Numerical implementation of an invariant scheme for one-dimensional shallow water equations in Lagrangian coordinates

Various approaches to a finite-difference modeling of continuum mechanics' equations are discussed. The numerical implementation of a new invariant finite-difference scheme for one-dimensional shallow water equations in Lagrangian (potential) and mass Lagrangian coordinates is presented. The new scheme possesses the local conservation laws of energy, mass, center of mass and momentum. Some known exact solutions that do not contain essential discontinuities are considered as test problems. Calculations are carried out for several examples of various conservative schemes with artificial viscosity. The graphic illustrations of the obtained solutions are given where the conservation laws are checked on the solutions.