Abstract
The use of an adaptive step size selection significantly reduces the
computational effort for the numerical solution of the generalized nonlinear
Schrödinger equation (GNLSE). The most commonly employed adaptive step
size method is based on the estimation of the local error by applying step
size doubling and local extrapolation. While this method works well in combination
with the globally second-order split-step Fourier (SSF) integration scheme,
it can be significantly improved when the highly accurate fourth-order Runge-Kutta
in the Interaction Picture (RK4IP) method is used for integration, which was
recently introduced into the nonlinear optics field. It is demonstrated that
the local error can then be estimated using a conservation quantity error
(CQE) without the necessity of step size doubling. The CQE method for solving
the GNLSE is explained in detail, and in addition the concept is transferred
to the normal nonlinear Schrödinger equation and extended to include
linear loss. The RK4IP-CQE combination proves to be the most efficient algorithm
for the modeling of ultrashort pulse propagation in optical fiber, reducing
the computational effort by up to ${\sim
50}\%$ relative to the local error method.
© 2009 IEEE
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