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Numerical solution of the time‐dependent Schrödinger equation with a position‐dependent effective mass is challenging to compute due to the presence of the non‐constant effective mass. To tackle the problem we present operator splitting‐based numerical methods. The wavefunction will be propagated either by the Krylov subspace method‐based exponential integration or by an asymptotic Green's function‐based time propagator. For the former, the wavefunction is given by a matrix exponential whose associated matrix–vector product can be approximated by the Krylov subspace method; and for the latter, the wavefunction is propagated by an integral with retarded Green's function that is approximated asymptotically. The methods have complexity O(NlogN)$$ O\left(N\log N\right) $$ per step with appropriate algebraic manipulations and fast Fourier transform, where N$$ N $$ is the number of spatial points. Numerical experiments are presented to demonstrate the accuracy, efficiency, and stability of the methods.

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The distributed Kaczmarz algorithm is an adaptation of the standard Kaczmarz algorithm to the situation in which data is distributed throughout a network represented by a tree. We isolate substructures of the network and study convergence of the distributed Kazmarz algorithm for relatively large relaxation parameters associated to these substructures. If the system is consistent, then the algorithm converges to the solution of minimal norm; however, if the system is inconsistent, then the algorithm converges to an approximated least-squares solution that is dependent on the parameters and the network topology. We show that the relaxation parameters may be larger than the standard upper-bound in literature in this context and provide numerical experiments to support our results.

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