The high-order Hermite discrete correction function method for surface-driven electromagnetic problems

The Hermite-Taylor method evolves all the variables and their derivatives through order $m$ in time to achieve a $2m+1$ order rate of convergence. The data required at each node of the staggered Cartesian meshes used by this method makes the enforcement of boundary and interface conditions challenging. In this work, we propose a novel correction function method, referred to as the discrete correction function method, which provides all the data required by the Hermite method near the surface where a condition is enforced. The flexibility of the resulting Hermite-Taylor discrete correction function method is demonstrated by considering a wide range of problems, including those with variable coefficients, discontinuous solutions at the interface, and generalized sheet transition conditions. Although the focus of this work is on Maxwell's equations, this high-order method can be adapted to other linear wave systems. Several numerical examples in two space dimensions are performed to verify the properties of the proposed method, including long-time simulations.