Numerical Methods in Computational Finance. Daniel J. Duffy
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alt="normal upper Delta t"/>, then there exist constants
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 solution is given by:
In this case the transient solution is the exponential term, and this decays very fast (especially when the constant a is large) for increasing t. The steady-state solution is a constant, and this is the value of the solution when t is infinity. The transient solution is called the complementary function, and the steady-state solution is called the particular integral (when