Finite differences method for shallow water equations in a class of discontinuous functions

In this study a finite difference method for solving initial boundary value problem for the one-dimensional nonlinear system of differential equations describing shallow water flows in a class of discontinuous functions is suggested. In order to find the numerical solution of the problem, known as main problem, a special auxiliary problem is introduced. The degree of smoothness of the solution of the auxiliary problem is higher than the smoothness of the solution of the main problem. Moreover, the suggested auxiliary problem makes it possible to write out the effective and higher order numerical algorithms. Some results of numerical experiments are demonstrated.