A Case Study of the Validity of Finite Difference Approximations in Solving Dynamic Stability Problems

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Abstract
The differential equation governing the problem of baroclinic stability has singularities when friction is omitted and leads, therefore, to a multivalued solution. For an analytic solution to the stability problem, theory on hydrodynamic stability has shown how to select the physically correct solution satisfying the inviscid equation, while there is little guidance in the literature on how to proceed if finite difference methods are used. Since there are some doubts as to the validity of the use of a straightforward finite difference approach where singularities are involved, the authors have made a case study in an attempt to clarify the matter. The case selected admits an analytic solution in terms of elementary functions, and it is possible to obtain a closed-form solution for the corresponding finite-difference equation. The results of the study show that the finite difference approximation is valid on one side of the singular point but not on the other, even if the mesh size approaches zero. It is, therefore, concluded that use of finite difference methods can give erroneous results if singularities occur in the differential equation to be solved.

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