While reading some literature on PDE solvers I came across the term pseudo time-stepping today. It seems to be a common term, however I failed to find a good definition or an introductionary article for it.
Hence: What is pseudo time-stepping, and how is it usually used?
Pseudo time-stepping, probably better known as pseudo-transient continuation, is the technique of solving for the steady-state solution of time-evolving partial differential equations by setting an initial guess and using a time-stepper to evolve the solution forward. It tends to succeed where standard globalization strategies fail by taking advantage of the natural structure of the problem.
My advisor, David E. Keyes, coauthored a few papers discussing pseudo-transient continuation. I don't think any of them are purely introductory, but his writing style is very clear. You might try: Convergence Analysis of Pseudo-Transient Continuation for a more comprehensive description and discussion.
Already one good answer is available here, I just want to highlight some things, dual time stepping scheme uses pseudo time in addition to real time (so in your equations two time parameters will come one real and one pseudo). Real time act as a new dimension to the equations and generally discretised implicitly (refer paper). Real time step size should be such that, it resolves the flow features satisfactorily and, pseudo time step size is depend on the general stability condition. At every real time step, pseudo time iterations are performed to find out converged solution. This method is used in both incompressible and compressible flow calculations. It is also commonly used for the moving mesh problems (refer paper). All the best.
