This project focus on numerical sound propagation over irregular terrain using a body-fitted grid in cylindrical coordinates, assuming symmetry leading to a 2-dimensional spatial discretization.

The first task is to set up the continuous (mathematical) model for the problem, to analyze stability and well-posedness using the energy method.

The second task is to develop a semi-discrete finite difference approximation of the corresponding continuous problem by combining narrow-stencil summation-by-parts (SBP) operators [1] and the Simultaneous Approximation Term (SAT) method [2] to implement the boundary and interface conditions. It is imperative to use finite difference approximations that do not allow growth in time|a property termed \strict stability". The SBP-SAT method makes it possible to exactly mimic the continuous energy estimate and thus proving strict stability. There is a published bench-mark problem [3], simulating sound propagation over an irregular domain. The final task of this project is to do a simulation of this bench-mark problem using the newly developed finite difference method, and do a comparison of earlier results that utilizes other numerical techniques. Today, there are large deviations between the present results. The strength of the SBP-SAT technique is that we can derive strict stability proofs, and thus prove convergence. Hence, we expect that our method will provide the "correct" answer to this bench-mark problem.

Contact: Ken Mattsson
email: ken.mattsson@it.uu.se

References:
[1] K. Mattsson. Summation by parts operators for finite difference approximations of second-derivatives with variable coffcients. Journal of Scientific Computing, To appear, 2011.
[2] K. Mattsson and M. H. Carpenter. Stable and accurate interpolation operators for high-order multi-block finite-difference methods. SIAM J. Sci Comput., 32(4):2298{2320, 2010.
[3] S. Parakkal, KE. Gilbert, D. Xiao, and HE. Bass. A generalized polar coordinate method for sound propagation over large-scale irregular terrain. J Acoust Soc Am., 128(5):2573{2580, 2010.