What is the best way to simulate the short rate $r(t)$ in a simple one factor Hull White process? Suppose I have

$$
dr(t) = (\theta(t)-\alpha r(t))dt+\sigma dW\_t
$$

where $\theta(t)$ is calibrated to swap curve, constants $\alpha$ and $\sigma$ are calibrated to caps using closed form solution for zero-coupon bond options. The best way I can think to do it is an Euler discretisation, that is:

$$
r(t+\Delta t) = r(t) + \theta(t)\Delta t - \alpha r(t) \Delta t + \sigma \sqrt {\Delta t} Z
$$
where $Z \sim N(0,1)$. In this case, I need $t$ to go from 0 to 10 years, ideally in 0.25 increments. But with Euler, I'd need to use small $\Delta t$, so perhaps 0.025 or less? Once I have string of $r(t)$, I can easily calculate $P(t,T)$ zero coupon bonds.

Appreciate any other ideas or if someone could point me in the right direction. I'm quite new to rates modelling!