Rewrite differential equation

Question

If $h(t+1) = g \cdot h(t) - dh(t)/dx$, then how can you rephrase it to $dh/dt = \dots$ ?

It's an advection equation describing the evolution of a fluid h over time.

Thanks

Answer 1

$$ \begin{align} h(t+1) &= g\cdot h(t) + \frac{d h(t)}{dx} &\\ \frac{d h(t)}{dx} &= h(t+1) - g\cdot h(t) & \text{(reorder terms.)}\\ \frac{d h(t)}{dt}\cdot\frac{d t}{dx} &= h(t+1) - g\cdot h(t) & \text{(chain rule.)}\\ \frac{d h(t)}{dt} &= \dot{x}\cdot\left[ h(t+1) - g\cdot h(t)\right] & \frac{d x}{dt} = \dot{x}\\ \end{align} $$

Disclaimer: I am assumming that you can apply chain rule here, of course.