t Subscript n Baseline right-parenthesis plus normal epsilon Subscript n Baseline comma n equals 0 comma ellipsis comma normal k minus 1 2nd Row 1st Column Blank 2nd Column Blank 3rd Column sigma-summation Underscript j equals 0 Overscript k Endscripts left-parenthesis alpha Subscript j Baseline upper X Subscript n minus j Baseline minus normal upper Delta Subscript t Baseline beta Subscript j Baseline f left-parenthesis upper X Subscript n minus j Baseline right-parenthesis right-parenthesis equals normal epsilon Subscript n comma Baseline k less-than-or-equal-to n less-than-or-equal-to StartFraction b minus a Over normal upper Delta t EndFraction period EndLayout"/>
If the multistep method is stable and satisfies:
where C is a positive constant independent of
In all cases we define the norm
We state this theorem in more general terms: consistency and stability of a multistep scheme are sufficient for convergence.
Finally, the discussion in this section is also applicable to systems of ODEs. For more discussions, we recommend Henrici (1962) and Lambert (1991).
Finally, we present four finite difference schemes for the IVP (2.31) and their generating polynomials as defined by Equations (2.34):
We recommend that you verify the results using the forms of the generating polynomials for one-step and two-step methods, respectively. The general forms are:
2.6 STIFF ODEs
We now discuss special classes of ODEs that arise in practice and whose numerical solution demands special attention. These are called stiff systems whose solutions consist of two components; first, the transient solution that decays quickly in time, and second, the steady-state solution that decays slowly. We speak of fast transient and slow transient, respectively. As a first example, let us examine the scalar linear initial value problem:
(2.40)
whose exact
