Diagonalization and simultaneous symmetrization of the gas-dynamic matrices

Source abstract

The hyperbolicity of the unsteady, inviscid, gas-dynamic equations implies the existence of a similarity transformation which diagonalizes an arbitrary linear combination Σ k j A j \Sigma {k_j}{A_j} of coefficient matrices A j {A_j} . The matrix T that accomplishes this transformation is given explicitly, and the spectral norms of T and T − 1 {T^{ - 1}} are computed. It is also shown that the individual matrices A j {A_j} are simultaneously symmetrized by the same similarity transformation. Applications of the transformations T and T − 1 {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.