I have a problem of the following form to be solved numerically. Can someone suggest a numerical scheme that can be used for this problem? $$\frac{∂y}{∂t}= -C \frac{∂}{∂x} \left[\sqrt\frac{∂y}{∂x}\right]$$ where C is a constant [$L^2/T$]
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The equation can be written as $$ \frac{\partial y}{\partial t} = -\frac{C}{2\sqrt{\frac{\partial y}{\partial x}}} \frac{\partial^2 y}{\partial x^2} $$
Using a simple Euler's method for the advance in time and handling explicitly the coefficient in front of the second derivative, you can have a scheme of the form $$ y_i^{(j+1)} =y_i^{(j)}+\delta t \dfrac{C}{2 \sqrt{\dfrac{y_{i+1}^{(j)}-y_{i-1}^{(j)}}{2 \delta x}}} \dfrac{y_{i+1}^{j+1}-2y_i^{(j+1)}+y_{i-1}^{(j+1)}}{\delta x^2}, $$
which amounts to solve a linear system for each time step. If this does not work properly, you can try other variants, namely with respect to the advance in time.
PierreCarre
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Thank you very much! I will try this. A small clarification, why did you chose to handle the term inside the square root explicitly? – Nadi Oct 07 '21 at 22:11
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@Nadi Treating this term explicitly leads to the solution of a linear system in each time step, instead of a nonlinear one. If you want to treat it implicitly, you need to devise some form of fixed point iteration to solve the corresponding nonlinear system. – PierreCarre Oct 07 '21 at 22:14