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In this paper, we propose a wavelet collocation splitting (WCS) method, and a Fourier pseudospectral splitting (FPSS) method as comparison, for solving one-dimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics. The two methods can preserve the intrinsic properties of original problems as much as possible. The splitting technique increases the computational efficiency. Meanwhile, the error estimation and some conservative properties are investigated. It is proved to preserve the charge conservation exactly. The global energy and momentum conservation laws can be preserved under several conditions. Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

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In this paper, we consider the numerical methods preserving single or multiple conserved quantities, and these methods are able to reach high order of strong convergence simultaneously based on some kinds of projection methods. The mean-square convergence orders of these methods under certain conditions are given, which can reach order 1.5 or even 2 according to the supporting methods embedded in the projection step. Finally, three numerical experiments are taken into account to show the superiority of the projection methods.

In this paper, we construct stochastic symplectic Runge-Kutta (SSRK) methods of high strong order for Hamiltonian systems with additive noise. By means of colored rooted tree theory, we combine conditions of mean-square order 1.5 and symplectic conditions to get totally derivative-free schemes. We also achieve mean-square order 2.0 symplectic schemes for a class of second-order Hamiltonian systems with additive noise by similar analysis. Finally, linear and non-linear systems are solved numerically, which verifies the theoretical analysis on convergence order. Especially for the stochastic harmonic oscillator with additive noise, the linear growth property can be preserved exactly over long-time simulation.