Autonomization refers to the rewriting of a non-autonomous differential equation system into an autonomous differential equation system in numerical mathematics .
description Let the initial value problem be with
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With
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{\ displaystyle u '(t) = f (t, u (t)) \ quad {\ text {with}} \ quad u (t_ {0}) = u_ {0}}
given.
Then the goal is to convert this non-autonomous DGL into an autonomous, i.e. H. Transferring DGL that is independent of time (= invariant against time translation) (" autonomizing ").
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{\ displaystyle t}
We can restrict ourselves to the beginning , since there must be a solution at any given time if the function can be autonomous .
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{\ displaystyle t_ {0} = 0}
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{\ displaystyle u (t)}
In the numerical method, the points in time are correspondingly omitted. That is, it applies to the explicit Euler method
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{\ displaystyle u_ {n + 1} = u_ {n} + h_ {n} f (u (t_ {n}))}
for an autonomous differential equation.
Autonomy If we now rewrite the non-autonomous DGL
in an autonomous DGL
, then we get an additional dimension.
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{\ displaystyle u '(t) = f (t, u (t)), u \ in \ mathbb {R} ^ {d}, t \ in [t_ {0}, T]}
U
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{\ displaystyle U '(t) = F (U (t)), U \ in \ mathbb {R} ^ {d + 1}}
Define
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{\ displaystyle \ tau = t-t_ {0}, U = {\ begin {pmatrix} u \\\ tau \ end {pmatrix}}, F (U) = {\ begin {pmatrix} f (\ tau + t_ {0}, u) \\ 1 \ end {pmatrix}}}
Then the relationship applies:
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∗
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{\ displaystyle {\ frac {dU} {d \ tau}} = {\ frac {d} {d \ tau}} {\ begin {pmatrix} u \\\ tau \ end {pmatrix}} = {\ begin { pmatrix} {\ frac {du} {d \ tau}} \\ 1 \ end {pmatrix}} = {\ begin {pmatrix} {\ frac {du} {dt}} {\ frac {dt} {d \ tau }} \\ 1 \ end {pmatrix}} = {\ begin {pmatrix} f (t, u) * 1 \\ 1 \ end {pmatrix}} {\ underset {t = \ tau + t_ {0}} { =}} {\ begin {pmatrix} f (\ tau + t_ {0}, u) \\ 1 \ end {pmatrix}} = F (U)}
Examples 1. The Euler methods (explicit, implicit), the Heun method and the Trapez method are all invariant to autonomization.
2. The "crooked" Euler method (note time step in )
is not invariant to autonomization.
f
{\ displaystyle f}
u
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{\ displaystyle u_ {n + 1} = u_ {n} + h_ {n} f (t_ {n + 1}, u_ {n})}
Individual evidence
↑ a b Kloeden, PE: Script for the lecture: "Numerical Methods for Differential Equations". p. 36 ff. , Archived from the original January 27, 2018 ; accessed on January 30, 2018 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1 @ 2 Template: Webachiv / IABot / www.math.uni-frankfurt.de
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![{\ displaystyle u '(t) = f (t, u (t)), u \ in \ mathbb {R} ^ {d}, t \ in [t_ {0}, T]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a913877eb0eaf072eacf013ea64a0aab5455ffd)
