The analytical solution of shallow water equations by the Rothe’s method
For a one-dimensional system of nonlinear hyperbolic partial differential equations - the shallow water equations in conservative form, a proof of unique solvability by the Rothe’s methodic obtained. The differential-difference problem of the Rothe’s method is transformed so that a system of ordinary differential equations is obtained, which has a conservative form similar to the system of nonlinear shallow water equations. Since the behavior of the system is determined by eigenvalues, a homogeneous system of ordinary differential equations (without the right part) is investigated. From such a system follows a square equation that characterizes its unstable behavior at singular points. The results are presented by a theorem and a proof is given confirming the validity of the conclusions. An formula is derived that generalizes the Courant-Friedrichs-Lewy condition on the nonlinear shallow water equations. The time interval is limited by the inequality due to the singularity, but the distance to the center of the unstable node calculated by the formula.