The time-dependent Schrödinger equation is discretized in space by a sparse grid pseudo-spectral method. The Strang splitting for the resulting evolutionary problem features first or second order convergence in time, depending on the smoothness of the potential and of the initial data. In contrast to the full grid case, where the frequency domain is the working place, the proof of the sufficien...

We study uniform bounds associated with the Allen–Cahn equation and its numerical discretization schemes. These uniform bounds are different from, and weaker than, the conventional energy dissipation and the maximum principle, but they can be helpful in the analysis of numerical methods. In particular, we show that finite difference spatial discretization, like the original continuum model, sha...

The goal of this paper is to provide an analysis of the “toolkit” method used in the numerical approximation of the time-dependent Schrödinger equation. The “toolkit” method is based on precomputation of elementary propagators and was seen to be very efficient in the optimal control framework. Our analysis shows that this method provides better results than the second order Strang operator spli...

In this paper a technique is suggested to avoid order reduction when using Strang method to integrate nonlinear Schrödinger equation subject to time-dependent Dirichlet boundary conditions. The computational cost of this technique is negligible compared to that of the method itself, at least when the timestepsize is fixed. Moreover, a thorough error analysis is given as well as a modification o...

We present a method for computing the dimension of C 1 piecewise polynomials on a triangulated polygonal domain in the plane. Our results verify a conjecture of Strang in a large number of cases. For fourth{degree piecewise polynomials we deene, inductively, a nodal basis on quite general meshes.