Diagonalization and simultaneous symmetrization of the gas-dynamic matrices

Abstract:

The hyperbolicity of the unsteady, inviscid, gas-dynamic equations implies the existence of a similarity transformation which diagonalizes an arbitrary linear combination $\Sigma {k_j}{A_j}$ of coefficient matrices ${A_j}$. The matrix T that accomplishes this transformation is given explicitly, and the spectral norms of T and ${T^{ - 1}}$ are computed. It is also shown that the individual matrices ${A_j}$ are simultaneously symmetrized by the same similarity transformation. Applications of the transformations T and ${T^{ - 1}}$ and their norms include the well-posedness of the Cauchy problem, linear stability theory for finite-difference approximations, construction of difference schemes based on characteristic relations, and simplification of the solution of block-tridiagonal systems that arise in implicit time-split algorithms.