Split-step methods for the solution of the nonlinear Schro¨dinger equation
SIAM Journal on Numerical Analysis
Compact high-order schemes for the Euler equations
Journal of Scientific Computing
A history of Runge-Kutta methods
Applied Numerical Mathematics - Special issue on selected keynote papers presented at 14th IMACS World Congress, Atlanta, NJ, July 1994
Iterative and Parallel Performance of High-Order Compact Systems
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
Finite Differences And Partial Differential Equations
Finite Differences And Partial Differential Equations
A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrödinger equation
Journal of Computational and Applied Mathematics
The use of compact boundary value method for the solution of two-dimensional Schrödinger equation
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
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 Schrodinger 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 is well worth the advantages of having the schemes compact, and their ease of implementation makes their use very useful for practical implementations.