High-order/spectral methods for transient wave equations

In this thesis, several novel numerical methods are developed to solve two important systems, Maxwell's equations and the acoustic equations. These methods belong to two categories: finite-difference time-domain (FDTD) based methods and spectral time-domain based methods. They are shown by numerical experiments to have some distinguished features and behave better in most applications than conventional approaches. It is well known that the staircasing approximation in the conventional FDTD method for modeling curved objects is limited in accuracy to the first order. Two new methods, the enlarged cell method (ECM) to model curved conducting objects, and the staggered upwind embedded boundary (SUEB) method to model material interfaces, are put forward to overcome such large approximation errors. The ECM does not need to reduce the time step to ensure numerical stability for small irregular cells near the boundary. Therefore, it is more efficient than the recently-proposed conformal FDTD, and moreover, it incurs a smaller dispersion error. The SUEB method embeds the boundary in a simple staggered Cartesian grid and employs upwind conditions to correctly represent the physical conditions of the interfaces. Both methods only require modifying the finite-difference stencils around the boundaries to preserve the second-order accuracy. As a result, they are much more efficient than the conventional FDTD for curved objects. Recently spectral time-domain methods, such as the well-known discontinuous Galerkin method (DGM), have drawn abundant attention due to their exponential convergence. Unfortunately, the accuracy is usually dominated by geometry representation rather than spectral approximation. For the widely-used linear simplex grids, the first-order approximation to curved geometries leads to at best a second-order solution no matter how accurate the schemes are applied. For this reason we introduced an efficient staggered second-order time-stepping technique instead of high-order Runge-Kutta methods to avoid the waste of time-integration effort. In addition, to achieve higher-order accuracy, the DGM on a more accurate grid, the quadratic (second-order) simplex grid, is developed. To further maximize the advantages of the high-order/spectral methods, a novel framework is developed to combine a series of spectral methods with different orders and elements. Such a hybrid spectral method is optimal for complex problems.