2014/05/13 by Alexander Zlotnik, Zlotnik, A., Ilya Zlotnik +1
Engineering · Mathematics · #35Q41 #65M12 #65M60 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Numerical Analysis (math.NA) #Numerical methods for differential equations #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.1405.3147
openalex publication_date 2014/05/13 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
We consider the Cauchy problem for the 1D generalized Schrödinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transparent boundary conditions (TBCs) has recently been constructed. Now we engage the Richardson extrapolation to improve significantly the accuracy in time step. To study its properties, we give results of numerical experiments and enlarged practical error analysis for three typical examples. The resulting method is able to provide high precision results in the uniform norm for reasonable computational costs that is unreachable by more common 2nd order methods in either space or time step. Comparing our results to the previous ones, we obtain much more accurate results using much less amount of both elements and time steps.