The delocalization or scrambling of quantum information has emerged as a central ingredient in the understanding thermalization isolated many-body systems. Recently, significant progress been made analytically by modeling non-integrable systems stochastic systems, lacking Hamiltonian picture, while honest dynamics are frequently limited to small system sizes due computational constraints. In th...

In this paper we discuss the issue of conservation and convergence to weak solutions of several global schemes, including the commonly used compact schemes and spectral collocation schemes, for solving hyperbolic conservation laws. It is shown that such schemes, if convergent boundedly ahnost everywhere, will converge to weak solutions. The results are extensions of the classical Lax-Wendroff t...

In this paper, we study the preservation of quadratic conservation laws of Runge-Kutta methods and partitioned Runge-Kutta methods for Hamiltonian PDEs and establish the relation between multi-symplecticity of Runge-Kutta method and its quadratic conservation laws. For Schrödinger equations and Dirac equations, the relation implies that multi-sympletic RungeKutta methods applied to equations wi...

It is emphasized that the collapse postulate of standard quantum theory can violate conservation of energy-momentum and there is no indication from where the energy-momentum comes or to where it goes. Likewise, in the Continuous Spontaneous Localization (CSL) dynamical collapse model, particles gain energy on average. In CSL, the usual Schrödinger dynamics is altered so that a randomly fluctuat...