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

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

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

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