Numerical solution of a hydrodynamics problem with a nonlinear source function in a class of discontinuous functions

In this paper, a new numerical method is proposed for solving the generalized Buckley-Leverett problem, which describes the movement of two-phase mixtures in Bazhenov bed sediments in a class of discontinuous functions. For this aim, an auxiliary problem that has some advantages over the main problem and is equivalent to the main problem in a definite sense is introduced. By developing an original finite difference method obtained via advantages of the auxiliary problem, it is proposed that the efficient and economical numerical algorithms from a computer point of view for finding the weak solution of the main problem. In addition to these, the auxiliary problem also allows us to obtain the weak solution in a class of discontinuous functions, which accurately describes all physical properties of the problem.