Question

The trapezoidal rule is ___________

(a) stable when Δ t>1

(b) stable when Δ t<1

(c) always stable

(d) never stable

The question was posed to me during an interview for a job.

This is a very interesting question from Transient Flows in chapter Transient Flows of Computational Fluid Dynamics

Answer 1

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 $\Delta t$ in relation to the system's properties.

• For the trapezoidal rule, stability is typically guaranteed when the time step $\Delta t$ is small enough, particularly for problems involving diffusion and convection. The method is conditionally stable, meaning that the size of $\Delta t$ must be chosen appropriately for the specific problem to avoid instability.
• It is more stable for smaller values of $\Delta t$. If the time step $\Delta t$ is too large, numerical instability can occur. In general, the method is considered stable when $\Delta t$ is sufficiently small.

Thus, the trapezoidal rule is stable when $\Delta t$ is smaller than a certain threshold value, typically when $\Delta t<1$ for many practical problems.

Correct Answer: (b) stable when Δt < 1