Atmospheric Radiation Boundary Conditions for the Helmholtz Equation

This work offers some contributions to the numerical study of acoustic waves propagating in the Sun and its atmosphere. The main goal is to provide boundary conditions for outgoing waves in the solar atmosphere where it is assumed that the sound speed is constant and the density decays exponentially with radius. Outgoing waves are governed by a Dirichlet-to-Neumann map which is obtained from the factorization of the Helmholtz equation expressed in spherical coordinates. For the purpose of extending the outgoing wave equation to axisymmetric or 3D cases, different approximations are implemented by using the frequency and/or the angle of incidence as parameters of interest. This results in boundary conditions called Atmospheric Radiation Boundary Conditions (ARBC) which are tested in ideal and realistic configurations. These ARBCs deliver accurate results and reduce the computational burden by a factor of two in helioseismology applications.

[1]  L. Nirenberg,et al.  Lectures on linear partial differential equations , 1973 .

[2]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[3]  M. Gunzburger,et al.  Boundary conditions for the numerical solution of elliptic equations in exterior regions , 1982 .

[4]  Chris S. Hanson,et al.  Atmospheric-radiation boundary conditions for high-frequency waves in time-distance helioseismology , 2017, 1709.02156.

[5]  Marc Duruflé,et al.  High Order Finite Element Method for solving Convected Helmholtz equation in radial and axisymmetric domains. Application to Helioseismology , 2016 .

[6]  R. Djellouli,et al.  Performance assessment of a new class of local absorbing boundary conditions for elliptical- and prolate spheroidal-shaped boundaries , 2009 .

[7]  Xavier Antoine,et al.  Fast approximate computation of a time-harmonic scattered field using the on-surface radiation condition method , 2001 .

[8]  A. Kosovichev.  Advances in Global and Local Helioseismology: An Introductory Review , 2011, 1103.1707.

[9]  S. Schot,et al.  Eighty years of Sommerfeld's radiation condition , 1992 .

[10]  Victor Péron,et al.  Equivalent boundary conditions for acoustic media with exponential densities. Application to the atmosphere in helioseismology , 2019, Appl. Math. Comput..

[11]  S. M. Chitre,et al.  The Current State of Solar Modeling , 1996, Science.

[12]  Rabia Djellouli,et al.  Three-dimensional approximate local DtN boundary conditions for prolate spheroid boundaries , 2010, J. Comput. Appl. Math..

[13]  Xavier Antoine,et al.  Bayliss-Turkel-like radiation conditions on surfaces of arbitrary shape , 1999 .

[14]  Chris S. Hanson,et al.  Computational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models with flows , 2016, 1611.01666.

[15]  L. Gizon,et al.  Constructing and Characterising Solar Structure Models for Computational Helioseismology , 2011, 1105.0219.

[16]  H. Spruit,et al.  Local Helioseismology: Three-Dimensional Imaging of the Solar Interior , 2010, 1001.0930.