New answers tagged finite-volume
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Usually you speak of a $n$'th order (accurate) method if your Taylor truncation error is of order $n+1$. This means your approximation is accurate up to order $n$ terms, and your errors are of order $n+1$. However, in FVM methods you often have no easy way of obtaining the truncation error of your formulation, since you reconstruct the numerical fluxes $F$ ...
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For methods that discretize simultaneously space and time, such as Lax-Wendroff or Upwind with Forward/Backward Euler, you can manually compute the truncation errors (for a sufficiently regular $u$). For instance, the former (LW) is both second order accurate in space and time, while the latter (Euler-Upwind) is only first order accurate.
To obtain overall ...
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With reference to the figure below, the equations are discretized in the region enclosed within the red dotted lines. And, the blue colored line denotes the subscript 2 in your equations.
$$
\frac{\partial u^2}{\partial x}|_2 = \frac{\left({u_e^2 - u_w^2}\right)}{\frac{(\Delta x_1 + \Delta x_2)}{2}} \\
\frac{\partial uv}{\partial y}|_2 = \frac{\left({u_s v_s ...
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