Skip to main content

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