An hr-adaptive method for the cubic nonlinear Schrödinger equation

Abstract

The nonlinear Schrödinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients. Adaptive moving mesh methods (r-adaptive) attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features. This approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest. Mesh refinement methods (h-adaptive), where the mesh is locally coarsened or refined, is an alternative adaptive strategy which is popular for time-independent problems. In this paper, we consider the effectiveness of a combined method (hr-adaptive) to solve the NLSE in one space dimension. Simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches. The method is also shown to control the spatial error based on the user's input error tolerance. Evidence is also presented indicating second-order spatial convergence using a novel monitor function to generate the adaptive moving mesh.

Novel energy dissipative method on the adaptive spatial discretization for the Allen–Cahn equation*

“…We also use this method as a reference to prove the theorem. In a recent study, [23] an hr adaptive method is proposed to get the spatial and temporal adaptation effect and then to solve the nonlinear Schrödinger equation (NSE). Nevertheless, the proposed method only involves r-refinement and explores the adaptation in space.…”

hr-adaptivity for nonconforming high-order meshes with the target matrix optimization paradigm

“…In this work, we focus on hr−adaptivity; rp− and hrp−adaptivity will be explored in future papers. A survey of the literature shows that hr−adaptivity methods have been effective for mesh optimization with applications ranging from the Schrödinger equations [15] to ocean modeling [16]. These methods typically employ an error-based function that is minimized by moving the nodal positions (r−adaptivity) followed by refining/derefining each element (h−adaptivity) by elemental operations such as node insertion, node removal, or edge swapping.…”

“…A non-exhaustive list of publications and recent advances on the subject of hr−adaptivity is given by [15,16,17,18,19,20,21,22,23,24]. All of the existing methods hr− methods are either restricted to low-order meshes (first-or second-order), specific element types (simplices or quadrilaterals/hexahedra), or specific h−refinement types (typically isotropic).…”