The FTCS scheme for heat equation isQuestion 24Answera.conditionally convergentb. always convergentc.divergent

To determine the behavior of the FTCS (Forward Time Centered Space) scheme for the heat equation, we first need to understand its stability properties.

1. Break Down the Problem
   The FTCS scheme can be evaluated for convergence by investigating its stability and consistency. The key factors to consider are the time step size, spatial discretization, and the relation between them.

2. Relevant Concepts

   • The heat equation can be expressed as: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
   • The FTCS discretization leads to the update formula: $u_i^{n+1} = u_i^n + \frac{\Delta t}{\Delta x^2} (u_{i+1}^n - 2u_i^n + u_{i-1}^n)$
   • Stability condition for the FTCS scheme is usually given by the criterion: $\frac{\Delta t}{\Delta x^2} \leq \frac{1}{2 \alpha}$
   • The FTCS scheme is conditionally stable, meaning it converges under certain conditions related to the size of $\Delta t$ in relation to $\Delta x$.

3. Analysis and Detail

   • If the stability condition is satisfied, the FTCS scheme is convergent. However, if the condition is violated, the scheme may produce divergent results.
   • Thus, the FTCS scheme is not always convergent; it depends on the ratio of $\Delta t$ to $\Delta x^2$.
   • In practical applications, this means one must choose their discretization values carefully to ensure convergence.

4. Verify and Summarize
   After analyzing the properties of the FTCS scheme, we can summarize that it is conditionally convergent due to its dependence on the time step and grid spacing. Thus, it can become divergent without the proper conditions met.

Final Answer

The FTCS scheme for the heat equation is a. conditionally convergent.