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We develop a high accuracy power series method for solving partialdifferential equations with emphasis on the nonlinear Schr\"odinger equations.The accuracy and computing speed can be systematically and arbitrarilyincreased to orders of magnitude larger than those of other methods. Machineprecision accuracy can be easily reached and sustained for long evolution timeswithin rather short computing time. In-depth analysis and characterisation forall sources of error are performed by comparing the numerical solutions withthe exact analytical ones. Exact and approximate boundary conditions areconsidered and shown to minimise errors for solutions with finite background.The method is extended to cases with external potentials and coupled nonlinearSchr\"odinger equations.

High accuracy power series method for solving scalar, vector, and inhomogeneous nonlinear Schr\"odinger equations