In the paper i'm currently working with I don't understand the role of the term $C^m$ in the following semi-implicit numerical Euler scheme they use, which consists of following two recurrence relations, where $A^m = A^0 + B^m$ ($A^0$ is a constant) is the value of $A$ at the $m$-th timestep. $z, \eta, \omega, D$ and $\gamma$ are constants, $\Delta t$ is the numerical timestep: \begin{align} \left[ 1 + \omega \Delta t - \frac{\eta D \Delta t}{z} \nabla^2 \right] B^{m+1} &= B^m + \epsilon D \Delta t \rho^m A^m \\ \left[ 1 + C^m - \frac{D \Delta t}{z} \nabla^2 \right] \rho^{m+1} &= \bigg[ 1 + C^m - f(A^m) \bigg] \rho^m - \frac{2 D \Delta t}{z} \frac{\nabla \rho^m \cdot \nabla A^m}{A^m} + \gamma \Delta t \end{align} where $C^m$ is the global maximum of $f(A^m)$, defined as \begin{equation} f(A) = \frac{2 D \Delta t}{z} \left[ \frac{\nabla^2 A}{A} - \frac{(\nabla A)^2}{A^2} \right] + A \Delta t \end{equation} The first equation can actually be easily derived from the following PDE system by using forward Euler method for the non-linear terms and backward Euler for the linear terms of \begin{align} \frac{\partial B}{\partial t} &= \frac{\eta D}{z} \nabla^2 B - \omega B + \epsilon D \rho A \end{align} When I do the same with the PDE of $\rho$, i.e. $$ \frac{\partial \rho}{\partial t} = \frac{D}{z} \nabla \cdot \left[ \nabla \rho - \frac{2 \rho}{A} \nabla A \right] - \rho A + \gamma $$ it leads to the upper time-stepping equation for $\rho$, just without the $C^m$-term on both sides of the equation, i.e. \begin{align} \left[ 1 - \frac{D \Delta t}{z} \nabla^2 \right] \rho^{m+1} &= \bigg[ 1 - f(A^m) \bigg] \rho^m - \frac{2 D \Delta t}{z} \frac{\nabla \rho^m \cdot \nabla A^m}{A^m} + \gamma \Delta t \end{align}

My question: Why is $C^m$ added on both sides of the equation? Which role does $C^m$, the spatial maximum of $f(A^m)$, play? Has it the purpose of stabilizing the dynamics? If so, how does it work?

Does that fact that since $C^m$ is the maximum of $f(A^m)$, $C_{i,j}^m - f_{i,j}(A^m) > 0$ for all grid points $(i,j)$ on the right-hand side of the $rho$ equation, have any importance?

Thanks a lot in advance. I'm happy to provide more information if necessary!


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