SOR update

Substituting $c=\frac{a+b}{6a+2b}$, $d=\frac{a}{6a+2b}$, $e=\frac{b}{4(6a+2b)}$ we obtain:

$$\left. \begin{array}{ll} \mathbf{p}= & \hat{\mathbf{f}}_{i,j,k}+c... ...athbf{v}^{(m+1)}_{i,j,k-1}+\mathbf{v}^{(m)}_{i,j,k+1}\right) \end{array}\right.$$ (19)

Setting $\mathbf{v}:=(r s t)^T$ we may write

$$\begin{array}{ll} q_{x}=e & \left( s^{(m+1)}_{i-1,j-1,k}+s^{(... ...-s^{(m)}_{i,j-1,k+1}-s^{(m+1)}_{i,j+1,k-1}\right) , \end{array}$$ (20)

hence for the residual vector

$$\mathbf{r}_{i,j,k}=\omega \left( \mathbf{p}+\mathbf{q}-\mathbf{v}^{m}_{i,j,k}\right) ,$$ (21)

and the SOR update of $\mathbf{v}_{i,j,k}$ is given by

$$\mathbf{v}^{(m+1)}_{i,j,k}=\mathbf{v}^{(m)}_{i,j,k}+\mathbf{r}_{i,j,k}.$$ (22)