The convergence of the horizontal line method for the continuity equation with discontinuous data (Q1058834)

The mixed problem for the evolution transport equation - the simplest equation of hyperbolic type - is approximated by the method of lines using the discretization of the time derivative of backward differences. The convergence of the approximate solution (without examination of the rate of convergence) to the exact solution of the differential problem is proved in the case of piecewise continuous boundary and initial data. The explicit form of the original problem and its approximation scheme is essentially used in the proof.