Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8377Research ArticleMathematical and Computational Oil Spills ModellingTrepachevaA Vatrepacheva@sfedu.ruBurtykaPh Bfburtyka@sfedu.ruComputer Center of Southern Federal University15022014228128608092016Copyright © 2014,2014The model of oil spill evolution on the water surface based on quasi-linear convection-diffusion equation is considered. To solve the latter the method of finite differences is used. Two-dimensional problem is reduced to one-dimensional by alternating direction method. For one-dimensional case traditional difference approximations are briefly discussed. And also difference schemes obtained by computer algebra based method of automatic generation of difference schemes are considered. For generated schemes the order of approximation and linear numerical dissipation are estimated. For explicit schemes stability conditions are briefly discussed. In the absence of convection a numerical comparison of traditional implicit difference scheme and one implicit scheme generated automatically with similarity solution of quasi-linear diffusion equation is carried out. This comparison shows that generated implicit scheme allows to obtain the relative error less than for traditional scheme. Using obtained implicit scheme evaluation of oil slick thickness changes in the presence of oil evaporation is done. Two different models were used to estimate oil evaporation rate. The first one is based on hypothesis that oil evaporation is regulated by air boundary layer. The second assumes that oil evaporation is regulated by oil diffusion. Calculations show that the choice of model essentially influences on oil slick thickness and oil total volume changes in time.quasilinear advection-diffusion equationfinite difference methodGrorbner basisdifference schemes generationoil spillsoil evaporationквазилинейное уравнение конвекции-диффузииметод конечных разностейгенерация разностных схембазис Грёбнераиспарение нефти[Fingas M. Oil Spills Science and Technology: Prevention, Response and Cleanup. - Burlington: Elseivier, 2011.][Tkalich P. A CFD Solution of Oil Spill Problems // Environ. Modeling and Software. - 2006. - Vol. 21. - Pp. 271-282.][Роуч П. Вычислительная гидродинамика. - М.: Мир, 1980.][Zadeh E., Hejazi K. Eulerian Oil Spills Model using Finite-Volume Method with Moving Boundary and Wet-Dry Fronts // Model. Simul. Eng. - 2012. - Vol. 2012. - P. 33.][Тихонов А.Н., Самарский А.А. Уравнения математической физики. - М.: Наука, 1977.][Gerdt V.P., Blinkov Y.A., Mozzhilkin V.V. Gr¨obner Bases and Generation of Difference Schemes for Partial Differential Equations // Symmetry, Integrability and Geometry: Methods and Applications. - 2006. - Vol. 2. - P. 26.][Gerdt V.P., Robertz D. Maple Package for Computing Gr¨obner Bases for Linear Recurrence Relations // Nuclear Instruments and Methods in Physics Research. - 2006. - Vol. 2. - P. 26.][Gerdt V.P., Blinkov Y.A. On Computer Algebra-Aided Stability Analysis of Difference Schemes Generated by Means of Gr¨obner Bases // Computer Algebra and Differential Equations, Acta Academiae Aboensis, Ser. B. - 2007. - Vol. 67, No 2. - Pp. 168-177.][Stiver W., Mackay D. Evaporation Rate of Spills of Hydrocarbons and Petroleum Mixtures // Computer Algebra and Differential Equations, Acta Academiae Aboensis, Ser. B. - 1984. - Vol. 18. - Pp. 834-840.]