  w          i                )        =        (        1        +        h                  κ                +                              1            2                          (        h                  κ                          )          2                )                  w          i

I am having trouble finding the region of absolute stability for modified Euler method: w^_(i+1)=w_i+hf(t_i,w_i) w_(i+1)=w_i+h/2[f(t_i,w_i)+f(t_(i+1),w^_(i+1))].

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