Several weeks ago, there was a discussion here about the time accurate simulation. One of the people said that using pseudo-time approach, faster convergence to steady state is possible. If this is the case, why should we care about physical time step (which is missing in this method)? And, as such, what is the benefit of the time accurate approach?

Why should we care about the physical time step? For steady-state simulations we don't.

Time accurate computation, as the term suggests, will give you a solution for the time history of the flow, starting from an initial condition. If done right, the solution will be 'accurate' at any time, from the first to the last time step in your computation. In this case we are not only looking for a final solution, but the full evolution of the flow is of interest.

A steady-state solution can be considered a solution for t->infinity, or in some periodic cases as an approximate 'time-averaged' solution. The path that leads to that solution is irrelevant. The solution has to be accurate in the end only, not at every iteration step, as the evolution of the flow is not of interest in the steady-state computation.

There are many methods that can be applied. Pseudo-time iteration is applicable to both steady-state and unsteady computations.

Just to add on Mani's reply.. For unsteady computations, when one uses dual time stepping algorithm, the time based term (outer loop) is added as a source term and residual is driven to zero (inner loop). So pseudo-time iteration in this case is applicable for the inner loop.