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

Advances in Aerodynamics

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

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 \( \frac{\partial f}{\partial x} \)	\( -R\Gamma \left( xy\cos xy+\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)	\( -R\Gamma \left( xy\cos xy+\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)
Gradient reconstruction	error of \( \frac{\partial f}{\partial y} \)	\( -\frac{\Gamma}{2}\left(\cos y-x\sin xy\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)	\( -\frac{R\Gamma}{2\left(1+{\Gamma}^2\right)}\left(\left({y}^2+{x}^2{\Gamma}^2\right)\cos xy+\sin x+{\Gamma}^2\sin y\right){h}_{\theta }+O\left({h}_{\theta}^2\right) \)