Numerical research of the Barenblatt--Zheltov--Kochina stochastic model

S. I. Kadchenko ; E. A. Soldatova ; S. A. Zagrebina

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 9 (2016) no. 2, pp. 117-123

Résumé

At present, investigations of Sobolev-type models are actively developing. In the solution of applied problems the results allowing to get their numerical solutions are very significant. In the article the algorithm for numerical solving of the initial boundary value problem is developed. The problem describes the pressure distribution of the homogeneous fluid in the horizontal layer in the circle. The layer is opened by a vertical well of a small radius. In our research we suppose that random disturbing loads have an influence on the fluid. The problem was solved under two assumptions. Firstly, we suppose that an unstable fluid flow is axially symmetric, and secondly, that in initial moment the pressure in the layer is constant. After the process of the discretization we modify the original model to the Cauchy problem for the system of ordinary differential equations. For the numerical solution we use algorithms based on explicit one-step formulas of the Runge–Kutta type with the seventh-order accuracy and with the selection of the integration step. We also use the scheme of the eighth-order accuracy to evaluate the calculation accuracy on each steps of time. According to the results of this control, we choose the time-step. A lot of numerical experiments have shown high numerical efficiency of the algorithm that we use to solve the investigated initial-boundary problem.

Détail

Keywords: stochastic Sobolev type equation; numerical solution; Barenblatt–Zheltova–Kochina model; Cauchy problem.