Consider a system of PDEs $$ \begin{cases} u_t = \nabla \cdot (D(u)\nabla u) + \frac{c}{K_U+c}u-ku\\ c_t = d_c\Delta c -\frac{\nu_U c}{K_U + c}u \end{cases} $$ with some boundary conditions. Here, $D(u)$ is a diffusion coefficient which depends on $u$; $K_U$, $\nu_U$ and $k$ are some constants. $D(u)$ can be defined as, for example,

$$D(u):=\delta \frac{u^\alpha}{(1-u)^\beta},$$ with $\alpha,\beta,\delta$ being some constants.

After this system of PDEs is discretized using the finite volume method one obtains the following system of ODEs

$$\begin{cases}\frac{d\vec{U}}{dt}=\underline{D(\vec{U})}\vec{U}+\underline{R_U(\vec{C})}\vec{U}\\ \frac{d\vec{C}}{dt}=\underline{L}\vec{C}-\underline{R_C(\vec{C})}\vec{U}+\vec{b} \end{cases}$$

where the underlined letters are matrices and $\vec{b}$ is a vector containing some terms from (unspecified here) the boundary conditions.

As we can see, the matrix $\underline{D(\vec{U})}$ depends on the vector for which a numerical method will solve this system of equations, that is the vector $\vec{U}$.

But then how can such a system be solved linearly if it will contain non-linear terms? I.e. what am I missing?


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