Summary
Variational principles for elliptic boundary-value problems as well as linear initial-value problems have been derived by various investigators. For initial-value problems Tonti and Reddy have used a convolution type of bilinear form of the functional for the time-like coordinate. This introduces a certain amount of directionality thereby reflecting the initial-value nature of the problem. In the present investigation the methods of Tonti and Reddy are used to derive the appropriate variational formulation for the transonic flow problem. A number of linear and non-linear examples have been investigated. As a test for the existence of directionality, finite-differences are used to discretize the variational integral. For initial-value problems of wave equation and diffusion equation type, fully implicit finite-difference approximations are recovered. The small-disturbance transonic equation leads to the Murman and Cole differencing theory; when applied to the full potential-flow equations, the rotated difference scheme due to Jameson is obtained.
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An extended version of this paper was first presented at the Bat-Sheva International Seminar on Finite Elements for Non-Elliptic Problems, Tel-Aviv, Israel, July 1977.
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Khosla, P.K., Rubin, S.G. Applications of a variational method for mixed differential equations. J Eng Math 15, 185–200 (1981). https://doi.org/10.1007/BF00042779
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DOI: https://doi.org/10.1007/BF00042779