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Table 5 Discretization errors of gradient reconstruction and face midpoint value approximation

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

Discretization errors GG-Cell (curved quads.) LSQ (curved quads.)
Face midpoint value approximation face AB \( -\frac{1}{2}R\Gamma \left(\cos y-x\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
face BC \( O\left({h}_{\theta}^2\right) \)
face CD \( -\frac{1}{2}R\Gamma \left(\cos y-x\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
face DA \( O\left({h}_{\theta}^2\right) \)
Gradient reconstruction error of \( \frac{\partial f}{\partial x} \) \( -R\Gamma \left( xy\cos xy+\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \) \( -R\Gamma \left( xy\cos xy+\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
error of \( \frac{\partial f}{\partial y} \) \( -\frac{\Gamma}{2}\left(\cos y-x\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \) \( -\frac{R\Gamma}{2\left(1+{\Gamma}^2\right)}\left(\left({y}^2+{x}^2{\Gamma}^2\right)\cos xy+\sin x+{\Gamma}^2\sin y\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)