Table 1.

Raw residuals for least-squares formulation with Eq. (6)

Raw residuals	Expressions
RNF(fHb)	$$\frac{\partial f_{Hb}}{\partial t}+\mathbf{u}·\nabla f_{Hb}-r_{{\mathrm{\Delta }}_{Hb}}^l(1-f_{Hb})$$
RS1(C)	$$C\left(\frac{\partial C}{\partial t}+\mathbf{u}·\nabla C\right)-\frac{{\kappa }^2{r}_{{\mathrm{\Delta }}_{Hb}}^l(1-f_{Hb}^b-C^2)}{2}$$
RS1(C)	$$\{\begin{array}{l}\frac{\partial C}{\partial t}+\mathbf{u}·\nabla C-\frac{r_{{\mathrm{\Delta }}_{Hb}}^l}{2}\left(\frac{{\kappa }^2(1-f_{Hb}^b)}{C}-C\right)(\text{S}2\text{formulation})\hfill \\ \frac{\partial C}{\partial t}+\mathbf{u}·\nabla C-\frac{r_{{\mathrm{\Delta }}_{Hb}}^l}{2}\left(\frac{{\kappa }^2(1-f_{Hb}^b)}{{(C^4+{\epsilon }_r^4)}^{1/4}}-C\right)(\text{regularized}\text{S}2\text{formulation})\hfill \end{array}$$