I have a particular nonlinear PDE of the form
$u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t))$
Where f is some nonlinear function. With boundary/initial conditions
$u(0,t)=u(L,t)=0, u(x,0)\approx0$
Where L is the length of the region considered. The equation is homogeneous so the initial conditions cannot be zero everywhere. Thus the interior points need only be close to zero, just not exactly zero.
My first step to use finite difference on the derivative
$u_{xx}(x_l)=\frac{1}{\Delta x^2}(u_{l+1}-2u_l+u_{l-1})$
Giving now a system of stiff nonlinear ODE
$\textbf u_t=iD\textbf u+\textbf f(x_l,\textbf u)$
Where D is is a matrix with -2 on the diagonal, 1 on the first off diagonals, zeros everywhere else, and all divided by the spatial step size squared.
I'm interested in using the HHT-$\alpha$ integrator, but I don't understand how to apply it to my system. Specifically I was interested in using this paper (https://link.springer.com/article/10.1007/s12206-019-0208-2). Could anyone explain it to me? How do I find the M and Q matricies? They have several distinct jacobians, which one do I find if I take the jacobian of the right hand side of my equation? How do I find the other jacobian(s)?
When they perform the stability analysis on the simple equation $u_t=\lambda u$ I don't see how they even obtain the resultant equations. Considering this integrator was designed for rigid body mechanics I'm having difficulty translating the method to a more generic system of ODE.
Does anyone know of some other integrators that might be useful? Take note that I've tried rosenbrock and exponential integrators with limited success.
Some properties that might be useful to know: The real and imaginary solutions constantly oscillate from positive to negative in time. It can be shown there is no steady state solution for $u$. However, the term $|u|^2$ does reach a steady state.