We propose finite difference schemes for multidimensional quasilinear
parabolic systems whose main feature is the introduction of correctors
which control the second-order terms with mixed derivatives.
We show that with these correctors the schemes inherit physically
relevant properties present at the continuous level, such as
the existence of invariant domains and/or the nonincrease of
the total amount of entropy.
The analysis is performed with some general tools that could be used
also in the analysis of finite volume methods of flux vector splitting
type for first-order hyperbolic problems on unstructured meshes.
Applications to the compressible Navier-Stokes system are given.
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