A numerical study of nonlinear waves arising in a one-dimensional model of a fluidized bed. (English) Zbl 0662.76030

Christie, Ian; Ganser, G. H.

J. Comput. Phys. 81, No. 2, 300-318 (1989).

A third-order nonlinear problem, which under certain circumstances approximates the flow in a 1-dimensional fluidized bed, is solved numerically. A Petrov-Galerkin finite element method, with piecewise linear trial functions and cubic spline test functions, is used for the discretization in space. If product approximation is applied to the numerical representaion of the nonlinear terms, then the scheme is fourth-order in space in the finite difference sense. Second-order backward differentiation is used for the time stepping. It is found that, while certain values of the time step may produce stable results, a reduction in the time step introduces instability. This is confirmed by a von Neumann stability analysis of a simplified case and is also shown to be reasonable in view of the continuous problem which contains stable and unstable modes. A set of numerical experiments is presented.

MSC:

76B99	Incompressible inviscid fluids
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M99	Basic methods in fluid mechanics

Keywords:

third-order nonlinear problem; Petrov-Galerkin finite element method; cubic spline test functions; product approximation; Second-order backward differentiation; von Neumann stability analysis

References:

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