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 [itex]j = i[/itex] from the sum? And what are the boundary conditions?

After discretizing in space, you must end up with [tex]
\dot{\mathbf{f}} = M\mathbf{f} + \mathbf{g}[/tex] where [itex]f_i(t)= f(x_i, t)[/itex] etc. with the matrix [itex]M[/itex] depending on how you do the numerical integration. For the trapezoid rule with [itex]x_n = \frac{n}{N - 1} = n \Delta x[/itex], [tex]
\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})[/tex] so that [tex]
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}[/tex] You may need to adjust rows [itex]0[/itex] and [itex]N - 1[/itex] in order to enforce a boundary condition.

 

[ergospherical]

Thanks, I'm making progress...

 

[pasmith]

Another alternative is to use Gauss-Legendre quadrature, [tex]
\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[/tex] which is exact for polynomials of order up to [itex]2N - 1[/itex] in [itex]x'[/itex]. The [itex]z_i \in [-1,1][/itex] are then the Gauss-Legendre points with [itex]x_i = \frac12(z_i + 1)[/itex] and [tex]
M_{ij} = \frac12 w_jh(x_i,x_j).[/tex]