Finite Difference discretization of squared transient term

Question

If we were to discretize with Finite Difference the following transient term:

$$\frac{\partial u^2}{\partial t},$$

would it be okay to write

$$\frac{(u^{j+1})^2-(u^j)^2}{\Delta t}.$$

Or does this not hold?

Answer 1

That would be one possibility. Noting that $\frac{\partial}{\partial t} u^2 = 2 u \frac{\partial u}{\partial t}$, you can also try another forward difference: $$ \frac{\partial}{\partial t} u^2 \approx \frac{2 u^j}{\Delta t}(u^{j+1}-u^j). $$