Implicit DG Method for Time Domain Maxwell's Equations Involving Metamaterials (Advances in Applied Mathematics and Mechanics, 2021)
An implicit discontinuous Galerkin method is introduced to solve the time-domainMaxwell's equations in metamaterials. The Maxwell's equations in metamaterialsare represented by integral-differential equations. Our scheme is based on discontinuousGalerkin method in spatial domain and Crank-Nicolson method in temporaldomain. The fully discrete numerical scheme is proved to be unconditionally stable.When polynomial of degree at most $p$ is used for spatial approximation, our scheme isverified to converge at a rate of $\mathcal{O}(τ^2+h^{p+1/2})$. Numerical results in both 2D and 3Dare provided to validate our theoretical prediction.