# Monte Carlo Wavefunction Methods

• A

## 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.

For multistep methods and in the stiff case, the total error may be much larger, and additional stability conditions beyond the order are needed to get the same conclusion.

The possible conclusions about the global error are again more complicated (and not easy to summarize) in case of using these methods for stochastic differential equations, as in your case. (But all this has nothing to do with physics.)

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• Raptor112