Improved error bounds of time-splitting Hermite spectral method for the highly-oscillatory Gross-Pitaevskii equation

We establish improved error bounds for the time-splitting Hermite pseudospectral method applied to the highly-oscillatory Gross-Pitaevskii equation on unbounded domains, which corresponds to the strongly confined Gross-Pitaevskii equation. By combining refined Hermite spectral approximation estimates with an extension of the regularity compensation oscillation technique to unbounded domains, we establish an improved error bound of O ( N d 6 − m 2 + ε 2 τ 2 ) O(Nd/6-m/2+ε ^2τ ^2) in L 2 L^2 -norm up to the long time O ( 1 / ε 2 ) O(1/ε ^2) . Here, 0 > ε ≪ 1 0>ε 1 is a dimensionless parameter, N N denotes the spatial parameter, τ τ is the time step size, d d is the dimension and m m is an integer depending on the spatial regularity of the exact solution. Compared to prior results O ( N 1 + d 4 − m 2 + τ 2 ) O(N1+d/4-m/2+τ ^2) in the work of Shen and Wang [Found. Comput. Math. 13 (2013), pp. 99–137], our work demonstrates improved temporal estimates and sharper spatial estimates. Extensions to higher order time-s