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At a glance I too would expect this problem to be pretty well behaved, because Hermitian eigendecomposition itself is pretty well behaved (real eigenvalues, orthogonal eigenvectors). One place where things might go wrong, is calling a routine/algorithm that is unaware of the Hermitian-ness of your input. In LAPACK, the routine you'd probably want is ZHEEV, ...


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Usually you speak of a $n$'th order (accurate) method if your Taylor truncation error is of order $n+1$. This means your approximation is accurate up to order $n$ terms, and your errors are of order $n+1$. However, in FVM methods you often have no easy way of obtaining the truncation error of your formulation, since you reconstruct the numerical fluxes $F$ ...


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