On Optimal Convergence Rates for the Nonlinear Schrödinger Equation with a Wave Operator via Localized Orthogonal Decomposition
This work provides a numerical method for solving a specific nonlinear PDE, but it appears incremental as it applies an existing LOD technique to a new equation variant.
The authors tackled the two-dimensional time-dependent nonlinear Schrödinger equation with a wave operator by developing a Localized Orthogonal Decomposition method, proving it preserves conservation laws and achieves unconditional optimal-order superconvergent L^p error estimates.
In this paper, we develop a Localized Orthogonal Decomposition (LOD) method for the two-dimensional time-dependent nonlinear Schrödinger equation with a wave operator. We prove that our method preserves conservation laws and admits a unique numerical solution; furthermore, we obtain unconditional (i.e., time-step restriction-free) optimal-order superconvergent \(L^p\) error estimates. To complement the theoretical analysis, we present a series of numerical simulations that verify the analytical results and further illustrate structural aspects of the problem.