Regarding explicit numerical schemes based on finite differences:

Question

BlackTom AI · Accepted Answer

The question asks about explicit numerical schemes based on finite differences. In your prompt, the list of answer options is missing, so we will first acknowledge that gap and then discuss the core concepts that such options would typically address.

A typical true/false statement about explicit finite-difference schemes is that they are stable only under certain conditions on the time step Δt or the spatial step Δx (often summarized by a CFL-type condition). For many parabolic problems, such as the heat equation discretized with an explicit scheme (e.g., forward in time, centered in space), stability is not unconditional; it depends on a relationship between Δt and Δx. Specifically, there is a maximum allowable Δt relative to Δx (and physical parameters like diffusion coefficients) that ensures the scheme remains stable. If Δt is chosen too large for a given Δx, the amplification factors exceed 1 in magnitude, leading to numerical instability. Conversely, choosing Δt small enough (or Δx large enough) typically yields a stable scheme.

To unpack this further, consider a canonical example: the explicit FTCS scheme for the 1D heat equation. A von Neumann (Fourier) stability analysis shows that the amplification factor G depends on θ = kΔx and the ratio r = αΔt/(Δx^2). The scheme is stable if 0 ≤ r ≤ 1/2, which translates into a constraint on Δt for a given Δx: Δt ≤ (Δx^2)/(2α). This is a concrete manifestation of the general statement that explicit schemes are conditionally stable: stability hinges on satisfying a condition that ties together time and space discretizations.

With that in mind, let’s consider how the (unknown) options might fare:
- If an option asserts that explicit schemes are stable for any choice of Δt or Δx, that would be incorrect, because stability generally requires the Δt–Δx relationship to meet a critical bound; ignoring this leads to instability in practice.
- An option claiming stability is guaranteed as long as Δt or Δx meet some condition would be partially correct, but we would want specifics (e.g., a CFL-like bound), since “some condition” is too vague without a concrete inequality.
- An option stating that stability does not depend on problem parameters (like diffusion coefficients) would be misleading, because the critical bound usually involves those parameters (e.g., α in the heat equation).
- An option suggesting unconditional stability for an explicit scheme would be incorrect, as unconditional stability is generally a property of implicit schemes, not explicit ones.

In summary, the central idea you’d expect in the true or accurate statements about explicit finite-difference schemes is the dependence of stability on a time-step constraint relative to the spatial discretization and problem parameters. The lack of provided answer choices prevents a line-by-line evaluation of each option, but the standard, well-supported principle is that explicit schemes are conditionally stable and require a CFL-type relation between Δt and Δx (and problem coefficients) to ensure stability.

AMME2000 BMET2960 BMET9960 (ND) Week 5 Pre-work

Regarding explicit numerical schemes based on finite differences:

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