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Table 2 Discretization errors of gradient reconstruction and face midpoint value approximation (quads.)

From: Accuracy analysis of gradient reconstruction on isotropic unstructured meshes and its effects on inviscid flow simulation

Discretization errors GG-Cell LSQ-basic
Face midpoint value approximation face AB \( \frac{1}{8}\left(-{x}^2\cos xy-\sin y\right){dx}^2+O\left({dx}^3\right) \)
face BC \( \frac{1}{8}\left(-{y}^2\cos xy-\sin x\right){dx}^2+O\left({dx}^3\right) \)
face CD \( \frac{1}{8}\left(-{x}^2\cos xy-\sin y\right){dx}^2+O\left({dx}^3\right) \)
face AD \( \frac{1}{8}\left(-{y}^2\cos xy-\sin x\right){dx}^2+O\left({dx}^3\right) \)
Gradient reconstruction error of \( \frac{\partial f}{\partial x} \) \( \frac{1}{6}\left(\cos x-{y}^3\sin xy\right){dx}^2+O\left({dx}^3\right) \) \( \frac{1}{6}\left(\cos x-{y}^3\sin xy\right){dx}^2+O\left({dx}^3\right) \)
error of \( \frac{\partial f}{\partial y} \) \( \frac{1}{6}\left(\cos y-{x}^3\sin xy\right){dx}^2+O\left({dx}^3\right) \) \( \frac{1}{6}\left(\cos y-{x}^3\sin xy\right){dx}^2+O\left({dx}^3\right) \)
  1. note:dx in the equations is the grid spacing (side length of the quadrilateral) in the x-direction
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