I am a bit a newbie when to PDEs and especially the numerical solution of such. I am looking for some advice on how to integrate an equation which has derivative terms, integral terms and some external input, which depends on time.
The equation is the following:
$\frac{\partial u(x,t)}{\partial t} = - k \frac{\partial u(x,t)}{\partial x} + \gamma \frac{\partial^2 u(x,t)}{\partial^2 x} + u(x,t)[x - \hat\lambda ]\hat\lambda^{-1}(dN(t) - \hat\lambda)$
$x\in (X_{inf},X_{sup}), \quad u(x,0) = u_0, \quad u(x_{inf},t) = u(x_{sup},t) = 0$
As you may notice the first two terms are drift and diffusion typical of a Fokker planck.
Note:
- $dN(t)= 1, \forall t \in T=(t_1, t_2, \ldots , t_p ), \quad 0 \text{ otherwise}$
- Furthermore, note that $t_i - t_j\leq 10^{-6}$ for some $i,j$ (which imposes the constraint of having to discretise time with a time step of at least $\Delta t\leq10^{-6})$
- $\hat\lambda(x) = \int x u(x,t) dx$.
I have the approached the problem by writing an explicit solver in Matlab. However, as it is well known explicit solvers suffer of numerical instability, and the parameters $\gamma, k$, as well as $X_{inf},X_{sup}$ and $\Delta x$ the space discretisation step) which I use -and cannot modify- lead to numerical instability. To be sure I have verified the Courant conditions (which in the considered case depends also on the size of the grid $X: x \in (X_{inf}, X_{sup})$) for the Fokker planch terms and they are not full filled.
As the integration of this equation is not the core of my research I would not really want to spend too much time in writing an implicit solver to tackle the issue, unless it was the only option.
Is ther any other pre written solver or alternative approach I may use?
As for now I have tried:
- The matlab standard solver which does not allow integral terms
- Chebfun which does not seem to do be able to read the data from $dN(t)$
Thank very much in advance!