Thomas Hagstrom

High-order Absorbing Boundary Conditions for anisotropic and convective wave equations

High-order Absorbing Boundary Conditions (ABCs), applied on a rectangular artificial computational boundary that truncates an unbounded domain, are constructed for a general two-dimensional linear scalar time-dependent wave equation which represents acoustic wave propagation in anisotropic and subsonically convective media. They are extensions of the construction of Hagstrom, Givoli and Warburton for the isotropic stationary case. These ABCs are local, and involve only low-order derivatives owing to the use of auxiliary variables on the artificial boundary. The accuracy and well-posedness of these ABCs is analyzed. Special attention is given to the issue of mismatch between the directions of phase and group velocities, which is a potential source of concern. Numerical examples for the anisotropic case are presented, using a finite element scheme.

The Double Absorbing Boundary method

A new approach is devised for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. The new method, called the Double Absorbing Boundary (DAB) method, is based on truncating the unbounded domain to produce a finite computational domain @W, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and inside the layer bounded by them, and participate in the numerical scheme. The DAB method is first introduced in general terms, using the 2D scalar time-dependent wave equation as a model. Then it is applied to the 1D Klein-Gordon equation, using finite difference discretization in space and time, and to the 2D wave equation in a wave guide, using finite element discretization in space and dissipative time stepping. The computational aspects of the method are discussed, and numerical experiments demonstrate its performance.

A hybrid Hermite-discontinuous Galerkin method for hyperbolic systems with application to Maxwell's equations

A high order discretization strategy for solving hyperbolic initial-boundary value problems on hybrid structured-unstructured grids is proposed. The method leverages the capabilities of two distinct families of polynomial elements: discontinuous Galerkin discretizations which can be applied on elements of arbitrary shape, and Hermite discretizations which allow highly efficient implementations on staircased Cartesian grids. We demonstrate through numerical experiments in 1+1 and 2+1 dimensions that the hybridized method is stable and efficient.

The solution of the scalar wave equation in the exterior of a sphere

We derive new, explicit representations for the solution to the scalar wave equation in the exterior of a sphere, subject to either Dirichlet or Robin boundary conditions. Our formula leads to a stable and high-order numerical scheme that permits the evaluation of the solution at an arbitrary target, without the use of a spatial grid and without numerical dispersion error. In the process, we correct some errors in the analytic literature concerning the asymptotic behavior of the logarithmic derivative of the spherical modified Hankel function. We illustrate the performance of the method with several numerical examples.

Grid stabilization of high-order one-sided differencing II: Second-order wave equations

We demonstrate the stable boundary closure of difference methods of order up through 16 for the solution of wave equations in second order form. Our method combines the introduction of 1-2 judiciously placed subcell grid points near the boundary with minimal-stencil, one-sided difference operators of the same order as the interior scheme. The method is tested on a variety of problems including the scalar wave equation discretized on mapped grids and overlapping composite grids, as well as an integrable nonlinear system.

Solving PDEs with Hermite Interpolation

We examine the use of Hermite interpolation, that is interpolation using derivative data, in place of Lagrange interpolation to develop high-order PDE solvers. The fundamental properties of Hermite interpolation are recalled, with an emphasis on their smoothing effect and robust performance for nonsmooth functions. Examples from the CHIDES library are presented to illustrate the construction and performance of Hermite methods for basic wave propagation problems.

On Galerkin difference methods

Energy-stable difference methods for hyperbolic initial-boundary value problems are constructed using a Galerkin framework. The underlying basis functions are Lagrange functions associated with continuous piecewise polynomial approximation on a computational grid. Both theoretical and computational evidence shows that the resulting methods possess excellent dispersion properties. In the absence of boundaries the spectral radii of the operators for the first and second derivative matrices are bounded independent of discretization order. With boundaries the spectral radius of the first order derivative matrix appears to be bounded independent of discretization order, and grows only slowly with discretization order for problems in second-order form.

An energy-based discontinuous Galerkin discretization of the elastic wave equation in second order form

Abstract We present an application of our general formulation (Appelo and Hagstrom (2015) [12]) to construct energy based, arbitrary order accurate, discontinuous Galerkin spatial discretizations of the linear elastic wave equation. The resulting methods are stable and, depending on the choice of numerical flux, conserve or dissipate the elastic energy. The performance of the method is demonstrated for problems with manufactured and exact solutions. Applications to more realistic problems are also presented. Implementations of the methods are freely available at Appelo and Hagstrom (2015) [19].

Double absorbing boundaries for finite-difference time-domain electromagnetics

Abstract We describe the implementation of optimal local radiation boundary condition sequences for second order finite difference approximations to Maxwells equations and the scalar wave equation using the double absorbing boundary formulation. Numerical experiments are presented which demonstrate that the design accuracy of the boundary conditions is achieved and, for comparable effort, exceeds that of a convolution perfectly matched layer with reasonably chosen parameters. An advantage of the proposed approach is that parameters can be chosen using an accurate a priori error bound.

Extension of the Lorenz-Mie-Debye method for electromagnetic scattering to the time-domain

In this paper, we extend the frequency domain Lorenz-Mie-Debye formalism for the Maxwell equations to the time-domain. In particular, we show that the problem of scattering from a perfectly conducting sphere can be reduced to the solution of two scalar wave equations - one with Dirichlet boundary conditions and the other with Robin boundary conditions. An explicit, stable, and high-order numerical scheme is then developed, based on our earlier treatment of the scalar case. This new representation may provide some insight into transient electromagnetic phenomena, and can also serve as a reference solution for general purpose time-domain software packages.