Development and theoretical analysis of the energy stable nonsymmetric time-domain schemes for the 2D transverse electric mode of nonlinear Maxwell's equations

In this paper, we focus on the construction and theoretical analysis of the energy-stable nonsymmetric time-domain schemes of the 2D transverse electric mode of Maxwell's equations incorporating the linear Lorentz effect, the third-order nonlinear instantaneous Kerr and delayed Raman effects. The leap-frog scheme for temporal discretization and the nonsymmetric scheme based on the unstaggered grids in space are employed to develop the NSTD schemes, which are of second-order accuracy in time and high-order accuracy in space. Specifically, for the proposed NSTD(2,3) scheme, the energy-stable property and error estimate in the $L_2$ norm are demonstrated. Numerical examples verify the energy-stable property, convergence as well as the efficiency of the developed NSTD schemes.