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Linearized numerical stability bounds for solving the nonlineartime-dependent Schr\"odinger equation (NLSE) using explicit finite-differencingare shown. The bounds are computed for the fourth-order Runge-Kutta scheme intime and both second-order and fourth-order central differencing in space.Results are given for Dirichlet, modulus-squared Dirichlet, Laplacian-zero, andperiodic boundary conditions for one, two, and three dimensions. Our approachis to use standard Runge-Kutta linear stability theory, treating thenonlinearity of the NLSE as a constant. The required bounds on the eigenvaluesof the scheme matrices are found analytically when possible, and otherwiseestimated using the Gershgorin circle theorem.

Numerical Stability of Explicit Runge-Kutta Finite-Difference Schemes for the Nonlinear Schr\"odinger Equation