In CFD textbooks [5,23-27], one often replaces the lefthand...

In CFD textbooks $[5,23-27],$ one often replaces the lefthand sides of Eqs. $(8.102 b \text { and } c)$ with the following two expressions, respectively: \[ \frac{\partial}{\partial x}\left(u^{2}\right)+\frac{\partial}{\partial y}(v u) \text { and } \frac{\partial}{\partial x}(u v)+\frac{\partial}{\partial y}\left(v^{2}\right) \] Are these equivalent expressions, or are they merely simplified approximations? Either way, why might these forms be better for finite difference purposes?

In CFD textbooks $[5,23-27],$ one often replaces the lefthand sides of Eqs. $(8.102 b \text { and } c)$ with the following two expressions, respectively:
\[
\frac{\partial}{\partial x}\left(u^{2}\right)+\frac{\partial}{\partial y}(v u) \text { and } \frac{\partial}{\partial x}(u v)+\frac{\partial}{\partial y}\left(v^{2}\right)
\]
Are these equivalent expressions, or are they merely simplified approximations? Either way, why might these forms be better for finite difference purposes?

Key Concepts

Equivalence and Approximation in Numerical Schemes

This concept addresses whether alternate expressions of the same terms in a PDE are mathematically equivalent or are simplified through approximations. While the reformulated expressions might appear different, they are derived based on properties of the differential operators to maintain the conservation structure of the original equations. However, in the process of discretization, some equivalences in the continuous equations may only hold approximately, yet these forms are preferred in finite difference schemes because they ease the implementation of boundary conditions and preserve important conservation laws.

Finite Difference Discretization

This concept involves approximating the derivatives in a PDE with differences defined on a grid. In finite difference methods, rewriting flux terms as derivatives is advantageous because it allows for a symmetric and consistent discretization that aligns well with the physical conservation properties. Expressing the equations in divergence form can simplify the formulation of the numerical scheme, often leading to improved stability and accuracy, particularly near discontinuities or in regions with sharp gradients.

Conservative Form of Equations

This concept refers to writing partial differential equations (PDEs) in a form where the divergence (or flux) of certain quantities appears explicitly. In fluid dynamics and other areas, the conservative form ensures that physical conservation laws, such as mass, momentum, or energy, are fulfilled over discrete volumes, which can be particularly important when applying numerical methods. By expressing the equations in terms of derivatives of fluxes, one can more easily and accurately track the net transfer of quantities across control volume boundaries.

Please give Ace some feedback