Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability
The authors study the problem of an infinite horizontal layer of fluid of uniform thickness, heated from below, rotating uniformly and subject to a uniform magnetic field acting in the same direction as gravity. Instability considered as overstability may be modelled by an eigth-order ordinary differential equation. Taking into account the uniform magnetic field, the ordinary convection can be modelled by a tenth-order differential equation, while overstability can be described by a twelfth-order differential equation. Two methods are applied in all three cases. In the first approach, a direct second-order numerical method is developed by approximating the derivatives in the model differential equations by second-order finite differences. In the second approach, the differential equations are transformed into equivalent second-order systems, and well-known finite difference second-order and fourth-order methods are used to obtain eigenvalues. By comparing the results given by the present theory with those known from the literature, a very good agreement is observed.
scientific article; zbMATH DE number 784503
scientific article