A discontinuous Galerkin method for nonlinear biharmonic Schr\"{o}dinger equations

TL;DR

This paper introduces a fully discrete ultra-weak discontinuous Galerkin scheme combined with Crank--Nicolson time discretization for nonlinear biharmonic Schr"odinger equations, emphasizing efficiency, stability, and conservation properties.

Contribution

It develops a more efficient, unconditionally stable DG scheme that preserves key physical invariants and provides optimal error estimates for nonlinear biharmonic Schr"odinger equations.

Findings

The scheme is unconditionally stable without penalty terms.

It preserves mass and Hamiltonian conservation.

Numerical results confirm theoretical error estimates.

Abstract

This paper proposes and analyzes a fully discrete scheme that discretizes space with an ultra-weak local discontinuous Galerkin scheme and time with the Crank--Nicolson method for the nonlinear biharmonic Schr\"odinger equation. We first rewrite the problem into a system with a second-order spatial derivative and then apply the ultra-weak discontinuous Galerkin method to the system. The proposed scheme is more computationally efficient compared with the local discontinuous Galerkin method because of fewer auxiliary variables, and unconditionally stable without any penalty terms; it also preserves the mass and Hamiltonian conservation that are important properties of the nonlinear biharmonic Schr\"odinger equation. We also derive optimal L2-error estimates of the semi-discrete scheme that measure both the solution and the auxiliary variable with general nonlinear terms. Several numerical studies demonstrate and support our theoretical findings.