## Need a Newton-Krylov guy?

2004-06-09
2004-06-16

• Todd Chisholm
2004-06-09

Hi All,

I'm just finishing up a PhD at the University of Toronto Institute for Aerospace studies.  I've been working on Newton-Krylov methods for quite a while now.  I'd like to contribute if I could.  I've lots of C++ programming experience as well.

Most of my experience is in structured finite-difference methods, but the majority of the Newton-Krylov techniques we use are applicable to finite-volume as well.

Todd

• x-flow
2004-06-15

Hello Tod,

I am not a developer of this project yet, but I also hope to contribute in the future.

Can you explain better the Newton-krylov technique? I seen on the net that it can be used in metallurgical processes such as casting and welding and this might interest me. By the way, what is the title of your phD?

• Todd Chisholm
2004-06-16

Hi x-flow,

The details of an efficient N-K algorithm are pretty complex, especially for flows using a turbulence model.  The overall idea is pretty simple, though.  We use Newtons method (actually, it's usually Implicit Euler with an adjustable time step that tends to Newtons method) to solve the Navier-Stokes equations (or whatever nonlinear eqn's you have, especially if they are time-dependent).  The 'trick' is that the linear system formed at each Newton iteration is only solved approximately, by using an iterative linear solver.  This is pretty old technology, as SOR and other stationary linear solvers have been around forever.  The new (at least it was pretty new when I started my PhD!) technique is the Krylov solver, which is a non-stationary linear iterative solver, potentially much more efficient than SOR-type methods.

This is all fine, but the problem is that Krylov methods are notoriously difficult to apply with poorly conditioned systems, as most CFD systems are.  There is a number of hoops to jump through to get it working efficiently and robustly.  Once it's going, though, it can be very fast.

My thesis is going to be "A Fullly Couple Newton-Krylov Solver with a One-Equation Turbulence Model" - it's not quite done yet, though.  I've a couple of papers if you are interested.

If you want more of the gory details, let me know.  I can send a copy of the thesis once it's ready, too.  Next week, crossing fingers!

T