Linear Multistep Method - Definitions

Definitions

Numerical methods for ordinary differential equations approximate solutions to initial value problems of the form

The result is approximations for the value of at discrete times :

where h is the time step (sometimes referred to as ).

Multistep methods use information from the previous s steps to calculate the next value. In particular, a linear multistep method uses a linear combination of and to calculate the value of y for the desired current step. Thus, a linear multistep method is a method of the form

\begin{align} & y_{n+s} + a_{s-1} y_{n+s-1} + a_{s-2} y_{n+s-2} + \cdots + a_0 y_n \\ & \qquad {} = h \bigl( b_s f(t_{n+s},y_{n+s}) + b_{s-1} f(t_{n+s-1},y_{n+s-1}) + \cdots + b_0 f(t_n,y_n) \bigr), \end{align}

The coefficients and determine the method. The designer of the method chooses the coefficients, balancing the need to get a good approximation to the true solution against the desire to get a method that is easy to apply. Often, many coefficients are zero to simplify the method.

One can distinguish between explicit and implicit methods. If, then the method is called "explicit", since the formula can directly compute . If then the method is called "implicit", since the value of depends on the value of, and the equation must be solved for . Iterative methods such as Newton's method are often used to solve the implicit formula.

Sometimes an explicit multistep method is used to "predict" the value of . That value is then used in an implicit formula to "correct" the value. The result is a predictor–corrector method.

Read more about this topic:  Linear Multistep Method

Famous quotes containing the word definitions:

    “The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
    —Pamela Patrick Novotny (20th century)

    “What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
    —G.C. (Georg Christoph)

    “Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
    There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
    —Edmond De Goncourt (1822–1896)