Stable Explicit Schemes for Equations of the Schrödinger Type

Abstract

Most conventional explicit finite difference schemes, e.g. Euler’s scheme, for solving the parabolic equation of Schrödinger type $u_t = iu_{xx} $ are unconditionally unstable. This difficulty can be overcome by introducing a dissipative term to the conventional explicit schemes. Based on this approach, we derive a class of new explicit finite difference schemes which are conditionally stable, spans two time levels and are $O(k,h^2 )$ accurate. We also determine the schemes from this class that have the least restrictive stability requirements. It is interesting to note that the analog of the Lax–Wendroff scheme is unstable.