The space discretization of the one-dimensional barotropic quasi-gas dynamic system of equations and the energy balance equation

For the barotropic quasi-gas dynamic system of equations, the law of non-increasing total energy is valid. But even in the spatially one-dimensional case, for its standard discretizations the validity of this law cannot be provided since there appear mesh disbalance terms. We propose a new conservative symmetric in space discretization of this system, for which the energy balance equation of the proper form is derived and non-increasing of the total energy is guaranteed (that takes place even in the presence of the potential mass force). Important elements of the method are non-standard space average of the density depending on the state function and discretization of the derivative of this function. The results are valid for any non-uniform mesh. As an important special case, the results are valid for a regularized (quasi-gas dynamic) system of shallow water equations in the general case of non-flat bottom; moreover, here the non-standard discretizations become standard ones but the method is still new. It is the well-balanced in a sense.