# Monte Carlo Wavefunction Methods

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## Main Question or Discussion Point

From the Theory of Open Quantum Systems; the Euler scheme is given by:

$\psi_{k+1} = \psi_{k} + D_1(\psi_k)\Delta t + D_2(\psi_k) \Delta W_k$

and is a scheme of order 1. What does the order of convergence mean? From my understanding higher order schemes require fewer interations to give a decent approximation. Is there anything more than that?

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A. Neumaier
Order $p$ means that the error per step is $O(\Delta t^{p+1})$.
As a consequence, for a nonstiff differential equation, the total error after intergation with a single-step method over a time of order 1 is $O(\Delta t^{p})$. If $\Delta t$ is sufficiently small this makes higher order methods far more useful, in the sense that to get the same global accuracy, far bigger (and hence far fewer) steps can be taken.