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Suppose you have a limit for the accuracy you need to acquire. Using, say RK4, will allow you to achive this with less time steps than, say Euler.

Is there a particular scenario where using the Euler method will be more (computationaly) efficient, given the accuracy limit?

turnip
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  • Artificially Euler is one of simplest methods I can think of for Integration, so I'd imagine one could build a scenario of very questionable accuracy and fast computation with it. To be fair I don't have an exact example on hand, so I'll defer an answer to someone else. Another thing is that if a curve is very well behaved (mostly linear) then again Euler would be very accurate on it compared to most, and there you can see gains in time without trading accuracy by using it. Consider diff EQs such as dy/dx = x/y – Sidharth Ghoshal Nov 02 '15 at 13:50
  • This is not a very straightforward question. First there are issues of stability. High order methods often have additional stability problems; for instance there is the well-developed theory of linear multistep methods, which shows that a stable linear multistep method can only have order at most 2. – Ian Nov 02 '15 at 13:52
  • (Cont.) Second there are issues of floating point arithmetic. High order methods tend to get that way through subtraction of nearly equal numbers. Eventually the error in the true calculation gets beaten out by the floating point error, at which point your results begin to get worse as you increase the order. The "sweet spot" for double precision is right around 4th order. (Also, among RK methods, the amount of additional work per step per additional order begins to increase when you get past 4th order. This is more important historically than it is now.) – Ian Nov 02 '15 at 13:53

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Note that RK4 is convergent of order 4 provided that the right-hand side $f$ of the ODE $$\dot{y} = f(t,y)$$ is sufficiently smooth. If $f$ is only continuously differentiable, RK4 and the explicit Euler method share the same linear convergence rate. In this case, you might prefer the explicit Euler method.

Albe
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