Finite Difference Time Domain Methods for the Dirac Equation Coupled with the Chern-Simons Gauge Field

We introduce finite difference time domain (FDTD) methods for solving the Dirac equation coupled with the Chern-Simons gauge field and provide their error estimates. To discretize the spinor equation, we utilize well-known FDTD schemes, including the Crank-Nicolson method, leap-frog method, and semi-implicit methods. On the other hand, to discretize the gauge equations, we mainly employ the Lorenz gauge condition and derive Crank-Nicolson or leap-frog type finite difference methods for the gauge equations. We establish the error estimates for the introduced FDTD methods and prove the second-order accuracy both in space and time. Numerical examples are also presented to validate the second-order convergence and illustrate the dependencies of the numerical solutions on the parameters or gauge conditions.