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Diagonalization and simultaneous symmetrization of the gas-dynamic matrices


Authors: R. F. Warming, Richard M. Beam and B. J. Hyett
Journal: Math. Comp. 29 (1975), 1037-1045
MSC: Primary 76.35
DOI: https://doi.org/10.1090/S0025-5718-1975-0388967-5
MathSciNet review: 0388967
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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.


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DOI: https://doi.org/10.1090/S0025-5718-1975-0388967-5
Keywords: Hyperbolic partial differential equations, initial-value problems, inviscid flows, construction and stability of finite-difference schemes, similarity transformation
Article copyright: © Copyright 1975 American Mathematical Society

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