Nonlinear grid mapping applied to an FDTD-based, multi-center 3D Schrödinger equation solver

We developed a straightforward yet effective method of increasing the accuracy of grid-based partial differential equation (PDE) solvers by condensing computational grid points near centers of interest. We applied this “nonlinear mapping” of grid points to a finite-differenced explicit implementation of a time-dependent Schrödinger equation solver in three dimensions. A particular multi-center mapping was developed for systems with multiple Coulomb potentials, allowing the solver to be used in complex configurations where symmetry cannot be used for simplification. We verified our method by finding the eigenstates and eigenenergies of the hydrogen atom and the hydrogen molecular ion (H2+) and comparing them to known solutions. We demonstrated that our nonlinear mapping scheme – which can be readily added to existing PDE solvers – results in a marked increase in accuracy versus a linear mapping with the same number of (or even much fewer) grid points, thus reducing memory and computational requirements by orders of magnitude.

► Uniform and orthogonal 3D grids are computationally heavy.
► Proposed nonlinear mapping of spatial coordinates concentrate points.
► Simple and easy to implement in existing codes and other PDE solvers.
► Suitable for 3D and many-centers, applied to a FDTD Schrödinger solver.
► H, H2+ FDTD simulations show increased accuracy and reduction in running time.