Hi,

Firstly, I would like mention that I am not much experienced with concept of preconditioning ...I would like to know how local preconditioning can be used to obtain time dependent solution using explicit methods.(concept of pseudo time). As far as my concern local preconditioning does not affect the steady state solution, nevertheless time dependent solutions accuracy is lost...Now i am again restating my questions that how can we get time accurate solution using local preconditioning....

I am not sure I understand your question. Usually by "preconditioning" one means solving:

L*V*dw/dt = R(w)

rather than

V*dw/dt = R(w)

for the semidiscrete system. In this case you are NOT solving the original equations, except if your original equations are steady, when dw/dt = 0. So the short answer is you cannot do this "time accurately" because you are not solving the "right" equations.

However, you could implement something like this for example in a dual time stepping scheme, where an outer iteration is driven by a sequence of steady state calculations with a modified residual. See Jameson's work for details.

Bob

yeah when you use a dual time stepping scheme you can pull out the whole bag of tricks on the inner iterations (preconditioning, multigrid, local time stepping etc)

There are always a lot of preconditioning techniques in the sence of dual-time stepping procedure both for incompressible and low-Mach number compressible flows. You can see the J. Comput. Physics for the last three years. It seems me, the most popular local preconditioning uses local (varying in space) pseudo-time step. This can accelerate convergence of sub-iterations, but time accuracy (in physical time) of converged (in pseudo-time) solution does not depend on such details. The problem is, that artificial-compressibility methods calculate numerical fluxes in compressible way, so that converged solution (one for steady problems or one for any physical time step for non-steady problems) does depend on artificial compressibility parameter etc. You can not do much with it.

With best regards