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thedustbustr

quickie re. explicit euler

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Is it possible to use a Taylor series approximation instead of a first derivative tangent approximation for greater accuracy in explicit euler integration?

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yes. and doing so gives you various higher order integrators - ie. midpoint is O(2), and the standard is runge kutta order 4 O(4). both integrators can be derived from the taylor series expansion. the trick is that in practice multiple evaluations of the first derivative are is used to approximate the higher order derivatives, instead of using those quantities directly, eg. acceleration, derivative of acceleration, its derivative etc. this is because you dont typically know these values, you just have your function f(x) you are solving for and its derivative f'(x)

i have written a simple article explaining how to implement RK4 which you may find useful. i dont derive it from the taylor's series, but i do describe it algorithmically in a way that makes implementation easy for programmers:

http://www.gaffer.org/articles/Integration.html

you can find the source code to accompany this article here:

http://69.55.229.29/downloads/GamePhysics.zip

in my opinion a higher order integrator is a must when working with spring systems (when you can afford it). probably not good for massive particle systems etc, but its great for articulated bodies and so on, lets you really ramp up the spring k's without introducing instability compared to say ex-euler.

cheers

[Edited by - Gaffer on December 31, 2004 2:08:33 AM]

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Quote:
Original post by thedustbustr
Is it possible to use a Taylor series approximation instead of a first derivative tangent approximation for greater accuracy in explicit euler integration?


Two comments:

a) A truncated Taylor series approximation is the theoretical basis for that first derivative tangent approximation. For a forward difference Taylor series expansion truncated after the first-order term, they are exactly the same thing.

b) Any other approximation would be something other than explicit Euler (as Gaffer points out).

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