Adams methods
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| - | + | == Introduction == | |
| - | Adams methods begin by the integral approach, | + | Adams' methods are a subset of the family of [[multi-step methods]] used for the [[numerical integration]] of initial value problems in [[Ordinary Differential Equations]] (ODE's). [[Multi-step methods]] benefit from the fact that the computations have been going on for some time, and use previously computed values of the solution (BDF methods), or the right hand side (Adams' methods), to approximate the solution at the next step. |
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| + | Adams' methods begin by the integral approach, | ||
:<math> | :<math> | ||
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:<math> | :<math> | ||
| - | y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = \int_{t_n}^{t_{n+1}} f(t,y(t)) dt | + | y(t_{N+1}) = y(t_{n}) + \int_{t_n}^{t_{n+1}} y^\prime (t) dt = y(t_{n}) + \int_{t_n}^{t_{n+1}} f(t,y(t)) dt |
</math> | </math> | ||
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| + | Since <math>f</math> is unknown in the interval <math>t_n</math> to <math>t_{n+1}</math> it is approximated by an [[interpolating]] [[polynomial]] <math>p(t)</math> using the previously computed steps <math>t_{n},t_{n-1},t_{n-2} ...</math> and the current step at <math>t_{n+1}</math> if an implicit method is desired.{{fact}} | ||
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| + | == References == | ||
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| + | ? | ||
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| + | {{stub}} | ||
Latest revision as of 18:53, 13 August 2007
Introduction
Adams' methods are a subset of the family of multi-step methods used for the numerical integration of initial value problems in Ordinary Differential Equations (ODE's). Multi-step methods benefit from the fact that the computations have been going on for some time, and use previously computed values of the solution (BDF methods), or the right hand side (Adams' methods), to approximate the solution at the next step.
Adams' methods begin by the integral approach,
Since 
is unknown in the interval 
to 
it is approximated by an interpolating polynomial 
using the previously computed steps 
and the current step at if an implicit method is desired.
References
?
