A Two-Step High-Order Compact Scheme for the Laplacian Operator and its Implementation in an Explicit Method for Integrating the Nonlinear Schr\"odinger Equation

TL;DR

This paper introduces a simple, high-order compact finite-difference scheme for the Laplacian operator, demonstrating its accuracy, efficiency, and ease of implementation in simulating the nonlinear Schrödinger equation.

Contribution

The paper presents a novel two-step high-order compact scheme for the Laplacian, showing its equivalence to non-compact schemes and its practical advantages in NLSE simulations.

Findings

Achieves fourth-order accuracy in NLSE simulations.

Comparable computational cost to non-compact schemes.

Stable and efficient for practical use.

Abstract

We describe and test an easy-to-implement two-step high-order compact (2SHOC) scheme for the Laplacian operator and its implementation into an explicit finite-difference scheme for simulating the nonlinear Schr\"odinger equation (NLSE). Our method relies on a compact `double-differencing' which is shown to be computationally equivalent to standard fourth-order non-compact schemes. Through numerical simulations of the NLSE using fourth-order Runge-Kutta, we confirm that our scheme shows the desired fourth-order accuracy. A computation and storage requirement comparison is made between the 2SHOC scheme and the non-compact equivalent scheme for both the Laplacian operator alone, as well as when implemented in the NLSE simulations. Stability bounds are also shown in order to get maximum efficiency out of the method. We conclude that the modest increase in storage and computation of the 2SHOC schemes are well worth the advantages of having the schemes compact, and their ease of implementation makes their use very useful for practical implementations.