Question

The extra term added while discretizing the transient term of a flow with density ρ and flow variable φ using the first-order explicit Euler scheme is _________

(a) \(\Delta t\frac{\partial^2(\rho\phi)}{\partial t^2}\)

(b) \(-\Delta t\frac{\partial^2(\rho\phi)}{\partial t^2}\)

(c) \(\frac{\Delta t}{2}\frac{\partial^2(\rho\phi)}{\partial t^2}\)

(d) \(-\frac{\Delta t}{2}\frac{\partial^2(\rho\phi)}{\partial t^2}\)

This question was posed to me in a national level competition.

The question is from Transient Flows topic in division Transient Flows of Computational Fluid Dynamics

Answer 1

Correct option is (d) \(-\frac{\Delta t}{2}\frac{\partial^2(\rho\phi)}{\partial t^2}\)

Easiest explanation: While using the first-order explicit Euler scheme, an extra term called the numerical anti-diffusion occurs. This term can be obtained by using the Taylor series expansion. The term is

\(-\frac{\Delta t}{2}\frac{\partial^2(\rho\phi)}{\partial t^2}\).