The correct answer is:
(b) stable when Δt < 1
Explanation:
The trapezoidal rule (also known as the Crank-Nicolson method in the context of finite difference schemes for time-dependent problems) is a time-stepping method that is used to solve ordinary differential equations (ODEs) and partial differential equations (PDEs) numerically. It is a second-order accurate method, which is implicit, and is often used in transient flow problems.
The stability of the trapezoidal rule depends on the time step size Δt\Delta tΔt in relation to the system's properties.
- For the trapezoidal rule, stability is typically guaranteed when the time step Δt\Delta tΔt is small enough, particularly for problems involving diffusion and convection. The method is conditionally stable, meaning that the size of Δt\Delta tΔt must be chosen appropriately for the specific problem to avoid instability.
- It is more stable for smaller values of Δt\Delta tΔt. If the time step Δt\Delta tΔt is too large, numerical instability can occur. In general, the method is considered stable when Δt\Delta tΔt is sufficiently small.
Thus, the trapezoidal rule is stable when Δt\Delta tΔt is smaller than a certain threshold value, typically when Δt<1\Delta t < 1Δt<1 for many practical problems.
Correct Answer: (b) stable when Δt < 1