I am trying to write my own SIMPLEC based code for solving a general 2-D flow problem on a rectangular domain. When solving for velocities from the guessed pressure using the momentum equations, I need to invert a matrix that is tri-diagonal. How I can I be assured that the matrix is strictly diagonally dominant? How can I be assured that my differencing is numerically stable? Very often I get large oscillations in my solution (mostly, when convection is much larger than diffusion). Is there some good book that describes the numerical theory of Navier Stokes? I have been reading Patankar and Malalasekara.

1) check condition

[Sum(Anb)]/Ap <= 1.

It's a wideknown rule from Patankar

Also look Lectures Notes by David Apsley and Lars Davidson

2) Check scheme of approximation

3) Add under-relaxation

4) drop me the copy of Your code

And check B.C.

Before solving something very complicated - run next test cases

1) free flow with free boundaries

2) free flow over the flat plate

3) free developed flow between two plates

One good test for stability for all kinds of schemes is the test of Runge for different levels of approximation. If the estimation values are equal at all points then your scheme is numericaly stable (you must prove this for both stationary and non - stationary cases). If there are any variations in this value then you have some additional approximation error (like scheme viscosity) due to the approximation scheme - not to the mechanical equations, there are some methods that can correct this.

there is a book by Peric on CFD ,

i found that book better help than Malas. or patanker.

check that out if it helps.

Just another reference in addition to the aforementioned ones: Hirsch C. "Numerical computation of internal and external flows, Volume 1: Fundamentals of numerical discretisation, Volume 2: Computational methods for inviscid and viscous flows", John Willey & Sons, 1990

This is one of the best references for CFD numerics.

Regards George