Efficient Numerical Solution of the Density Profile Equation in Hydrodynamics

Abstract

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.

The structure of a set of positive solutions to Dirichlet BVPs with time and space singularities

“…This software is useful for the approximation of numerous singular boundary value problems important for applications, see e.g. [4], [9], [12], [17].…”

“…Therefore, results derived for equation (1.1a) also apply for the modified equation (t −a v ′ (t)) ′ = g(t, v(t), v ′ (t)). Such type of models arises in the study of phase transitions of Van der Waals fluids [3], [8], [12], [14], [18], in population genetics, in models for the spatial distribution of the genetic composition of a population [6], [7], in the homogenenous nucleation theory [1], in relativistic cosmology in description of particles which can be treated as domains in the universe [15], and in the nonlinear field theory [9], in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces [5].…”

Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: The primal–dual formulation and a posteriori error estimation

“…In many applications, second order singular models, cf. [4], [5], [10], [19], and [30], assume the forms…”

Fractional BVPs with strong time singularities and the limit properties of their solutions

“…The operator on the left hand side of (1) has a strong singularity in the time variable at = 0 because Operators of such type were studied in population genetics [9], in the homogeneous nucleation theory [1], in relativistic cosmology [18], in the nonlinear field theory [13] and also in phase transitions of Van der Waals fluids [3,12,17] (1) holds for ∈ (0 1]. In [22,23] were proved that if is a solution of (1), then (0) = 0.…”