A discontinuous Galerkin method for nonlinear biharmonic Schrödinger equations

This paper proposes and analyzes an ultra-weak local discontinuous Galerkin scheme for one-dimensional nonlinear biharmonic Schr\"{o}dinger equations. We develop the paradigm of the local discontinuous Galerkin method by introducing the second-order spatial derivative as an auxiliary variable instead of the conventional first-order derivative. The proposed semi-discrete scheme preserves a few physically relevant properties such as the conservation of mass and the conservation of Hamiltonian accompanied by its stability for the targeted nonlinear biharmonic Schr\"{o}dinger equations. We also derive optimal $L^2$-error estimates of the scheme that measure both the solution and the auxiliary variable. Several numerical studies demonstrate and support our theoretical findings.