Two dimensional bisection algorithm and shooting method for linear analysis of equilibrium stability in convection processes

A numerical algorithm for finding the critical numbers of the linear stability problem of mechanical equilibrium in the study of heat and mass transfer processes is developed. As an example, we consider the plane horizontal layer of a three-component liquid with the Soret effect subjected to vertical heating and gravity; the layer boundaries are rigid. To find the critical numbers of the problem, it is necessary to solve a boundary value problem for ordinary differential equations. In the shooting method, the boundary value problem is reduced to the Cauchy problem, and the eigenvalues are being picked (“shooted”) until the solution of the Cauchy problem satisfies the boundary conditions on both boundaries. At the last step of the algorithm implementation, we obtain a determinant, which must be equal to zero. This determinant is a function of the critical numbers, which we are looking for, the numerical solution of this function is traditionally carried out using the secant method, Newton’s method, etc. However, these methods, when solving real problems of heat and mass transfer, in some cases turn out to be ineffective, especially in those situations where oscillatory disturbances are present in the spectrum of perturbations. The two-dimensional analogue of the bisection method is usually less efficient than the methods mentioned above. However, as demonstrated by this research, in some cases when solving specific physical problems, this approach turns out to be the best choice.