Question

Is there any limitation on the size of the time step \(\Delta t\) in the solution of transient heat conduction problems using \((a)\) the explicit method and \((b)\) the implicit method?

Short answer

Answer: The explicit method has a limitation on the time step size (Δt) to maintain stability, dictated by the stability criterion (Δt ≤ (Δx)^2 / (2α)). In contrast, the implicit method does not have any limitations on the time step size and is unconditionally stable.

Step-by-step solution

(a) Time step limitation for the explicit method

The explicit method, also known as the Forward Time Centered Space (FTCS) method, is conditionally stable, which means that the time step Δt has a limitation for the solution to remain stable. This limitation is dictated by the stability criterion (or Courant-Friedrichs-Lewy (CFL) condition): Δt ≤ (Δx)^2 / (2α), where Δx is the spatial step and α is the thermal diffusivity. The explicit method is simple and easy to implement, but its stability constraint puts a limitation on both the spatial and time steps.

(b) Time step limitation for the implicit method

The implicit method, also known as the Backward Time Centered Space (BTCS) method, is unconditionally stable. It means that there is no limitation on the size of the time step Δt for the solution to remain stable, regardless of the values of Δx and α. The implicit method is generally more computationally expensive compared to the explicit method, due to the need to solve a system of linear equations. However, its unconditional stability is advantageous when larger time steps are needed to simulate the transient heat conduction process. In conclusion, the explicit method for solving transient heat conduction problems has a limitation on the size of the time step Δt to maintain stability, whereas the implicit method does not have any such limitation.