Richardson extrapolation with operator based interpolation

(even, even) points

${\tau }_{\text{Extrapo}}$

(odd, even) points

${\tau }_{\text{Extrapo}}+\frac{7}{48}\left[\frac{1}{24}\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)-\frac{1}{40}\left(u_{x^6}+u_{y^6}\right)\right]h^6$

(even, odd) points

${\tau }_{\text{Extrapo}}+\frac{7}{48}\left[\frac{1}{24}\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)-\frac{1}{40}\left(u_{x^6}+u_{y^6}\right)\right]h^6$

(odd, odd) points

${\tau }_{\text{Extrapo}}+\frac{8}{48}\left[\frac{1}{24}\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)-\frac{1}{40}\left(u_{x^6}+u_{y^6}\right)\right]h^6$

Richardson extrapolation with multiple coarse grid computation

(even, even) points

${\tau }_{\text{Extrapo}}$

(odd, even) points

${\tau }_{\text{Extrapo}}+\frac{1}{12}\left\{4\times \frac{1}{24}\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)-\frac{1}{40}\left(u_{x^6}+16\times u_{y^6}\right)\right\}{h}^6$

(even, odd) points

${\tau }_{\text{Extrapo}}+\frac{1}{12}\left\{4\times \frac{1}{24}\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)-\frac{1}{40}\left(16\times u_{x^6}+u_{y^6}\right)\right\}{h}^6$

(odd, odd) points

$\begin{array}{l}{\tau }_{\text{Extrapo}}+\{\frac{1}{30}\left[4\times \frac{1}{24}\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)-\frac{1}{40}\left(u_{x^6}+16\times u_{y^6}\right)\right]\\ +\frac{1}{30}\left[4\times \frac{1}{24}\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)-\frac{1}{40}\left(16\times u_{x^6}+u_{y^6}\right)\right]\\ +\frac{1}{20}\left[\frac{1}{24}\left(u_{x^4{y}^2}+u_{x^2{y}^4}\right)-\frac{1}{40}\left(u_{x^6}+u_{y^6}\right)\right]\}{h}^6\end{array}$

Completed Richardson extrapolation

(even, even) points

${\tau }_{\text{Extrapo}}$

(odd, even) points

${\tau }_{\text{Extrapo}}+\left\{\frac{1}{24}{u}_{x^4{y}^4}+\frac{1}{120}\left(u_{x^6{y}^2}+u_{x^2{y}^6}\right)-\frac{1}{120}\left(u_{x^8}+u_{y^8}\right)\right\}{h}^6$

(even, odd) points

${\tau }_{\text{Extrapo}}+\left\{\frac{1}{24}{u}_{x^4{y}^4}+\frac{1}{120}\left(u_{x^6{y}^2}+u_{x^2{y}^6}\right)-\frac{1}{120}\left(u_{x^8}+u_{y^8}\right)\right\}{h}^6$

(odd, odd) points

${\tau }_{\text{Extrapo}}+\left\{\frac{1}{24}{u}_{x^4{y}^4}+\frac{1}{120}\left(u_{x^6{y}^2}+u_{x^2{y}^6}\right)-\frac{1}{120}\left(u_{x^8}+u_{y^8}\right)\right\}{h}^6$