Well-posed initial conditions and numerical methods for one-dimensional models of liquid dynamics in a horizontal capillary
In this paper, we undertake a mathematical and numerical study of liquid dynamics models in a horizontal capillary. In particular, we prove that the classical model is ill-posed at initial time, and we recall two different approaches in order to define a well-posed problem. Moreover, for an academic test case, we compare the numerical approximations, obtained by an adaptive initial value problem solver based on an one-step one-method approach, with new asymptotic solutions. This is a possible way to validate the adaptive numerical approach for its application to real liquids.
KeywordsWell-posed initial conditions Horizontal capillary Asymptotic solutions Adaptive numerical method
Mathematics Subject Classification34A12 65L05 76D45
The first author gratefully thanks Chris Budd for fruitful discussions concerning the asymptotic solutions while he was visiting the Department of Mathematics, University of Bath, United Kingdom. This work was supported by the University of Messina and by the GNCS (Computational Science National Group) of the INDAM (Italian Institute of High Mathematics).
- Budd C, Huang H (2008) Private communication, BathGoogle Scholar
- Fazio R, Jannelli A (2010) Ill and well-posed one-dimensional models of liquid dynamics in a horizontal capillary. In: De Bernardis E, Spligher R, Valenti V (eds) Applied and industrial mathematics in Italy III. World Scientific, Singapore, pp 353–364Google Scholar