Table 8 Discretization errors of gradient reconstruction and face midpoint value approximation (reg. Double-split tri)

Face midpoint value approximation

face AB

\( \frac{1}{72}\left(-{x}^2\cos xy-\sin y\right){dx}^2+O\left({dx}^3\right) \)

–

face BC

\( \frac{1}{12}\left(\cos x-\cos y+\left(x-y\right)\sin xy\right) dx+O\left({dx}^2\right) \)

face AC

\( \frac{1}{12}\left(-\cos x-\cos y+\left(x+y\right)\sin xy\right) dx+O\left({dx}^2\right) \)

Gradient reconstruction

error of \( \frac{\partial f}{\partial x} \)

\( \frac{1}{3}\left(\cos x-y\sin xy\right)+O(dx) \)

\( \frac{1}{3}\left(- xy\cos xy-\sin xy\right) dx+O\left({dx}^2\right) \)

error of \( \frac{\partial f}{\partial y} \)

\( \frac{1}{3}\left(-\cos y+x\sin xy\right)+O(dx) \)

\( \frac{1}{18}\left(-\left({x}^2+2{y}^2\right)\cos xy-2\sin x-\sin y\right) dx+O\left({dx}^2\right) \)