Question The Neumann Boundary Value in PDE

[tony21248772]

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[lballabio]

The Neumann condition fixes the value $V$ of the first derivative on the boundary. On a finite-difference grid, the derivative at the (e.g., left) boundary is the one-sided numerical derivative $\displaystyle\frac{u_1 - u_0}{h}$ where $h$ is the distance between the corresponding nodes on the grid. So the Neumann condition can be written as

$$\frac{u_1 - u_0}{h} = V$$

and in order to enforce it, in applyAfterApplying we modify the value of $u_0$ by setting it as

$$u_0 = u_1 - h \times V$$

which is equivalent to the above. Now, the code in applyAfterApplying seems to be missing the $h$, but that's because the documentation of the NeumannBC class says:

//! Neumann boundary condition (i.e., constant derivative)
/*! \warning The value passed must not be the value of the derivative.
             Instead, it must be comprehensive of the grid step
             between the first two points--i.e., it must be the
             difference between f[0] and f[1].

that is, the variable value_ stored in the class corresponds to $h \times V$.

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