Efficient higher-order finite-difference schemes for parabolic models

Implicit finite-difference schemes for use in parabolic equation models are developed. Like the familiar Crank-Nicolson scheme, which has hitherto been used almost exclusively for the solution of these equations, these schemes are unconditionally stable and use a computational molecule of only six points on two ''time'' levels. However, they are accurate to a higher order than the Crank-Nicolson scheme, thus allowing the solution grid to be coarser and the solution time to be (approximately) halved. Examples of computations on constant depth are shown, in which significant reductions in time and grid-point density are achieved, for two different parabolic models. The schemes are then extended to refraction and diffraction, and are shown to have a similar effect in this more general case too. It is recommended that finite-difference schemes based on these higher-order (or Hermitian) methods replace the more commonly used Crank-Nicolson scheme in all physical domain parabolic equation models, but especially in minimax (wide-angle) equation models.