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A propagation method for the time dependent Schr\"odinger equation wasstudied leading to a general scheme of solving ode type equations. Standardspace discretization of time-dependent pde's usually results in system of ode'sof the form u_t -Gu = s where G is a operator (matrix) and u is atime-dependent solution vector. Highly accurate methods, based on polynomialapproximation of a modified exponential evolution operator, had been developedalready for this type of problems where G is a linear, time independent matrixand s is a constant vector. In this paper we will describe a new algorithm forthe more general case where s is a time-dependent r.h.s vector. An iterativeversion of the new algorithm can be applied to the general case where G dependson t or u. Numerical results for Schr\"odinger equation with time-dependentpotential and to non-linear Schr\"odinger equation will be presented.

New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation