Laurence Halpern

Well-Posedness of one-way wave equations and absorbing boundary conditions

A one-way wave equation is a partial differential which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. The construction of such equations can be reduced to the approximation of the square root of (1-s sup 2) on -1, 1 by a rational function r(s) = p sub m (s)/q sub n(s). Those rational functions r for which the corresponding one-way wave equation is well-posed are characterized both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if r(s) interpolates the square root of (1-s sup 2) at sufficiently many points in (-1,1), then well-posedness is assured. It follows that absorbing boundary conditions based on Pade approximation are well-posed if and only if (m, n) lies in one of two distinct diagonals in the Pade table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived from Chebyshev or least-squares approximation.

Higher order paraxial wave equation approximations in heterogenous media

A new family of paraxial wave equation approximations is derived. These approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa ...

Optimal Schwarz Waveform Relaxation for the One Dimensional Wave Equation

We introduce a nonoverlapping variant of the Schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension. We derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains, independently of the length of the time interval. These optimal transmission conditions are in general nonlocal, but we show that the nonlocality depends on the time interval under consideration, and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed. We show that convergence in two iterations can be achieved independently of the number of subdomains in that case. The algorithm thus scales optimally with the number of subdomains, provided the time windows are chosen appropriately. For continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates. We then introduce a finite volume discretization which permits computations on nonmatching grids, and we prove convergence of the fully discrete Schwarz waveform relaxation algorithm. We finally illustrate our analysis with numerical experiments.

Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems

We study in this paper a new class of waveform relaxation algorithms for large systems of ordinary differential equations arising from discretizations of partial differential equations of advection reaction diffusion type. We show that the transmission conditions between the subsystems have a tremendous influence on the convergence speed of the waveform relaxation algorithms, and we identify transmission conditions with optimal performance. Since these optimal transmission conditions are expensive to use, we introduce a class of local transmission conditions of Robin type, which approximate the optimal ones and can be used at the same cost as the classical transmission conditions. We determine the transmission conditions in this class with the best performance of the associated waveform relaxation algorithm. We show that the new algorithm is well posed and converges much faster than the classical one. We illustrate our analysis with numerical experiments.

A homographic best approximation problem with application to optimized Schwarz waveform relaxation

We present and study a homographic best approximation problem, which arises in the analysis of waveform relaxation algorithms with optimized transmission conditions. Its solution characterizes in each class of transmission conditions the one with the best performance of the associated waveform relaxation algorithm. We present the particular class of first order transmission conditions in detail and show that the new waveform relaxation algorithms are well posed and converge much faster than the classical one: the number of iterations to reach a certain accuracy can be orders of magnitudes smaller. We illustrate our analysis with numerical experiments.

Parabolic wave equation approximations in heterogenous media

The properties of different variants of parabolic approximations of scalar wave equations are analyzed. These equations are of general form which includes those used in seismology, underwater acoustics and other applications. A new version of the parabolic approximation is derived for heterogeneous media. It has optimal properties with respect to wave reflection at material interfaces. The amplitudes of the reflected and transmitted waves depend continuously on the interface. Existence, uniqueness and energy estimates are proved.

Artificial boundary conditions for incompressible viscous flows

Artificial boundary conditions for the linearized incompressible Navier–Stokes equations are designed by approximating the symbol of the transparent operator. The related initial boundary value problems are well posed in the same spaces as the original Cauchy problem. Furthermore, error estimates for small viscosity are proved.

Absorbing boundary conditions for the discretization schemes of the one-dimensional wave equation

When computing a partial differential equation, it is often necessary to introduce artificial boundaries. Here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension, fitting to the discretization scheme and stable. Moreover, we give error estimates on the reflected part.

Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems

Artificial boundary conditions are devised for small incompletely parabolic perturbations of hyperbolic systems, which are local, consistent with the hyperbolic equation, well posed, and produce weak boundary layers. The general strategy is applied to the Navier–Stokes system.

Absorbing Boundary Conditions for the Wave Equation and Parallel Computing

Absorbing boundary conditions have been developed for truncating unbounded computational domains to bounded ones. The authors investigate the use of absorbing boundary conditions in domain decomposition algorithms for the wave equation. Moreover, they optimize the absorbing boundary conditions in order to obtain a good performance of the Schwarz waveform relaxation algorithm. The analysis given is confirmed by numerical experiments.