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Suppose

$u_t+\alpha u_x =\int\limits_{b-h}^{b} u(t,x) dx $

$u=u(x,t), (x,t)\in ([a,b]\times [0,T])$

with initial data $ u(x,0)=u_0(x)$, h is fixed xonstant. I want to solve this equation by using finite volume method (FVM). I can apply FVM for such type of conservation laws with source term like $ u_t+au_x=S(u,x,t)$. But if $S(u,x,t)=\int\limits_{b-h}^{b} u(t,x) dx $ I have no idea how to tackle the right-hand side of the above differential equation. Can you provide me some suggestion.

Sandy
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    In the present case, the source term $S(u,x,t) = \int_{b-h}^b u(t,x), dx$ is actually only dependent on time. Just proceed as usual with FV methods: integrate the equation in time over $[t_1,t_2]$ and in space over $[x_{i-1/2},x_{i+1/2}]$. – EditPiAf Sep 18 '17 at 10:20
  • What you are presenting is actually a balance law, not a conservation law. – Dan Doe Jan 12 '22 at 17:46

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