Efficient Numerical Solution of the Density Profile Equation in Hydrodynamics

We discuss the numerical treatment of a nonlinear second order boundary value problem in ordinary differential equations posed on an unbounded domain which represents the density profile equation for the description of the formation of microscopical bubbles in a non-homogeneous fluid. For an efficient numerical solution the problem is transformed to a finite interval and polynomial collocation is applied to the resulting boundary value problem with essential singularity. We demonstrate that this problem is well-posed and the involved collocation methods show their classical convergence order. Moreover, we investigate what problem statement yields favorable conditioning of the associated collocation equations. Thus, collocation methods provide a sound basis for the implementation of a standard code equipped with an a posteriori error estimate and an adaptive mesh selection procedure. We present a code based on these algorithmic components that we are currently developing especially for the numerical solution of singular boundary value problems of arbitrary, mixed order, which also admits to solve problems in an implicit formulation. Finally, we compare our approach to a solution method proposed in the literature and conclude that collocation is an easy to use, reliable and highly accurate way to solve problems of the present type.