Comparison of additional second-order terms in finite-difference Euler equations and regularized fluid dynamics equations

In recent years, an area of research in computational mathematics has emerged that is associated with the numerical solution of fluid flow problems based on regularized fluid dynamics equations involving additional terms with velocity, pressure, and body force. The inclusion of these functions in the additional terms has been physically substantiated only for pressure and body force. In this paper, the continuity equation obtained geometrically by Euler is shown to involve second-order terms in time that contain Jacobians of the velocity field and are consistent with some of the additional terms in the regularized fluid dynamics equations. The same Jacobians are contained in the inhomogeneous right-hand side of the wave equation and generate waves of pressure, density, and sound. Physical interpretations of the additional terms used in the regularized fluid dynamics equations are given.