vb.c

How to solve \psi_{xx} - c\psi = 0, with Baslisk.

The aforementioned equation is a version of the 1D Poisson-Helmholtz (PH) equation:

\displaystyle L(a) = \nabla\cdot (\alpha\nabla a) + \lambda a = b,

and hence we include:

#include "grid/multigrid1D.h"
#include "poisson.h"

Scalar fields to store our solution and the right-hand-side of the full PH equation (b) are initialized.

scalar psi[], s[];

And boundary conditions are set to arive at a non-trivial solution.

psi[left] = dirichlet(1);
psi[right] = neumann(0.);

The equation is solved in the main() function, using 128 cells;

int main() {
  init_grid (128);

We choose c = 2 and solve the equation via the poisson() user interface function.

  const scalar c[] = -2.;
  poisson (psi, s, lambda = c);

Finally, the solution is printed so that it may be plotted:

  foreach()
    printf("%g %g\n", x, psi[]);
}

Resulting in,

set xlabel 'x'
set ylabel '{/Symbol psi}'
set key off
set grid
set size ratio -1
plot 'out' w l lw 3

the result. (script)