Used to improve stability of iterative matrix solvers ## Generally Fraction of the field variable from the previous iteration is used for the current iteration. Using pressure as an example: $ p = \alpha p_{new}+(1-\alpha)p_{old} $ ## [[SIMPLE Algorithm]] Intended for steady flows (no time derivative) $ \frac{\partial\boldsymbol{U}}{\partial t}=\frac{\boldsymbol{U}_p^{i+1}-\boldsymbol{U}_p^{i}}{\Delta t} $ - Time derivative terms are always on diagonal of matrices - As $\Delta t \rightarrow 0$, $\frac{\partial\boldsymbol{U}}{\partial t}$ grows larger --> improves diagonal dominance - improves stability Since there is no time derivative in steady flows, ***under-relaxation*** is used to artificially improve diagonal dominance Under-relaxation term $0<\alpha\leq1$ introduced to the equations: $ a_P\boldsymbol{U}_P+\sum_Na_N\boldsymbol{U}_N=R_P $ - $P$ - diagonal terms - $N$ - off-diagonal terms $ \frac{1-\alpha}{\alpha}\boldsymbol a_p{U}_P+a_P\boldsymbol{U}_P+\sum_Na_N\boldsymbol{U}_N=R_P+\frac{1-\alpha}{\alpha}a_P\boldsymbol{U}^\text{old}_P $ $ \Big(\frac{1-\alpha}{\alpha}+1\Big)a_P\boldsymbol{U}_P+\sum_Na_N\boldsymbol{U}_N=R_P+\frac{1-\alpha}{\alpha}a_P\boldsymbol{U}^\text{old}_P $ $ \frac{1}{\alpha}a_P\boldsymbol{U}_P+\sum_Na_N\boldsymbol{U}_N=R_P+\frac{1-\alpha}{\alpha}a_P\boldsymbol{U}^\text{old}_P $ - diagonal term gets larger as $\alpha\rightarrow0$