  We demonstrate an efficient numerical method for obtaining unique solutions
to the Eilenberger equation for a mesoscopic or nanoscale superconductor. In
particular, we calculate the local density of states of a circular d-wave
island containing a single vortex. The "vortex shadow" effect is found to
strongly depend on the quasiparticle energy in such small systems. We show how
to construct by geometry quasiparticle trajectories confined in a finite-size
system with specular reflections at the boundary, and discuss the stability of
the numerical solutions even in the case of vanishing order parameter as for
nodal quasiparticles in a d-wave superconductor, or for quasiparticles passing
through the vortex center with zero energy.
