PDE Limitations of Godunov Methods
 

I understand that Godunov type methods are very popular for solving hyperbolic PDEs, but are they less applicable to solving parabolic and/or elliptic PDEs? Are there any possible limitations of these type of methods?

Thanks

Hyperbolic pde admit discontinuous solutions. Central difference-type and galerkin-type methods are not stable for such problems. Godunov method is based on solving Riemann problem, which has a discontinuity. You can construct a stable scheme by this approach.

Parabolic and elliptic pde do not have shocks, so it does not make sense to try to use Godunov method. You can construct stable central/galerkin-type schemes for such problems.