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I want to discretize and numerically solve the following PDE: \begin{equation} v(k)\dfrac{\partial f}{\partial x} + E(x)\dfrac{\partial f}{\partial k} = S\{f\} \end{equation} using finite volume (box integration) method. Since the direct discretization results in an oscillatory solution, I have used the upwind discretization scheme. This eliminates the oscillations and the solution satisfies the maximum principle. However, if I have my $x$-grid as different $x_i$ points with $i=1,2,...,N$, the upwind discretization schemes loses the connection between $f(x_1)$ (the boundary value) and $f(x_2)$ because I am always calculating the right-sided difference on the second grid node.

When I try to use the central difference on $x_2$, oscillations and instabilities return to the solution, and when I use the upwind scheme, the boundary points are disconnected from their adjacent nodes.

Any suggestions on how to solve this issue?

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    $\begingroup$ Usually well-proposed boundary condition do not result in such problem. If at x2 you use the right-sided difference, then v(k) must be there negative. Very likely v(k) is negative also at x1, so you should not prescribe a boundary condition there, but solve your PDE at x1. The boundary at x=x1 seems to be of outflow type where there is no need for boundary condition for this type of PDE. If you prescribe your solution at outflow boundary, it is exactly like you say - there is no connection between the boundary value and the inner value. $\endgroup$ – Peter Frolkovič Feb 26 '16 at 17:33

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