I am applying a 4th order Runge-Kutta code, using the method of lines, to solve the following:\begin{equation} \frac {\partial y_1}{\partial t} = y_2 y_3  - C_1 y_1 \end{equation}\begin{equation} \frac {\partial y_2}{\partial t} = y_3y_1 - C_2 y_2 \end{equation}\begin{equation} \frac {\partial y_3}{\partial z} = y_4 \end{equation}\begin{equation} \frac {\partial y_4}{\partial z} = \frac {\partial ^2 y_2}{\partial t^2} + \frac {\partial ^2 y_3}{\partial t^2} + y_1y_3 - y_2y_3 + y_3^2y_2 - y_4\end{equation}(where $ y_n$ are the solutions I am looking for, $ C_n $ are real constants)My code appears to work and solve the system when I just remove the $ y_4 $ term from equation (4), but otherwise, the solution goes to infinity. I have tried rescaling parameters/variables, decreasing the spatial intervals, and changing the boundary/initial conditions (except trivial ones), but these seem not to work. For testing purposes, I also changed equation (4) to:$$ \frac {\partial y_4}{\partial z} = \frac {\partial ^2 y_2}{\partial t^2} + \frac {\partial ^2 y_3}{\partial t^2} + y_1y_3 - y_2y_3 + y_3^2y_2 - \text {SCALE}*y_4 $$where SCALE = $10^{-5}$. With this change, the output did not go to infinity, although it had an unexpected output (namely there were frequent large oscillations). I could not change SCALE to values larger than $10^{-5}$, otherwise the output would go to infinity.RK4 seems to work well for almost everything except this one term in the 4th equation, and I was just wondering if you had any ideas? I was considering possibly applying an implicit method on only the 3rd and 4th equations, although due to the nonlinearity, I haven't been able to isolate for the $y_3$ when using higher order BDF methods.
Applying Runge-Kutta to nonlinear system of PDEs
pde;nonlinear equations;runge kutta;stiffness