In the context of discretising space and time for numerical calculations,
𝐴
𝑖
−
1
𝑛
+
1
and
𝐴
𝑖
+
1
𝑛
+
1
refer to estimates of parameter A at the same time, but at spatial locations differing by
2
Δ
𝑥
.

Question

In the context of discretising space and time for numerical calculations, 
𝐴
𝑖
−
1
𝑛
+
1
 and 
𝐴
𝑖
+
1
𝑛
+
1
 refer to estimates of parameter A at the same time, but at spatial locations differing by 
2
Δ
𝑥
.

BlackTom AI · Accepted Answer

The question concerns discretising space and time for numerical calculations and refers to two quantities: A_{i-1}^{n+1} and A_{i+1}^{n+1}. These are described as estimates of parameter A at the same time level (n+1) but located at spatial indices differing by 2Δx.

First, recall that in a discretised grid, the index i labels a spatial location spaced by Δx from its neighbours. The positions i-1 and i+1 are one grid step away to the left and right, respectively. The spatial separation between i-1 and i+1 is (i+1)Δx − (i−1)Δx = 2Δx. Therefore, the two points i-1 and i+1 are indeed at the same time slice (n+1) but separated in space by 2Δx, which matches the statement.

Now, examine possible subtleties:
- If the superscript n+1 is interpreted as a future time level, then both A_{i-1}^{n+1} and A_{i+1}^{n+1} share that same future time, so the claim about time alignment holds.
- If one expected a spatial stencil that involves A_i^{n+1} or multiple time levels, it would be a different configuration; however, the given expressions explicitly use i−1 and i+1 with the same time index, confirming the intended interpretation.

In general, a central difference in space at a fixed time uses values at i−1 and i+1 at that same time level. The spatial distance between those two points remains 2Δx, since neighboring grid points are spaced Δx apart. Thus the statement that A_{i-1}^{n+1} and A_{i+1}^{n+1} are estimates at the same time but at spatial locations separated by 2Δx is technically accurate.

Potential misconceptions to avoid:
- Thinking the spatial separation is Δx is common if one misreads the pair as (i, i+1) rather than (i−1, i+1); in that case the separation would be 2Δx, not Δx, so clarity about which indices are used is crucial.
- Assuming different time levels for the two points would contradict the given superscript n+1 being the same for both terms.
- Assuming a non-central or asymmetric stencil would involve different index patterns, but here both terms share the same time index and symmetric spatial offsets.

AMME2000 BMET2960 BMET9960 (ND) Week 5 Pre-work

In the context of discretising space and time for numerical calculations, 𝐴 𝑖 − 1 𝑛 + 1 and 𝐴 𝑖 + 1 𝑛 + 1 refer to estimates of parameter A at the same time, but at spatial locations differing by 2 Δ 𝑥 .

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