Many problems in science and engineering are modelled by partial differential equations on very large or unbounded spatial domains. For example, in geophysical seismic exploration one naturally deals with a large computational domain, the Earth's underground, and space discretisations have to be truncated in order to solve the problem numerically. Another example is the simulation of radio signals in the atmosphere. Computing accurate solutions to the governing equations, of which the Helmholtz equation is an important prototype, is challenging because artificial boundaries can introduce unwanted reflections and pollute the numerical results.
Mathematicians at The University of Manchester proposed a new construction of absorbing boundary layers for indefinite Helmholtz problems that provably achieves near-optimal absorption with a smallest possible thickness. The reduction in thickness is very important in particular for large-scale 3D simulations, where every additional grid point in the absorber increases the problem dimension by the size of a 2D problem. Near-optimal boundary layers significantly reduce the problem dimension and hence the time to solution, and they are now part of seismic simulations run at Schlumberger. The analytical construction in this paper is also of interest to a variety of other applications, including photonic crystals, biological cell communication, and whole-space problems of energy-driven pattern formation.
- The Helmholtz equation represents a time-independent form of the wave equation and Helmholtz problems typically arise in the context of physical problems related to wave phenomena, such as seismic geophysical exploration or cloaking of physical objects. In order to solve such ‘continuous’ equations on computers they need to be ‘discretised’, which in our case means that their solution is approximated on a finite grid of points.
- This discretisation process leads to a finite (but large) number of linear equations to be solved. The requirements of modern Helmholtz simulations can easily lead to more than 500 million of such equations. Our work aims at keeping this number smallest possible when problems on unbounded or large domains (like the Earth’s atmosphere) are being solved by the commonly used finite difference method, and this in turn is beneficial for applications because computer simulations can produce the required results faster.