A discontinuous Galerkin method for nonlinear biharmonic Schrödinger equations

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.