I've inherited a finite volume code that does a second-order discretization of flux terms for a set of mixed parabolic-elliptic equations with discontinuous diffusion coefficients. The impression I get is that Discontinuous Galerkin (DG) methods are preferable for this sort of problem; however, since I have some time constraints, and know more about finite volume methods than any type of finite element methods (or DG), for now, I'm forging ahead with finite volume methods. As a disclaimer, I don't know a lot about the finite volume method, either (I've read up to Chapter 8 in LeVeque's Finite Volume Methods for Hyperbolic Problems on my own), so feel free to correct any misconceptions I have here.
Since my colleagues will be interested in unsteady problems, and results will be compared against previous work in more regular geometries that used the method of lines with high order temporal discretizations, I would like to investigate using higher order spatial discretizations to enable the use of higher order time integrators.
Is there a good textbook-like or tutorial-like source that someone can recommend on ENO/WENO methods that covers limiters in more than one dimension? The internal interfaces between materials will create, in essence, transmission boundary conditions that I don't want to handle explicitly in the code. These interfaces will probably cause oscillations that will need to be addressed by a limiter. The best source I've seen is Essentially Non-oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws by C. W. Shu; the book by LeVeque seems to focus mostly on Lax-Wendroff schemes from a wave propagation standpoint (which isn't being used in the code), and the book by Toro seems to focus on higher order schemes in the one-dimensional case.
As a follow-up question: given that the problem is mainly diffusive, is it particularly important to use an strong stability-preserving (SSP) integrator for this type of problem? I know that for hyperbolic problems employing higher-order discretizations, preserving the total variation diminishing property is important, which is why SSP integrators are used. This problem isn't hyperbolic, so the only place where being total variation bounded might actually matter will be any spurious oscillations that arise as a result of the interfaces between materials of different diffusivities.