# Finite-differences overview

At the moment, there are three variations of finite-differences implemented:

• FiniteDifferences, standard, explicit, three-point stencil, uniform node distribution,

• StaggeredFiniteDifferences, explicit, variationally derived three-point stencil, both uniform and non-uniform distributions. The nodes are staggered, such that in the uniform case, they follow the formula

$r_j = (j-1/2)\rho$

where $\rho$ is the step-size. This scheme is useful in polar and spherical coordinates. See

• Schafer, K. J., Gaarde, M. B., Kulander, K. C., Sheehy, B., & DiMauro, L. F. (2000). Calculations of Strong Field Multiphoton Processes in Alkali Metal Atoms. AIP Conference Proceedings, 525(1), 45–58. http://dx.doi.org/10.1063/1.1291925

• Krause, J. L., & Schafer, K. J. (1999). Control of THz Emission from Stark Wave Packets. The Journal of Physical Chemistry A, 103(49), 10118–10125. http://dx.doi.org/10.1021/jp992144

• Koonin, S. E., & Meredith, D. C. (1990). Computational Physics, FORTRAN Version. Reading, Mass: Addison-Wesley.

for the derivation.

• ImplicitFiniteDifferences, also known as Compact Finite-Differences, where differential operators are approximated as

$\vec{f}^{(o)}\approx\mat{M}_o^{-1}\Delta_o\vec{f}$

At present, only three-point stencils and uniform grids are supported.

• Lele, S. K. (1992). Compact Finite Difference Schemes With Spectral-Like Resolution. Journal of Computational Physics, 103(1), 16–42. http://dx.doi.org/10.1016/0021-9991(92)90324-r

• Muller, H. G. (1999). An Efficient Propagation Scheme for the Time-Dependent Schrödinger equation in the Velocity Gauge. Laser Physics, 9(1), 138–148.

## Boundary conditions

At the moment, support for anything but Dirichlet0 boundary conditions is lacking. Since the library was developed mostly with calculation in atomic physics in mind, there is some special support for singular Coulomb potentials built-in, which will remain there until the functionality has been sufficiently generalized.