The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian coordinates. For equations in Lagrangian coordinates an invariant finite-difference scheme is constructed for all cases for which conservation laws exist in the differential model. Such schemes possess the difference analogues of the conservation laws of mass, momentum, energy, the law of center of mass motion for horizontal, inclined and parabolic bottom topographies. Invariant conservative difference scheme is tested numerically in comparison with naive approximation invariant scheme.

翻译：考虑拉格朗日坐标中的单维修改浅水方程。 它显示了拉格朗日坐标、质量拉格朗日变量和欧莱安坐标的对称与保护法之间的关系。 对于拉格朗日坐标的方程,为差异模型中存在保护法的所有情况制定了不变的有限差异计划。这种计划具有质量、动力、能源、物质运动中心对水平、倾斜和倾斜底部地形的定律等同质量、动力、能源、物质运动中心对水平、倾斜和抛角底部地形的定律等差异。 与天性偏差计划相比,对差异保守性差异计划进行了数字测试。