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The solution of nonlinear hyperbolic equation systems by the finite element method. (English) Zbl 0551.76002

The difficulties experienced in the treatment of hyperbolic systems of equations by the finite element method (or other) spatial discretization procedures are well known. In this paper a temporal discretization precedes the spatial one which in principle is considered along the characteristics to achieve a self adjoint form. By a suitable expansion, the original co-ordinates are preserved and combined with the use of a standard Galerkin process to achieve an accurate discretization. It is shown that the process is equivalent to the Taylor-Galerkin methods of J. Donea [Int. J. Numer. Methods Eng. 20, 101-119 (1984; Zbl 0524.65071)]. Several examples illustrate the accuracy and efficiency attainable in such problems as transport, shallow water equations, transonic flow etc.

MSC:

76M99 Basic methods in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76H05 Transonic flows

Citations:

Zbl 0524.65071
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References:

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