Non-reflecting boundary conditions
Journal of Computational Physics
Numerical simulation of gravity waves
Journal of Computational Physics
On nonreflecting boundary conditions
Journal of Computational Physics
Traveling water waves: spectral continuation methods with parallel implementation
Journal of Computational Physics
High-order boundary perturbation methods
Mathematical modeling in optical science
A Stable High-Order Method for Two-Dimensional Bounded-Obstacle Scattering
SIAM Journal on Scientific Computing
Error analysis and preconditioning for an enhanced DtN-FE algorithm for exterior scattering problems
Journal of Computational and Applied Mathematics
A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering
Journal of Computational Physics
Efficient stochastic Galerkin methods for random diffusion equations
Journal of Computational Physics
A rapid boundary perturbation algorithm for scattering by families of rough surfaces
Journal of Computational Physics
Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements
Journal of Scientific Computing
A Rigorous Numerical Analysis of the Transformed Field Expansion Method
SIAM Journal on Numerical Analysis
Exact non-reflecting boundary conditions
Journal of Computational Physics
A perfectly matched layer for the absorption of electromagnetic waves
Journal of Computational Physics
Journal of Scientific Computing
Hi-index | 31.45 |
The Method of Transformed Field Expansions (TFE) has been demonstrated to be a robust and highly accurate numerical scheme for simulating solutions of boundary value and free boundary problems from the sciences and engineering. As a Boundary Perturbation Method it builds highly accurate solutions based upon exact solutions in a simple, canonical, geometry and corrects these via Taylor series to fit the actual geometry at hand. The TFE method has significantly enhanced stability properties when compared with other Boundary Perturbation approaches, however, this comes at the cost of requiring a full volumetric discretization as opposed the surface formulation that other methods can realize. In this paper we outline two techniques for ameliorating this shortcoming, first by employing a Legendre Spectral Element Method to implement efficient, graded meshes, and second by utilizing an Artificial Boundary with a Transparent Boundary Condition placed quite close to the boundary of the domain. In this contribution we focus on the specific problem of simulating the Dirichlet-Neumann operator associated to Laplace's equation on a periodic cell (which arises in the water wave problem). While the details of our results are specific to this problem, the general conclusions are valid for the wider class of problems to which the TFE method can be applied. For each technique we discuss implementation details and display numerical results which support the conclusion that each of these techniques can greatly reduce the computational cost of using the TFE method.