Question

Which of these characteristics of the Crank-Nicolson scheme is affected by the non-uniform transient grids?

(a) Consistency

(b) Convergence

(c) Stability

(d) Accuracy

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

Origin of the question is Approaches for Non-uniform Time Steps in division Transient Flows of Computational Fluid Dynamics

Answer 1

In the Crank-Nicolson scheme, which is commonly used for solving transient partial differential equations in Computational Fluid Dynamics (CFD), the accuracy can be affected by the use of non-uniform transient (time) grids. The Crank-Nicolson method is second-order accurate in both time and space when the time steps are uniform. However, when non-uniform time steps are used, this can introduce errors and reduce the overall accuracy of the scheme.

Let's evaluate each option:

• (a) Consistency: Consistency relates to how well the discretized equations approximate the continuous equations, which is less affected by non-uniform grids.

• (b) Convergence: Convergence refers to whether the solution approaches the true solution as the grid is refined. While it can be influenced by non-uniform grids, accuracy is more directly impacted.

• (c) Stability: The Crank-Nicolson scheme is unconditionally stable for linear problems, even with non-uniform time steps.

• (d) Accuracy: This is the correct answer. Non-uniform time steps can disrupt the second-order temporal accuracy of the Crank-Nicolson scheme, reducing its effectiveness.

Correct Answer: (d) Accuracy