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I am using a box-scheme for solving partial differential equation. The function is approximated with: $$ f_p=\psi\cdot\left(\theta\cdot f_{j}^{k+1}+(1-\theta)\cdot f_{j}^{k} \right) + (1-\psi)\cdot \left(\theta\cdot f_{j+1}^{k+1}+(1-\theta)\cdot f_{j+1}^{k} \right) $$ Time derivative is approximated with: $$ \psi\frac{f_j^{k+1}-f_j^k}{\Delta t}+(1-\psi)\frac{f_{j+1}^{k+1}-f_{j+1}^k}{\Delta t} $$ Time spatial derivative is approximated with: $$ (1-\theta)\frac{f_{j+1}^{k}-f_j^k}{\Delta x}+\theta\frac{f_{j+1}^{k+1}-f_{j}^{k+1}}{\Delta x} $$

My question is how to approximate with box-scheme following PDE: $$ \frac{\partial f}{\partial t} + \frac{\partial}{\partial x} \left(\sqrt{c- \frac{\partial f}{\partial x}} \right) $$

I am stuck at that moment at which I approximated the outer spatial derivative: $$ \frac{\partial}{\partial x} \left(\sqrt{c- \frac{\partial f}{\partial x}} \right)\approx (1-\theta)\frac{\left(\sqrt{c- \frac{\partial f}{\partial x}} \right)_{j+1}^{k}-\left(\sqrt{c- \frac{\partial f}{\partial x}} \right)_j^k}{\Delta x}+\theta\frac{\left(\sqrt{c- \frac{\partial f}{\partial x}} \right)_{j+1}^{k+1}-\left(\sqrt{c- \frac{\partial f}{\partial x}} \right)_{j}^{k+1}}{\Delta x} $$ Where:

$c$ - constant, $t$ - time, $x$ - spatial variable, $j$ - space node index, $k$ - time node index.

What is the best method to approximate the derivatives inside the square root?

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