Fundamentals of Scientific Computing by Bertil Gustafsson

Author:Bertil Gustafsson
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(10.6)

which leads to the simple fourth order method

(10.7)

In order to find out about the stability domain for the test equation, g(u n+2) is replaced by λu n+2. It is easy to determine when a given one-step method is stable as we saw above, but here we encounter a new difficulty. When more time levels are involved, how do we analyze stability? We write a general difference equation as

The key to the analysis is the roots z j of the characteristic equation

which is formally obtained by substituting u n =z n and then dividing by z n . If all the roots are distinct, then the general solution has the form

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