The correct answer is:
(b) (ρC ΦC)^t + 1/2 (ρC ΦC)^t-Δt
Explanation:
In the second-order upwind Euler scheme for the finite volume approach, the term (ρCΦC)t+Δt/2(ρC ΦC)^{t+Δt/2}(ρCΦC)t+Δt/2 represents the average value of the quantity at time levels ttt and t−Δtt-Δtt−Δt. The method is designed to achieve second-order accuracy by using both the current and previous time steps.
The scheme typically takes the average of (ρCΦC)t(ρC ΦC)^t(ρCΦC)t and (ρCΦC)t−Δt(ρC ΦC)^{t-Δt}(ρCΦC)t−Δt, with the factor 1/2 to properly weight the values from the two time steps.
Thus, the expression becomes: (ρCΦC)t+Δt/2=(ρCΦC)t+12(ρCΦC)t−Δt(ρC ΦC)^{t + Δt/2} = (ρC ΦC)^t + \frac{1}{2} (ρC ΦC)^{t - Δt}(ρCΦC)t+Δt/2=(ρCΦC)t+21(ρCΦC)t−Δt
Correct Answer: (b) (ρC ΦC)^t + 1/2 (ρC ΦC)^t-Δt