convection dominated convection-diffusion problem

Kang, S. K.

I'm studying computational hydraulics. I want to develop 2D flood model as finite volume method. In depth-averaged 2D shallow water equation, the momentum equation has 2nd order diffusion terms which are sometimes ignored in modelling open channel flow. When diffusion terms are ignored (ie. eddy viscosity is set to zero) shallow water equations become hyperbolic, and usually used to model dam-break flow becuase of its hyperbolic property. Shock-capturing schemes are used with FVM or FDM.

When diffusive terms are not ignored the equations are parabloc and used to model circulation phenomena.

My Question :

When convection is dominant and diffusion is very small (but eddy viscosity is not zero), I want to use shock-capturing scheme (maybe TVD class) with FVM to simulate flood or dam-break flow. The governing equations are not strictly hyperbolic in this case. How should I solve this problem ??

Thanks for reading..

Regards