Time Integration

I’m trying to come up with a time-integration procedure for the following system -

\frac{\partial c}{\partial t} = D_c \nabla^2 c -\beta c + k\delta(\mathbf{x} - \mathbf{x}_a(t) ) = f(c,\nabla^2c,\mathbf{x}_a) \tag{1}
\frac{d\phi}{dt} = -(\kappa/\gamma_R)\begin{pmatrix}-\sin \phi \\ \cos\phi\end{pmatrix}\cdot\nabla = g(\phi,\nabla c)\tag{2}
\frac{d\mathbf{x}_a(t)}{dt} = v\begin{pmatrix} \cos\phi \\ \sin\phi \end{pmatrix} = h(\phi). \tag{3}

where \mathbf{x}_a(t) is the position of some particle, c is an ambient field and D_c, \beta, k, \kappa, \gamma_R and v are all positive constants.

The time-integration is severely-complicated by the presence of the \delta and the nonlinearity in (2). One idea is to use the following pseudo-implicit Euler scheme:

\frac{c^{n+1} - c^n}{dt} = f(c^{n+1},\nabla^2c^{n+1},\mathbf{x}_a^n) \\ \frac{\phi^{n+1}-\phi^n}{dt} = g(\phi^{n+1},\nabla c^{n+1}) \\ \frac{\mathbf{x}_a^{n+1} - \mathbf{x}_a^{n}}{dt} = h(\phi^{n+1})

however, the fact that I’m using \mathbf{x}_a^n instead of \mathbf{x}_a^{n+1} is a bit suspect and I’m not sure I could guarantee unconditional stability.

Does anyone have any ideas?

That looks like a semi implicit scheme, have you done a literature review on these sort of schemes for your type of problem? Also, your question doesn’t seem to be related to FEniCS itself. The community is happy to help, but we do not replace talking to your advisor and asking them for guidance.

I have looked in the literature - but I can’t find anything about semi-implicit schemes for coupled PDEs/ PDE-ODE systems. I’m currently in a chemistry group so my advisor isn’t much help - I thought it couldn’t hurt posting the question here.

Anyway, thank you. The question is rather misplaced so I might just go ahead and delete it.