Numerically solving a non-local PDE

[ergospherical]

I have a PDE to solve numerically on the region $x \in [0,1]$ and $t \in (0, \infty)$. It is of the form:$$\frac{\partial f(x,t)}{\partial t} = g(x,t) + \int_0^1 h(x, x') f(x', t) dx'$$The second term is the tricky part. The change in $f(x,t)$ at $x$ depends on the value $f(x',t)$ of every other point $x'$ in the space.

In discretized form, it is something like$$f(x_i,t_{j+1}) = f(x_i,t_j) + \Delta t \left[ g(x_i,t_i) + \sum_{j \neq i} h(x_i, x_j) f(x_j, t_i) \right]$$How can I apply an accurate scheme like Runge-Kutta to this system?

 

[pasmith]

Why do you exclude $j = i$ from the sum? And what are the boundary conditions?

After discretizing in space, you must end up with $$\dot{\mathbf{f}} = M\mathbf{f} + \mathbf{g}$$ where $f_i(t)= f(x_i, t)$ etc. with the matrix $M$ depending on how you do the numerical integration. For the trapezoid rule with $x_n = \frac{n}{N - 1} = n \Delta x$, $$\int_0^1 h(x,x')f(x',t)\,dx' \approx \frac12 \Delta x (h(x,x_0)f_0 + 2h(x,x_1)f_1 + \dots + h(x,x_{N-1})f_{N-1})$$ so that $$M_{ij} = \begin{cases} \frac12 \Delta x h(x_i,x_j) & j = 0, N - 1 \\ \Delta x h(x_i, x_j) & j = 1 , \dots, N - 2 \end{cases}$$ You may need to adjust rows $0$ and $N - 1$ in order to enforce a boundary condition.

 

[ergospherical]

Thanks, I'm making progress...

 

[pasmith]

Another alternative is to use Gauss-Legendre quadrature, $$\int_0^1 h(x,x')f(x',t)\,dx = \frac12\int_{-1}^1 h(x,\tfrac12(z + 1))f(\tfrac12(z + 1),t)\,dz \approx \sum_{j=0}^{N-1} \tfrac12 w_j h(x,\tfrac12(z_j + 1))f_j$$ which is exact for polynomials of order up to $2N - 1$ in $x'$. The $z_i \in [-1,1]$ are then the Gauss-Legendre points with $x_i = \frac12(z_i + 1)$ and $$M_{ij} = \frac12 w_jh(x_i,x_j).$$