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Numerically solving the Advection-diffusion equation with no-flux boundary condition leads to violation of mass conservation

For $\alpha, \beta = \mathrm{const}$ you can rewrite your equation $$ \frac{\partial C}{\partial t}=\alpha\frac{\partial^2 C}{\partial x^2}+\beta \frac{\partial C}{\partial x}, \qquad (1), $$ to $$ \...
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