Continuous-Time
In continuous-time form, the "Heston model" is written as
$$ dS_t = \mu S_t \,dt + \sqrt{\nu_t} S_t \, dW_t^S \\ d\nu_t = \kappa (\theta - \nu_t)\, dt + \xi \sqrt{v_t} \, dW_t^\nu $$
where $W_t^\nu$ and $W_t^S$ are Wiener processes with correlation $\rho$, $\mu$, $\theta$, $\kappa$, $\xi$ and $\rho$ are parameters, $S_t$ is the observable stock price process, and $\nu_t$ is the unobserved variance process.
Discrete-Time
The discrete-time version of this is usually my starting point:
$$ y_t = \varepsilon_t \exp(h_t / 2) \\ h_{t+1} = \mu(1 - \phi) + \phi h_t + \sigma_\eta \eta_t $$ where $\{\eta_t\}$ and $\{\varepsilon_t\}$ are correlated ($\rho$) mean zero unit variance normal variates, $y_t$ is the stock return ending at time $t$, and $h_t$ is the logarithm of the return's variance. $\mu$, $\phi$, $\sigma_\eta$, and $\rho$ are the parameters of the model.
Question
How is the former discretized into the latter?
I asked a similar question here, but it was for a simpler model. In the reference mentioned in that answer, it mentions that it's pretty typical to use one of four schemes that discretize a process. You use Euler schemes on each stated process separately, perhaps applying Ito's lemma to find the log diffusion, first. I've tried them, but they don't come out looking the same. Perhaps there is some approximation argument applied in the finance literature I don't know about?