High-frequency asymptotics for the numerical solution of the Helmholtz equation

It is often noted that the Helmholtz equation is extremely difficult to solve, in particular, for high-frequency solutions for heterogeneous media. Since stability for second-order discretization methods requires one to choose at least 10–12 grid points per wavelength, the discrete problem on the possible coarsest mesh is huge. In a realistic simulation, one is required to choose 20–30 points per wavelength to achieve a reasonable accuracy; this problem is hard to solve. This article is concerned with the high-frequency asymptotic decomposition of the wavefield for an efficient and accurate simulation for the high-frequency numerical solution of the Helmholtz equation. It has been numerically verified that the new method is accurate enough even when one chooses 4–5 grid points per wavelength.The work of the first author is supported in part by NSF grants DMS-0107210 and DMS-0312223