On Grid Method for Entropy Solution of the Problem of Simultaneous Motion of Two-Phase Fluid in a Natural Reservoir
Keywords:Buckley-Leverettâ€™s problem, Non-convex state function, Entropy solution, Numerical solution in a class of discontinuous functions
AbstractIn this paper a method for obtaining an exact and numerical solution of the initial and initial-boundary value problems for a first order partial differential equation with a non-convex state function is suggested, which models the macroscopic motion of the two- phase fluids in a porous medium. For this aim, an auxiliary problem is introduced such a way that it has some advantages over the main problem, and it is equivalent to the main problem in a definite sense. By use of this auxiliary problem it is proved that the exact and numerical solutions of the investigated problems satisfy the entropy condition in the sense of Oleinik. To make use of this auxiliary problem, a method for fixing the location of shock which appears in the solution of the main problem and its evolution in time is offered. The proposed auxiliary problem permits us also to prove convergence in the meaning of a numerical solution to the exact solution of the main problem. Besides, the auxiliary problem permits us to write the higher sensitive and the higher order numerical scheme with respect to time variable whose solution expresses all the physical properties of the problem accurately. Using the suggested algorithms, two laboratory experiments were carried out.
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