Abstract
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 representation 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.
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