Analysis of a fast method for solving the high frequency Helmholtz equation in one dimension
2011 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, no 3, 721-755 p.Article in journal (Refereed) Published
We propose and analyze a fast method for computing the solution of the high frequency Helmholtz equation in a bounded one-dimensional domain with a variable wave speed function. The method is based on wave splitting. The Helmholtz equation is split into one-way wave equations with source functions which are solved iteratively for a given tolerance. The source functions depend on the wave speed function and on the solutions of the one-way wave equations from the previous iteration. The solution of the Helmholtz equation is then approximated by the sum of the one-way solutions at every iteration. To improve the computational cost, the source functions are thresholded and in the domain where they are equal to zero, the one-way wave equations are solved with geometrical optics with a computational cost independent of the frequency. Elsewhere, the equations are fully resolved with a Runge-Kutta method. We have been able to show rigorously in one dimension that the algorithm is convergent and that for fixed accuracy, the computational cost is asymptotically just for a pth order Runge-Kutta method, where omega is the frequency. Numerical experiments indicate that the growth rate of the computational cost is much slower than a direct method and can be close to the asymptotic rate.
Place, publisher, year, edition, pages
2011. Vol. 51, no 3, 721-755 p.
Helmholtz equation, High frequency, Wave splitting
IdentifiersURN: urn:nbn:se:kth:diva-40647DOI: 10.1007/s10543-011-0315-7ISI: 000294463100013ScopusID: 2-s2.0-80052296077OAI: oai:DiVA.org:kth-40647DiVA: diva2:443874
FunderSwedish e‐Science Research Center
QC 201109272011-09-272011-09-202012-11-16Bibliographically approved