Appendix 1: The coefficients \(b_{1,p}\) used in Eq. (22) for the first stencils with \(j=1\)
The first 10 coefficients \(b_{1,p}\) (\(p=1,2,\ldots ,10\)) used in Eq. (22) are presented below in the case of the mesh aspect ratio \(b_y=1\). All coefficients \(b_{1,p}\) used in these formulas are given in the attached file ‘b-coef.nb’. For simplicity of notations, below we use that
$$\begin{aligned}&b_{1,i} =b_{i}\ (i=1,2,\ldots ,10),\quad k_{1,i} =k_{i},\quad {{{\bar{k}}}}_{1,i} = {{{\bar{k}}}}_{i}\ (i=1,2,\ldots ,25),\\ &q_{i} = q_{1,i},\ q_{i+9} = q_{2,i},\ q_{i+18} = q_{3,i},\ q_{i+27} = q_{4,i}\ (i=1,2,\ldots ,9). \end{aligned}$$
$$\begin{aligned} b_{1}= & {} a_{1} { k}_{1}+a_{10} { k}_{10}+a_{11} { k}_{11}+a_{12} { k}_{12}\\ &+a_{13} { k}_{13}+a_{14} { k}_{14}+a_{15} { k}_{15}\\ &+a_{16} {k}_{16} +a_{17} { k}_{17}+a_{18} { k}_{18}+a_{19} { k}_{19}\\ &+a_{2} { k}_{2}+a_{20} { k}_{20}+a_{21} { k}_{21}\\ &+a_{22} { k}_{22} +a_{23} { k}_{23}+a_{24} { k}_{24}+a_{25} { k}_{25}+a_{3} { k}_{3}+a_{4} { k}_{4}\\ &+a_{5} { k}_{5}+a_{6} { k}_{6}+a_{7} { k}_{7}+a_{8} { k}_{8}+a_{9} { k}_{9}\\ &+q_{1}+q_{2}+q_{3}+q_{4}+q_{5}+q_{6}+q_{7}+q_{8}+q_{9}\\ b_{2}= & {} a_{1} {{{\bar{k}}}}_{1}+a_{10} {{{\bar{k}}}}_{10}+a_{11} {{{\bar{k}}}}_{11}+a_{12} {{{\bar{k}}}}_{12}+a_{13} {{{\bar{k}}}}_{13}+a_{14} {{{\bar{k}}}}_{14}\\ &+a_{15} {{{\bar{k}}}}_{15}+a_{16} {{{\bar{k}}}}_{16}\\ &+a_{17} {{{\bar{k}}}}_{17}+a_{18} {{{\bar{k}}}}_{18}+a_{19} {{{\bar{k}}}}_{19}+a_{2} {{{\bar{k}}}}_{2}+a_{20} {{{\bar{k}}}}_{20}+a_{21} {{{\bar{k}}}}_{21}\\ &+a_{22} {{{\bar{k}}}}_{22}+a_{23} {{{\bar{k}}}}_{23}+a_{24} {{{\bar{k}}}}_{24}+a_{25} {{{\bar{k}}}}_{25}+a_{3} {{{\bar{k}}}}_{3}\\ &+a_{4} {{{\bar{k}}}}_{4}+a_{5} {{{\bar{k}}}}_{5}+a_{6} {{{\bar{k}}}}_{6}+a_{7} {{{\bar{k}}}}_{7}+a_{8} {{{\bar{k}}}}_{8}+a_{9} {{{\bar{k}}}}_{9}\\ &+q_{19}+q_{20}+q_{21}+q_{22}+q_{23}+q_{24}+q_{25}+q_{26}+q_{27} \end{aligned}$$
$$\begin{aligned} b_{3}=-a_{1} { k}_{1}-a_{10} { k}_{10}-a_{11} { k}_{11}-a_{12} { k}_{12}-a_{13} { k}_{13}\\ &-a_{14} { k}_{14}-a_{15} { k}_{15}-a_{16} { k}_{16}\\ &-a_{17} { k}_{17}-a_{18} { k}_{18}-a_{19} { k}_{19}-a_{2} { k}_{2}-a_{20} { k}_{20}-a_{21} { k}_{21}\\ &-a_{22} { k}_{22}-a_{23} { k}_{23}-a_{24} { k}_{24}-a_{25} { k}_{25}-a_{3} { k}_{3}-a_{4} { k}_{4}\\ &-a_{5} { k}_{5}-a_{6} { k}_{6}-a_{7} { k}_{7}-a_{8} { k}_{8}-a_{9} { k}_{9}\\ &+{ k}_{1}+{ k}_{10}+{ k}_{11}+{ k}_{12}+{ k}_{13}+{ k}_{14}\\ &+{ k}_{15}+{ k}_{16}+{ k}_{17}+{ k}_{18}+{ k}_{19}\\ &+{ k}_{2}+{ k}_{20}+{ k}_{21}+{ k}_{22}+{ k}_{23}+{ k}_{24}\\ &+{ k}_{25}+{ k}_{3}+{ k}_{4}+{ k}_{5}+{ k}_{6}\\ &+{ k}_{7}+{ k}_{8}+{ k}_{9}-q_{1}-q_{2}-q_{3}-q_{4}\\ &-q_{5}-q_{6}-q_{7}-q_{8}-q_{9}\end{aligned}$$
$$\begin{aligned} b_{4}= & {} -a_{1} {{{\bar{k}}}}_{1}-a_{10} {{{\bar{k}}}}_{10}-a_{11} {{{\bar{k}}}}_{11}-a_{12} {{{\bar{k}}}}_{12}-a_{13} {{{\bar{k}}}}_{13}\\ &-a_{14} {{{\bar{k}}}}_{14}-a_{15} {{{\bar{k}}}}_{15}-a_{16} {{{\bar{k}}}}_{16}\\ &-a_{17} {{{\bar{k}}}}_{17}-a_{18} {{{\bar{k}}}}_{18}-a_{19} {{{\bar{k}}}}_{19}-a_{2} {{{\bar{k}}}}_{2}-a_{20} {{{\bar{k}}}}_{20}-a_{21} {{{\bar{k}}}}_{21}\\ &-a_{22} {{{\bar{k}}}}_{22}-a_{23} {{{\bar{k}}}}_{23}-a_{24} {{{\bar{k}}}}_{24}-a_{25} {{{\bar{k}}}}_{25}\\ &-a_{3} {{{\bar{k}}}}_{3}-a_{4} {{{\bar{k}}}}_{4}-a_{5} {{{\bar{k}}}}_{5}-a_{6} {{{\bar{k}}}}_{6}-a_{7} {{{\bar{k}}}}_{7}-a_{8} {{{\bar{k}}}}_{8}\\ &-a_{9} {{{\bar{k}}}}_{9}+{{{\bar{k}}}}_{1}+{{{\bar{k}}}}_{10} +{{{\bar{k}}}}_{11}+{{{\bar{k}}}}_{12}+{{{\bar{k}}}}_{13}+{{{\bar{k}}}}_{14}\\ &+{{{\bar{k}}}}_{15}+{{{\bar{k}}}}_{16}+{{{\bar{k}}}}_{17}+{{{\bar{k}}}}_{18}+{{{\bar{k}}}}_{19}+{{{\bar{k}}}}_{2}\\ &+{{{\bar{k}}}}_{20}+{{{\bar{k}}}}_{21}+{{{\bar{k}}}}_{22}+{{{\bar{k}}}}_{23}+{{{\bar{k}}}}_{24}+{{{\bar{k}}}}_{25}+{{{\bar{k}}}}_{3}\\ &+{{{\bar{k}}}}_{4}+{{{\bar{k}}}}_{5}+{{{\bar{k}}}}_{6}+{{{\bar{k}}}}_{7}\\ &+{{{\bar{k}}}}_{8}+{{{\bar{k}}}}_{9}-q_{19}-q_{20}-q_{21}-q_{22}\\ &-q_{23}-q_{24}-q_{25}-q_{26}-q_{27}\end{aligned}$$
$$\begin{aligned} b_{5}= & {} -a_{1} ({\mathrm{d}}x_G+2) { k}_{1}-a_{10} ({\mathrm{d}}x_G-2) { k}_{10}-a_{11} ({\mathrm{d}}x_G+2) { k}_{11}\\ &-a_{12} ({\mathrm{d}}x_G+1) { k}_{12}-a_{13} {\mathrm{d}}x_G { k}_{13}\\ &+{ k}_{14} (a_{14}-a_{14} {\mathrm{d}}x_G)-a_{15} ({\mathrm{d}}x_G-2) { k}_{15}-a_{16} ({\mathrm{d}}x_G+2) { k}_{16}\\ &-a_{17} ({\mathrm{d}}x_G+1) { k}_{17}-a_{18} {\mathrm{d}}x_G { k}_{18}\\ &+{ k}_{19} (a_{19}-a_{19} {\mathrm{d}}x_G)-a_{2} ({\mathrm{d}}x_G+1) { k}_{2}\\ &-a_{20} ({\mathrm{d}}x_G-2) { k}_{20}-a_{21} ({\mathrm{d}}x_G+2) { k}_{21}\\ &-a_{22} ({\mathrm{d}}x_G+1) { k}_{22}-a_{23} {\mathrm{d}}x_G { k}_{23}+{ k}_{24} (a_{24}\\ &-a_{24} {\mathrm{d}}x_G)-a_{25} ({\mathrm{d}}x_G-2) { k}_{25}\\ &-a_{3} {\mathrm{d}}x_G { k}_{3}+{ k}_{4} (a_{4}-a_{4} {\mathrm{d}}x_G)-a_{5} ({\mathrm{d}}x_G-2) { k}_{5}\\ &-a_{6} ({\mathrm{d}}x_G+2) { k}_{6}-a_{7} ({\mathrm{d}}x_G+1) { k}_{7}-a_{8} {\mathrm{d}}x_G { k}_{8}+{ k}_{9} (a_{9}\\ &-a_{9} {\mathrm{d}}x_G)+d_{x,2} q_{2}+d_{x,3} q_{3}+d_{x,4} q_{4}+d_{x,5} q_{5}+d_{x,6} q_{6}\\ &+d_{x,7} q_{7}+d_{x,8} q_{8}+d_{x,9} q_{9}+n_{x,1} q_{10} ({\lambda _*}+2 {\mu _{*}})+n_{x,2} q_{11} ({\lambda _*}\\ &+2 {\mu _{*}})+n_{x,3} q_{12} ({\lambda _*}+2 {\mu _{*}})+n_{x,4} q_{13} ({\lambda _*}\\ &+2 {\mu _{*}})+n_{x,5} q_{14} ({\lambda _*}\\ &+2 {\mu _{*}})+n_{x,6} q_{15} ({\lambda _*}+2 {\mu _{*}})+n_{x,7} q_{16} ({\lambda _*}\\ &+2 {\mu _{*}})+n_{x,8} q_{17} ({\lambda _*}+2 {\mu _{*}})\\ &+n_{x,9} q_{18} ({\lambda _*}+2 {\mu _{*}})+{\lambda _*} n_{y,1} q_{28}\\ &+{\lambda _*} n_{y,2} q_{29}+{\lambda _*} n_{y,3} q_{30}+{\lambda _*} n_{y,4} q_{31}\\ &+{\lambda _*} n_{y,5} q_{32}+{\lambda _*} n_{y,6} q_{33}\\ &+{\lambda _*} n_{y,7} q_{34}+{\lambda _*} n_{y,8} q_{35}+{\lambda _*} n_{y,9} q_{36}\end{aligned}$$
$$\begin{aligned} b_{6}= & {} -a_{1} ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{1}-a_{10} ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{10}-a_{11} ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{11}\\ &-a_{12} ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{12}-a_{13} {\mathrm{d}}x_G {{{\bar{k}}}}_{13}+{{{\bar{k}}}}_{14} (a_{14}\\ &-a_{14} {\mathrm{d}}x_G)-a_{15} ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{15}-a_{16} ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{16}\\ &-a_{17} ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{17}-a_{18} {\mathrm{d}}x_G {{{\bar{k}}}}_{18}\\ &+{{{\bar{k}}}}_{19} (a_{19}-a_{19} {\mathrm{d}}x_G)-a_{2} ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{2}-a_{20} ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{20}\\ &-a_{21} ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{21}-a_{22} ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{22}\\ &-a_{23} {\mathrm{d}}x_G {{{\bar{k}}}}_{23}+{{{\bar{k}}}}_{24} (a_{24}\\ &-a_{24} {\mathrm{d}}x_G)-a_{25} ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{25}-a_{3} {\mathrm{d}}x_G {{{\bar{k}}}}_{3}\\ &+{{{\bar{k}}}}_{4} (a_{4}-a_{4} {\mathrm{d}}x_G)-a_{5} ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{5}-a_{6} ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{6}\\ &-a_{7} ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{7}-a_{8} {\mathrm{d}}x_G {{{\bar{k}}}}_{8}\\ &+{{{\bar{k}}}}_{9} (a_{9}-a_{9} {\mathrm{d}}x_G)+d_{x,2} q_{20}+d_{x,3} q_{21}\\ &+d_{x,4} q_{22}+d_{x,5} q_{23}+d_{x,6} q_{24}\\ &+d_{x,7} q_{25}+d_{x,8} q_{26}+d_{x,9} q_{27}+n_{x,1} {\mu _{*}} q_{28}\\ &+n_{x,2} {\mu _{*}} q_{29}+n_{x,3} {\mu _{*}} q_{30}+n_{x,4} {\mu _{*}} q_{31}\\ &+n_{x,5} {\mu _{*}} q_{32}+n_{x,6} {\mu _{*}} q_{33}\\ &+n_{x,7} {\mu _{*}} q_{34}+n_{x,8} {\mu _{*}} q_{35}+n_{x,9} {\mu _{*}} q_{36}\\ &+n_{y,1} {\mu _{*}} q_{10}+n_{y,2} {\mu _{*}} q_{11}+n_{y,3} {\mu _{*}} q_{12}\\ &+n_{y,4} {\mu _{*}} q_{13}+n_{y,5} {\mu _{*}} q_{14}+n_{y,6} {\mu _{*}} q_{15}+n_{y,7} {\mu _{*}} q_{16}\\ &+n_{y,8} {\mu _{*}} q_{17}+n_{y,9} {\mu _{*}} q_{18}\end{aligned}$$
$$\begin{aligned} b_{7}= & {} (a_{1}-1) ({\mathrm{d}}x_G+2) { k}_{1}+(a_{10}-1) ({\mathrm{d}}x_G-2) { k}_{10}\\ &+(a_{11}-1) ({\mathrm{d}}x_G+2) { k}_{11}\\ &+(a_{12}-1) ({\mathrm{d}}x_G+1) { k}_{12}+(a_{13}-1) {\mathrm{d}}x_G { k}_{13}\\ &+(a_{14}-1) ({\mathrm{d}}x_G-1) { k}_{14}+(a_{15}-1) ({\mathrm{d}}x_G-2) { k}_{15}\\ &+(a_{16}-1) ({\mathrm{d}}x_G+2) { k}_{16}+(a_{17}-1) ({\mathrm{d}}x_G+1) { k}_{17}\\ &+(a_{18}-1) {\mathrm{d}}x_G { k}_{18}+(a_{19}-1) ({\mathrm{d}}x_G-1) { k}_{19}\\ &+(a_{2}-1) ({\mathrm{d}}x_G+1) { k}_{2}+(a_{20}-1) ({\mathrm{d}}x_G-2) { k}_{20}\\ &+(a_{21}-1) ({\mathrm{d}}x_G+2) { k}_{21}+(a_{22}-1) ({\mathrm{d}}x_G+1) { k}_{22}\\ &+(a_{23}-1) {\mathrm{d}}x_G { k}_{23}+(a_{24}-1) ({\mathrm{d}}x_G-1) { k}_{24}\\ &+(a_{25}-1) ({\mathrm{d}}x_G-2) { k}_{25}+(a_{3}-1) {\mathrm{d}}x_G { k}_{3}+(a_{4}-1) ({\mathrm{d}}x_G-1) { k}_{4}\\ &+(a_{5}-1) ({\mathrm{d}}x_G-2) { k}_{5}+(a_{6}-1) ({\mathrm{d}}x_G+2) { k}_{6}\\ &+(a_{7}-1) ({\mathrm{d}}x_G+1) { k}_{7}+(a_{8}-1) {\mathrm{d}}x_G { k}_{8}\\ &+(a_{9}-1) ({\mathrm{d}}x_G-1) { k}_{9}-d_{x,2} q_{2}-d_{x,3} q_{3}\\ &-d_{x,4} q_{4}-d_{x,5} q_{5}-d_{x,6} q_{6}\\ &-d_{x,7} q_{7}-d_{x,8} q_{8}-d_{x,9} q_{9}\\ &-n_{x,1} q_{10} ({\lambda _{**}}+2 {\mu _{**}})\\ &-n_{x,2} q_{11} ({\lambda _{**}}+2 {\mu _{**}})-n_{x,3} q_{12} ({\lambda _{**}}\\ &+2 {\mu _{**}})-n_{x,4} q_{13} ({\lambda _{**}}\\ &+2 {\mu _{**}})-n_{x,5} q_{14} ({\lambda _{**}}+2 {\mu _{**}})-n_{x,6} q_{15} ({\lambda _{**}}\\ &+2 {\mu _{**}})-n_{x,7} q_{16} ({\lambda _{**}}\\ &+2 {\mu _{**}})-n_{x,8} q_{17} ({\lambda _{**}}\\ &+2 {\mu _{**}})-n_{x,9} q_{18} ({\lambda _{**}}+2 {\mu _{**}})\\ &-{\lambda _{**}} n_{y,1} q_{28}-{\lambda _{**}} n_{y,2} q_{29}-{\lambda _{**}} n_{y,3} q_{30}\\ &-{\lambda _{**}} n_{y,4} q_{31}-{\lambda _{**}} n_{y,5} q_{32}\\ &-{\lambda _{**}} n_{y,6} q_{33}-{\lambda _{**}} n_{y,7} q_{34}\\ &-{\lambda _{**}} n_{y,8} q_{35}-{\lambda _{**}} n_{y,9} q_{36} \end{aligned}$$
$$\begin{aligned} b_{8}= & {} (a_{1}-1) ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{1}+(a_{10}-1) ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{10}\\ &+(a_{11}-1) ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{11}+(a_{12}-1) ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{12}\\ &+(a_{13}-1) {\mathrm{d}}x_G {{{\bar{k}}}}_{13}+(a_{14}-1) ({\mathrm{d}}x_G-1) {{{\bar{k}}}}_{14}\\ &+(a_{15}-1) ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{15}+(a_{16}-1) ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{16}\\ &+(a_{17}-1) ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{17}+(a_{18}-1) {\mathrm{d}}x_G {{{\bar{k}}}}_{18}\\ &+(a_{19}-1) ({\mathrm{d}}x_G-1) {{{\bar{k}}}}_{19}+(a_{2}-1) ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{2}\\ &+(a_{20}-1) ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{20}+(a_{21}-1) ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{21}\\ &+(a_{22}-1) ({\mathrm{d}}x_G+1)
{{{\bar{k}}}}_{22}\\ &+(a_{23}-1) {\mathrm{d}}x_G {{{\bar{k}}}}_{23}+(a_{24}-1) ({\mathrm{d}}x_G-1) {{{\bar{k}}}}_{24}\\ &+(a_{25}-1) ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{25}+(a_{3}-1) {\mathrm{d}}x_G {{{\bar{k}}}}_{3}\\ &+(a_{4}-1) ({\mathrm{d}}x_G-1) {{{\bar{k}}}}_{4}+(a_{5}-1) ({\mathrm{d}}x_G-2) {{{\bar{k}}}}_{5}\\ &+(a_{6}-1) ({\mathrm{d}}x_G+2) {{{\bar{k}}}}_{6}+(a_{7}-1) ({\mathrm{d}}x_G+1) {{{\bar{k}}}}_{7}\\ &+(a_{8}-1) {\mathrm{d}}x_G {{{\bar{k}}}}_{8}+(a_{9}-1) ({\mathrm{d}}x_G-1) {{{\bar{k}}}}_{9}-d_{x,2} q_{20}-d_{x,3} q_{21}\\ &-d_{x,4} q_{22}-d_{x,5} q_{23}-d_{x,6} q_{24}-d_{x,7} q_{25}-d_{x,8} q_{26}-d_{x,9} q_{27}\\ &-n_{x,1} {\mu _{**}} q_{28}-n_{x,2} {\mu _{**}} q_{29}-n_{x,3} {\mu _{**}} q_{30}\\ &-n_{x,4} {\mu _{**}} q_{31}-n_{x,5} {\mu _{**}} q_{32}\\ &-n_{x,6} {\mu _{**}} q_{33}-n_{x,7} {\mu _{**}} q_{34}-n_{x,8} {\mu _{**}} q_{35}-n_{x,9} {\mu _{**}} q_{36}\\ &-n_{y,1} {\mu _{**}} q_{10}-n_{y,2} {\mu _{**}} q_{11}-n_{y,3} {\mu _{**}} q_{12}-n_{y,4} {\mu _{**}} q_{13}\\ &-n_{y,5} {\mu _{**}} q_{14}-n_{y,6} {\mu _{**}} q_{15}-n_{y,7} {\mu _{**}} q_{16}\\ &-n_{y,8} {\mu _{**}} q_{17}-n_{y,9} {\mu _{**}} q_{18} \end{aligned}$$
$$\begin{aligned} b_{9}= & {} -a_{1} ({\mathrm{d}}y_G+2) { k}_{1}-a_{10} ({\mathrm{d}}y_G+1) { k}_{10}-a_{11} {\mathrm{d}}y_G { k}_{11}\\ &-a_{12} {\mathrm{d}}y_G { k}_{12}-a_{13} {\mathrm{d}}y_G { k}_{13}-a_{14} {\mathrm{d}}y_G { k}_{14}\\ &-a_{15} {\mathrm{d}}y_G { k}_{15}+{ k}_{16} (a_{16}-a_{16} {\mathrm{d}}y_G)\\ &+{ k}_{17} (a_{17}-a_{17} {\mathrm{d}}y_G)+{ k}_{18} (a_{18}-a_{18} {\mathrm{d}}y_G)+{ k}_{19} (a_{19}\\ &-a_{19} {\mathrm{d}}y_G)-a_{2} ({\mathrm{d}}y_G+2) { k}_{2}+{ k}_{20} (a_{20}\\ &-a_{20} {\mathrm{d}}y_G)-a_{21} ({\mathrm{d}}y_G-2) { k}_{21}\\ &-a_{22} ({\mathrm{d}}y_G-2) { k}_{22}-a_{23} ({\mathrm{d}}y_G-2) { k}_{23}\\ &-a_{24} ({\mathrm{d}}y_G-2) { k}_{24}\\ &-a_{25} ({\mathrm{d}}y_G-2) { k}_{25}-a_{3} ({\mathrm{d}}y_G+2) { k}_{3}-a_{4} ({\mathrm{d}}y_G+2) { k}_{4}\\ &-a_{5} ({\mathrm{d}}y_G+2) { k}_{5}-a_{6} ({\mathrm{d}}y_G+1) { k}_{6}\\ &-a_{7} ({\mathrm{d}}y_G+1) { k}_{7}-a_{8} ({\mathrm{d}}y_G+1) { k}_{8}\\ &-a_{9} ({\mathrm{d}}y_G+1) { k}_{9}+d_{y,2} q_{2}+d_{y,3} q_{3}+d_{y,4} q_{4}\\ &+d_{y,5} q_{5}+d_{y,6} q_{6}+d_{y,7} q_{7}+d_{y,8} q_{8}+d_{y,9} q_{9}\\ &+n_{x,1} {\mu _{*}} q_{28}+n_{x,2} {\mu _{*}} q_{29}+n_{x,3} {\mu _{*}} q_{30}+n_{x,4} {\mu _{*}} q_{31}\\ &+n_{x,5} {\mu _{*}} q_{32}+n_{x,6} {\mu _{*}} q_{33}+n_{x,7} {\mu _{*}} q_{34}\\ &+n_{x,8} {\mu _{*}} q_{35}+n_{x,9} {\mu _{*}} q_{36}\\ &+n_{y,1} {\mu _{*}} q_{10}+n_{y,2} {\mu _{*}} q_{11}+n_{y,3} {\mu _{*}} q_{12}\\ &+n_{y,4} {\mu _{*}} q_{13}+n_{y,5} {\mu _{*}} q_{14}\\ &+n_{y,6} {\mu _{*}} q_{15}+n_{y,7} {\mu _{*}} q_{16}+n_{y,8} {\mu _{*}} q_{17}+n_{y,9} {\mu _{*}} q_{18} \end{aligned}$$
$$\begin{aligned} b_{10}= & {} -a_{1} ({\mathrm{d}}y_G+2) {{{\bar{k}}}}_{1}-a_{10} ({\mathrm{d}}y_G+1) {{{\bar{k}}}}_{10}-a_{11} {\mathrm{d}}y_G {{{\bar{k}}}}_{11}\\ &-a_{12} {\mathrm{d}}y_G {{{\bar{k}}}}_{12}-a_{13} {\mathrm{d}}y_G {{{\bar{k}}}}_{13}\\ &-a_{14} {\mathrm{d}}y_G {{{\bar{k}}}}_{14}-a_{15} {\mathrm{d}}y_G {{{\bar{k}}}}_{15}+{{{\bar{k}}}}_{16} (a_{16}\\ &-a_{16} {\mathrm{d}}y_G)+{{{\bar{k}}}}_{17} (a_{17}-a_{17} {\mathrm{d}}y_G)\\ &+{{{\bar{k}}}}_{18} (a_{18}-a_{18} {\mathrm{d}}y_G)+{{{\bar{k}}}}_{19} (a_{19}-a_{19} {\mathrm{d}}y_G)\\ &-a_{2} ({\mathrm{d}}y_G+2) {{{\bar{k}}}}_{2}+{{{\bar{k}}}}_{20} (a_{20}-a_{20} {\mathrm{d}}y_G)\\ &-a_{21} ({\mathrm{d}}y_G-2) {{{\bar{k}}}}_{21}-a_{22} ({\mathrm{d}}y_G-2) {{{\bar{k}}}}_{22}-a_{23} ({\mathrm{d}}y_G-2) {{{\bar{k}}}}_{23}\\ &-a_{24} ({\mathrm{d}}y_G-2) {{{\bar{k}}}}_{24}-a_{25} ({\mathrm{d}}y_G-2) {{{\bar{k}}}}_{25}-a_{3} ({\mathrm{d}}y_G+2) {{{\bar{k}}}}_{3}\\ &-a_{4} ({\mathrm{d}}y_G+2) {{{\bar{k}}}}_{4}-a_{5} ({\mathrm{d}}y_G+2) {{{\bar{k}}}}_{5}\\ &-a_{6} ({\mathrm{d}}y_G+1) {{{\bar{k}}}}_{6}-a_{7} ({\mathrm{d}}y_G+1) {{{\bar{k}}}}_{7}-a_{8} ({\mathrm{d}}y_G+1) {{{\bar{k}}}}_{8}\\ &-a_{9} ({\mathrm{d}}y_G+1) {{{\bar{k}}}}_{9}+d_{y,2} q_{20}+d_{y,3} q_{21}\\ &+d_{y,4} q_{22}+d_{y,5} q_{23}+d_{y,6} q_{24}+d_{y,7} q_{25}\\ &+d_{y,8} q_{26}+d_{y,9} q_{27}\\ &+{\lambda _*} n_{x,1} q_{10}+{\lambda _*} n_{x,2} q_{11}+{\lambda _*} n_{x,3} q_{12}\\ &+{\lambda _*} n_{x,4} q_{13}+{\lambda _*} n_{x,5} q_{14}+{\lambda _*} n_{x,6} q_{15}+{\lambda _*} n_{x,7} q_{16}\\ &+{\lambda _*} n_{x,8} q_{17}+{\lambda _*} n_{x,9} q_{18}\\ &+n_{y,1} q_{28} ({\lambda _*}+2 {\mu _{*}})+n_{y,2} q_{29} ({\lambda _*}+2 {\mu _{*}})\\ &+n_{y,3} q_{30} ({\lambda _*}+2 {\mu _{*}})+n_{y,4} q_{31} ({\lambda _*}\\ &+2 {\mu _{*}})+n_{y,5} q_{32} ({\lambda _*}\\ &+2 {\mu _{*}})+n_{y,6} q_{33} ({\lambda _*}+2 {\mu _{*}})\\ &+n_{y,7} q_{34} ({\lambda _*}+2 {\mu _{*}})+n_{y,8} q_{35} ({\lambda _*}\\ &+2 {\mu _{*}})+n_{y,9} q_{36} ({\lambda _*}+2 {\mu _{*}}) \end{aligned}$$
Appendix 2: The stencil coefficients for homogeneous materials
The stencils coefficients can be analytically found (see [38]) and for the first stencil they are (for convenience, the matrix form is used below for the representation of these coefficients for square meshes with \(b_y=1\)):
$$\begin{aligned}&\left( \begin{array}{ccccc} k_{1,21} &{} k_{1,22} &{} k_{1,23} &{} k_{1,24} &{} k_{1,25} \\ k_{1,16} &{} k_{1,17} &{} k_{1,18} &{} k_{1,19} &{} k_{1,20} \\ k_{1,11} &{} k_{1,12} &{} k_{1,13} &{} k_{1,14} &{} k_{1,15} \\ k_{1,6} &{} k_{1,7} &{} k_{1,8} &{} k_{1,9} &{} k_{1,10} \\ k_{1,1} &{} k_{1,2} &{} k_{1,3} &{} k_{1,4} &{} k_{1,5} \\ \end{array} \right) = \; \left| \begin{array}{cc} -\frac{593298 \lambda ^3+2618461 \lambda ^2 \mu +3545745 \lambda \mu ^2+1471382 \mu ^3}{36 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{2 \left( 99558 \lambda ^3+935431 \lambda ^2 \mu +1655370 \lambda \mu ^2+766547 \mu ^3\right) }{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ -\frac{3648141 \lambda ^3+12961562 \lambda ^2 \mu +14963715 \lambda \mu ^2+5544394 \mu ^3}{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{28 \left( 65178 \lambda ^3+468496 \lambda ^2 \mu +748695 \lambda \mu ^2+321302 \mu ^3\right) }{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ -\frac{5534814 \lambda ^3+18941123 \lambda ^2 \mu +21099985 \lambda \mu ^2+7567326 \mu ^3}{6 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{4 \left( 116166 \lambda ^3+2637137 \lambda ^2 \mu +5190940 \lambda \mu ^2+2508819 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} \\ -\frac{3648141 \lambda ^3+12961562 \lambda ^2 \mu +14963715 \lambda \mu ^2+5544394 \mu ^3}{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{28 \left( 65178 \lambda ^3+468496 \lambda ^2 \mu +748695 \lambda \mu ^2+321302 \mu ^3\right) }{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ -\frac{593298 \lambda ^3+2618461 \lambda ^2 \mu +3545745 \lambda \mu ^2+1471382 \mu ^3}{36 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{2 \left( 99558 \lambda ^3+935431 \lambda ^2 \mu +1655370 \lambda \mu ^2+766547 \mu ^3\right) }{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ \end{array} \right. \\ &\left. \begin{array}{ccc} -\frac{326514 \lambda ^3+2124173 \lambda ^2 \mu +3348835 \lambda \mu ^2+1424826 \mu ^3}{6 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{2 \left( 99558 \lambda ^3+935431 \lambda ^2 \mu +1655370 \lambda \mu ^2+766547 \mu ^3\right) }{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{593298 \lambda ^3+2618461 \lambda ^2 \mu +3545745 \lambda \mu ^2+1471382 \mu ^3}{36 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ \frac{5228286 \lambda ^3+5361652 \lambda ^2 \mu -6876310 \lambda \mu ^2-6365076 \mu ^3}{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} -\frac{28 \left( 65178 \lambda ^3+468496 \lambda ^2 \mu +748695 \lambda \mu ^2+321302 \mu ^3\right) }{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{3648141 \lambda ^3+12961562 \lambda ^2 \mu +14963715 \lambda \mu ^2+5544394 \mu ^3}{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ 1 &{} -\frac{4 \left( 116166 \lambda ^3+2637137 \lambda ^2 \mu +5190940 \lambda \mu ^2+2508819 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} -\frac{5534814 \lambda ^3+18941123 \lambda ^2 \mu +21099985 \lambda \mu ^2+7567326 \mu ^3}{6 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ \frac{5228286 \lambda ^3+5361652 \lambda ^2 \mu -6876310 \lambda \mu ^2-6365076 \mu ^3}{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} -\frac{28 \left( 65178 \lambda ^3+468496 \lambda ^2 \mu +748695 \lambda \mu ^2+321302 \mu ^3\right) }{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{3648141 \lambda ^3+12961562 \lambda ^2 \mu +14963715 \lambda \mu ^2+5544394 \mu ^3}{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ -\frac{326514 \lambda ^3+2124173 \lambda ^2 \mu +3348835 \lambda \mu ^2+1424826 \mu ^3}{6 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{2 \left( 99558 \lambda ^3+935431 \lambda ^2 \mu +1655370 \lambda \mu ^2+766547 \mu ^3\right) }{9 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{593298 \lambda ^3+2618461 \lambda ^2 \mu +3545745 \lambda \mu ^2+1471382 \mu ^3}{36 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ \end{array} \; \right| , \end{aligned}$$
(40)
$$\begin{aligned}&\left( \begin{array}{ccccc} {{{\bar{k}}}}_{1,21} &{} {{{\bar{k}}}}_{1,22} &{} {{{\bar{k}}}}_{1,23} &{} {{{\bar{k}}}}_{1,24} &{} {{{\bar{k}}}}_{1,25} \\ {{{\bar{k}}}}_{1,16} &{} {{{\bar{k}}}}_{1,17} &{} {{{\bar{k}}}}_{1,18} &{} {{{\bar{k}}}}_{1,19} &{} {{{\bar{k}}}}_{1,20} \\ {{{\bar{k}}}}_{1,11} &{} {{{\bar{k}}}}_{1,12} &{} {{{\bar{k}}}}_{1,13} &{} {{{\bar{k}}}}_{1,14} &{} {{{\bar{k}}}}_{1,15} \\ {{{\bar{k}}}}_{1,6} &{} {{{\bar{k}}}}_{1,7} &{} {{{\bar{k}}}}_{1,8} &{} {{{\bar{k}}}}_{1,9} &{} {{{\bar{k}}}}_{1,10} \\ {{{\bar{k}}}}_{1,1} &{} {{{\bar{k}}}}_{1,2} &{} {{{\bar{k}}}}_{1,3} &{} {{{\bar{k}}}}_{1,4} &{} {{{\bar{k}}}}_{1,5} \\ \end{array} \right) = \left| \begin{array}{cc} \frac{25 \left( 16818 \lambda ^3+49421 \lambda ^2 \mu +46425 \lambda \mu ^2+13822 \mu ^3\right) }{12 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} \frac{50 \left( 5319 \lambda ^3+14038 \lambda ^2 \mu +10980 \lambda \mu ^2+2261 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} \\ \frac{25 \left( 21453 \lambda ^3+62726 \lambda ^2 \mu +58815 \lambda \mu ^2+17542 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} \frac{50 \left( 74538 \lambda ^3+243115 \lambda ^2 \mu +259491 \lambda \mu ^2+90914 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} \\ 0 &{} 0 \\ -\frac{25 \left( 21453 \lambda ^3+62726 \lambda ^2 \mu +58815 \lambda \mu ^2+17542 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} -\frac{50 \left( 74538 \lambda ^3+243115 \lambda ^2 \mu +259491 \lambda \mu ^2+90914 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} \\ -\frac{25 \left( 16818 \lambda ^3+49421 \lambda ^2 \mu +46425 \lambda \mu ^2+13822 \mu ^3\right) }{12 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } &{} -\frac{50 (e+\mu ) \left( 5319 \lambda ^2+8719 \lambda \mu +2261 \mu ^2\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} \\ \end{array} \right. \\ &\quad \left. \begin{array}{ccc} 0 &{} -\frac{50 \left( 5319 \lambda ^3+14038 \lambda ^2 \mu +10980 \lambda \mu ^2+2261 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} -\frac{25 \left( 16818 \lambda ^3+49421 \lambda ^2 \mu +46425 \lambda \mu ^2+13822 \mu ^3\right) }{12 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ 0 &{} -\frac{50 \left( 74538 \lambda ^3+243115 \lambda ^2 \mu +259491 \lambda \mu ^2+90914 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} -\frac{25 \left( 21453 \lambda ^3+62726 \lambda ^2 \mu +58815 \lambda \mu ^2+17542 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} \\ 0 &{} 0 &{} 0 \\ 0 &{} \frac{50 \left( 74538 \lambda ^3+243115 \lambda ^2 \mu +259491 \lambda \mu ^2+90914 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} \frac{25 \left( 21453 \lambda ^3+62726 \lambda ^2 \mu +58815 \lambda \mu ^2+17542 \mu ^3\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} \\ 0 &{} \frac{50 (e+\mu ) \left( 5319 \lambda ^2+8719 \lambda \mu +2261 \mu ^2\right) }{4094838 \lambda ^3+69578991 \lambda ^2 \mu +133228095 \lambda \mu ^2+63715242 \mu ^3} &{} \frac{25 \left( 16818 \lambda ^3+49421 \lambda ^2 \mu +46425 \lambda \mu ^2+13822 \mu ^3\right) }{12 \left( 1364946 \lambda ^3+23192997 \lambda ^2 \mu +44409365 \lambda \mu ^2+21238414 \mu ^3\right) } \\ \end{array} \right| . \end{aligned}$$
(41)
Similarly, we can find 50 coefficients \(k_{2,i}\) and \({{{\bar{k}}}}_{2,i}\) (\(i=1,2,\ldots ,25\)) of the second stencil equation:
$$\begin{aligned}&\left( \begin{array}{ccccc} k_{2,21} &{} k_{2,22} &{} k_{2,23} &{} k_{2,24} &{} k_{2,25} \\ k_{2,16} &{} k_{2,17} &{} k_{2,18} &{} k_{2,19} &{} k_{2,20} \\ k_{2,11} &{} k_{2,12} &{} k_{2,13} &{} k_{2,14} &{} k_{2,15} \\ k_{2,6} &{} k_{2,7} &{} k_{2,8} &{} k_{2,9} &{} k_{2,10} \\ k_{2,1} &{} k_{2,2} &{} k_{2,3} &{} k_{2,4} &{} k_{2,5} \\ \end{array} \right) = \left( \begin{array}{ccccc} {{{\bar{k}}}}_{1,21} &{} {{{\bar{k}}}}_{1,22} &{} {{{\bar{k}}}}_{1,23} &{} {{{\bar{k}}}}_{1,24} &{} {{{\bar{k}}}}_{1,25} \\ {{{\bar{k}}}}_{1,16} &{} {{{\bar{k}}}}_{1,17} &{} {{{\bar{k}}}}_{1,18} &{} {{{\bar{k}}}}_{1,19} &{} {{{\bar{k}}}}_{1,20} \\ {{{\bar{k}}}}_{1,11} &{} {{{\bar{k}}}}_{1,12} &{} {{{\bar{k}}}}_{1,13} &{} {{{\bar{k}}}}_{1,14} &{} {{{\bar{k}}}}_{1,15} \\ {{{\bar{k}}}}_{1,6} &{} {{{\bar{k}}}}_{1,7} &{} {{{\bar{k}}}}_{1,8} &{} {{{\bar{k}}}}_{1,9} &{} {{{\bar{k}}}}_{1,10} \\ {{{\bar{k}}}}_{1,1} &{} {{{\bar{k}}}}_{1,2} &{} {{{\bar{k}}}}_{1,3} &{} {{{\bar{k}}}}_{1,4} &{} {{{\bar{k}}}}_{1,5} \\ \end{array} \right) ^T , \end{aligned}$$
(42)
$$\begin{aligned}&\left( \begin{array}{ccccc} {{{\bar{k}}}}_{2,21} &{} {{{\bar{k}}}}_{2,22} &{} {{{\bar{k}}}}_{2,23} &{} {{{\bar{k}}}}_{2,24} &{} {{{\bar{k}}}}_{2,25} \\ {{{\bar{k}}}}_{2,16} &{} {{{\bar{k}}}}_{2,17} &{} {{{\bar{k}}}}_{2,18} &{} {{{\bar{k}}}}_{2,19} &{} {{{\bar{k}}}}_{2,20} \\ {{{\bar{k}}}}_{2,11} &{} {{{\bar{k}}}}_{2,12} &{} {{{\bar{k}}}}_{2,13} &{} {{{\bar{k}}}}_{2,14} &{} {{{\bar{k}}}}_{2,15} \\ {{{\bar{k}}}}_{2,6} &{} {{{\bar{k}}}}_{2,7} &{} {{{\bar{k}}}}_{2,8} &{} {{{\bar{k}}}}_{2,9} &{} {{{\bar{k}}}}_{2,10} \\ {{{\bar{k}}}}_{2,1} &{} {{{\bar{k}}}}_{2,2} &{} {{{\bar{k}}}}_{2,3} &{} {{{\bar{k}}}}_{2,4} &{} {{{\bar{k}}}}_{2,5} \\ \end{array} \right) = \left( \begin{array}{ccccc} k_{1,21} &{} k_{1,22} &{} k_{1,23} &{} k_{1,24} &{} k_{1,25} \\ k_{1,16} &{} k_{1,17} &{} k_{1,18} &{} k_{1,19} &{} k_{1,20} \\ k_{1,11} &{} k_{1,12} &{} k_{1,13} &{} k_{1,14} &{} k_{1,15} \\ k_{1,6} &{} k_{1,7} &{} k_{1,8} &{} k_{1,9} &{} k_{1,10} \\ k_{1,1} &{} k_{1,2} &{} k_{1,3} &{} k_{1,4} &{} k_{1,5} \\ \end{array} \right) ^T , \end{aligned}$$
(43)
where the right-hand sides in Eqs. (42) and (43) are given by Eqs. (40) and (41) for the first stencil.
Appendix 3: The expression for the term \(p_1\) in Eqs. (28) and (29)
$$\begin{aligned} p_1= & {} (a_{10} { k}_{10} ({\mathrm{d}}x_G-2)^2+a_{1} ({\mathrm{d}}x_G+2)^2 { k}_{1} \\ &+a_{11} {\mathrm{d}}x_G^2 { k}_{11}+4 a_{11} { k}_{11} \\ &+4 a_{11} {\mathrm{d}}x_G { k}_{11}+a_{12} {\mathrm{d}}x_G^2 { k}_{12}+a_{12} { k}_{12}+2 a_{12} {\mathrm{d}}x_G { k}_{12} \\ &+a_{13} {\mathrm{d}}x_G^2 { k}_{13}+a_{14} {\mathrm{d}}x_G^2 { k}_{14}+a_{14} { k}_{14} \\ &-2 a_{14} {\mathrm{d}}x_G { k}_{14}+a_{15} {\mathrm{d}}x_G^2 { k}_{15} \\ &+4 a_{15} { k}_{15}-4 a_{15} {\mathrm{d}}x_G { k}_{15} \\ &+a_{16} {\mathrm{d}}x_G^2 { k}_{16}+4 a_{16} { k}_{16}+4 a_{16} {\mathrm{d}}x_G { k}_{16} \\ &+a_{17} {\mathrm{d}}x_G^2 { k}_{17}+a_{17} { k}_{17} \\ &+2 a_{17} {\mathrm{d}}x_G { k}_{17}+a_{18} {\mathrm{d}}x_G^2 { k}_{18} \\ &+a_{19} {\mathrm{d}}x_G^2 { k}_{19}+a_{19} { k}_{19}-2 a_{19} {\mathrm{d}}x_G { k}_{19} \\ &+a_{2} {\mathrm{d}}x_G^2 { k}_{2}+a_{2} { k}_{2} \\ &+2 a_{2} {\mathrm{d}}x_G { k}_{2}+a_{20} {\mathrm{d}}x_G^2 { k}_{20}+4 a_{20} { k}_{20}-4 a_{20} {\mathrm{d}}x_G { k}_{20} \\ &+a_{21} {\mathrm{d}}x_G^2 { k}_{21}+4 a_{21} { k}_{21} \\ &+4 a_{21} {\mathrm{d}}x_G { k}_{21}+a_{22} {\mathrm{d}}x_G^2 { k}_{22}+a_{22} { k}_{22} \\ &+2 a_{22} {\mathrm{d}}x_G { k}_{22}+a_{23} {\mathrm{d}}x_G^2 { k}_{23}+a_{24} {\mathrm{d}}x_G^2 { k}_{24} \\ &+a_{24} { k}_{24}-2 a_{24} {\mathrm{d}}x_G { k}_{24} \\ &+a_{25} {\mathrm{d}}x_G^2 { k}_{25}+4 a_{25} { k}_{25}-4 a_{25} {\mathrm{d}}x_G { k}_{25}+a_{3} {\mathrm{d}}x_G^2 { k}_{3} \\ &+a_{4} {\mathrm{d}}x_G^2 { k}_{4}+a_{4} { k}_{4}-2 a_{4} {\mathrm{d}}x_G { k}_{4}+a_{5} {\mathrm{d}}x_G^2 { k}_{5} \\ &+4 a_{5} { k}_{5}-4 a_{5} {\mathrm{d}}x_G { k}_{5} \\ &+a_{6} {\mathrm{d}}x_G^2 { k}_{6}+4 a_{6} { k}_{6} \\ &+4 a_{6} {\mathrm{d}}x_G { k}_{6}+a_{7} {\mathrm{d}}x_G^2 { k}_{7}+a_{7} { k}_{7} \\ &+2 a_{7} {\mathrm{d}}x_G { k}_{7}+a_{8} {\mathrm{d}}x_G^2 { k}_{8} \\ &+a_{9} {\mathrm{d}}x_G^2 { k}_{9}+a_{9} { k}_{9}-2 a_{9} {\mathrm{d}}x_G { k}_{9} \\ &+2 d_{x,2} {\lambda _*} n_{x,2} q_{11}+4 d_{x,2} n_{x,2} {\mu _{*}} q_{11} \\ &+2 d_{x,3} {\lambda _*} n_{x,3} q_{12} \\ &+4 d_{x,3} n_{x,3} {\mu _{*}} q_{12}+2 d_{x,4} {\lambda _*} n_{x,4} q_{13} \\ &+4 d_{x,4} n_{x,4} {\mu _{*}} q_{13}+2 d_{x,5} {\lambda _*} n_{x,5} q_{14} \\ &+4 d_{x,5} n_{x,5} {\mu _{*}} q_{14}+2 d_{x,6} {\lambda _*} n_{x,6} q_{15} \\ &+4 d_{x,6} n_{x,6} {\mu _{*}} q_{15}+2 d_{x,7} {\lambda _*} n_{x,7} q_{16} \\ &+4 d_{x,7} n_{x,7} {\mu _{*}} q_{16} \\ &+2 d_{x,8} {\lambda _*} n_{x,8} q_{17}+4 d_{x,8} n_{x,8} {\mu _{*}} q_{17} \\ &+2 d_{x,9} {\lambda _*} n_{x,9} q_{18}+4 d_{x,9} n_{x,9} {\mu _{*}} q_{18} \\ &+d_{x,2}^2 q_{2} +2 d_{x,2} {\lambda _*} n_{y,2} q_{29}+d_{x,3}^2 q_{3} \\ &+2 d_{x,3} {\lambda _*} n_{y,3} q_{30}+2 d_{x,4} {\lambda _*} n_{y,4} q_{31} \\ &+2 d_{x,5} {\lambda _*} n_{y,5} q_{32}+2 d_{x,6} {\lambda _*} n_{y,6} q_{33} \\ &+2 d_{x,7} {\lambda _*} n_{y,7} q_{34}+2 d_{x,8} {\lambda _*} n_{y,8} q_{35} \\ &+2 d_{x,9} {\lambda _*} n_{y,9} q_{36} \\ &+d_{x,4}^2 q_{4}+d_{x,5}^2 q_{5}+d_{x,6}^2 q_{6} \\ &+d_{x,7}^2 q_{7}+d_{x,8}^2 q_{8}+d_{x,9}^2 q_{9}). \end{aligned}$$
(44)
