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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.