## Appendix A
### Accuracy Analysis
We utilize the following [PERSON] expansions about \((i\,h,\,n\,k)\),  
\[\phi_{i}^{n+1}=\phi_{i}^{n}+k\,\phi_{t}+\frac{k^{2}}{2!}\phi_{tt}+\frac{k^{3}} {3!}\phi_{tt}+\ldots \tag{119}\]  
\[\phi_{i+1}^{n}=\phi_{i}^{n}+h\,\phi_{x}+\frac{h^{2}}{2!}\phi_{xx}+\ldots \tag{120}\]\[\phi_{i-1}^{n}=\phi_{i}^{n}-h\,\phi_{x}+\frac{h^{2}}{2!}\phi_{xx}+\ldots \tag{116}\]  
\[\phi_{i+1}^{n+1}=\phi_{i}^{n}+h\phi_{x}+k\,\phi_{t}+\frac{h^{2}}{2!}\phi_{xx}+2 \frac{hk}{2!}\phi_{xt}+\frac{k^{2}}{2!}\phi_{tt}+\frac{h^{3}}{3!}\phi_{xxx}+3 \frac{h^{2}k}{3!}\phi_{xxt}+3\frac{hk^{2}}{3!}\phi_{xtt}+\frac{k^{3}}{3!}\phi_{ttt }+\ldots \tag{117}\]  
\[\phi_{i-1}^{n+1}=\phi_{i}^{n}-h\phi_{x}+k\,\phi_{t}+\frac{h^{2}}{2!}\phi_{xx}+2 \frac{hk}{2!}\phi_{xt}+\frac{k^{2}}{2!}\phi_{tt}-\frac{h^{3}}{3!}\phi_{xxx}+3 \frac{h^{2}k}{3!}\phi_{xtxt}+3\frac{hk^{2}}{3!}\phi_{xtt}+\frac{k^{3}}{3!}\phi_ {ttt}+\ldots \tag{118}\]  
\[D_{i+1/2}=D_{i}+\frac{1}{2}h\,D^{\prime}+\frac{h^{2}}{4}D^{\prime\prime}+\ldots \tag{119}\]  
\[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]0  
We also need to form the following quantities,  
\[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]1  
\[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]2 \[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]3  
\[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]4 \[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]5  
\[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]6  
\[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]7  
\[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]8  
\[P_{k,h}c\,=\,\left(\frac{c_{i}^{n+1}-c_{i}^{n}}{k}\right) - \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n+1}-c_{i}^{n+1}}{h}-D_{ i-1/2}\frac{c_{i}^{n+1}-c_{i-1}^{n+1}}{h}}{h} \tag{118}\]9  
\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]0  
\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]1  
Using all necessary information, the new and old diffusion-related terms in (113) become  
\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]2  
\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]3\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]4  
\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]5  
\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]6  
Using the necessary Taylor series, the source term in (100) becomes  
\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]7 \[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]8  
Now, putting all the information from (101) to (102) together and dropping the indices, we have  
\[- \frac{1}{2}\frac{D_{i+1/2}\frac{c_{i+1}^{n}-c_{i}^{n}}{h}-D_{i-1/ 2}\frac{c_{i}^{n}-c_{i-1}^{n}}{h}}{h}\]9  
\[- \frac{1}{2}\rho c_{i}^{n+1}\left(1-\frac{c_{i}^{n+1}}{c_{\text{ max}}}\right)-\frac{1}{2}\rho c_{i}^{n}\left(1-\frac{c_{i}^{n}}{c_{\text{max}}} \right),\]0  
Notice that if we take a partial derivative of the original PDE with respect to \(t\), then we obtain  
\[- \frac{1}{2}\rho c_{i}^{n+1}\left(1-\frac{c_{i}^{n+1}}{c_{\text{ max}}}\right)-\frac{1}{2}\rho c_{i}^{n}\left(1-\frac{c_{i}^{n}}{c_{\text{max}}} \right),\]1  
Now, \(P\phi-P_{k,h}\phi=O(k^{2})+O(h^{2})\to 0\) as \(h,k\to 0\) clearly assume that the coefficients and partial derivatives in (103) are bounded.