×

Efficient and accurate algorithm for the full modal Green’s kernel of the scalar wave equation in helioseismology. (English) Zbl 1457.85005

Summary: In this work, we provide an algorithm to compute efficiently and accurately the full outgoing modal Green’s kernel for the scalar wave equation in local helioseismology under spherical symmetry. Due to the high computational cost of a full Green’s function, current helioseismic studies rely on single-source computations. However, a more realistic modelization of the helioseismic products (cross-covariance and power spectrum) requires the full Green’s kernel. In the classical approach, the Dirac source is discretized and one simulation gives the Green’s function on a line. Here, we propose a two-step algorithm which, with two simulations, provides the full kernel on the domain. Moreover, our method is more accurate, as the singularity of the solution due to the Dirac source is described exactly. In addition, it is coupled with the exact Dirichlet-to-Neumann boundary condition, providing optimal accuracy in approximating the outgoing Green’s kernel, which we demonstrate in our experiments. In addition, we show that high-frequency approximations of the nonlocal radiation boundary conditions can represent accurately the helioseismic products.

MSC:

85A15 Galactic and stellar structure
85A25 Radiative transfer in astronomy and astrophysics
85-10 Mathematical modeling or simulation for problems pertaining to astronomy and astrophysics
85-08 Computational methods for problems pertaining to astronomy and astrophysics
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
35J10 Schrödinger operator, Schrödinger equation
35L05 Wave equation
35A08 Fundamental solutions to PDEs

Software:

MUMPS; Arb
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. D. Agaltsov, T. Hohage, and R. G. Novikov, Global uniqueness in a passive inverse problem of helioseismology, Inverse Problems, 36 (2020), 055004. · Zbl 1471.35101
[2] S. Agmon, M. Klein, et al., Analyticity properties in scattering and spectral theory for Schrödinger operators with long-range radial potentials, Duke Math. J., 68 (1992), pp. 337-399. · Zbl 0818.34047
[3] P. R. Amestoy, I. S. Duff, J.-Y. L’Excellent, and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 15-41, https://doi.org/10.1137/S0895479899358194. · Zbl 0992.65018
[4] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749-1779, https://doi.org/10.1137/S0036142901384162. · Zbl 1008.65080
[5] H. Barucq, J. Chabassier, M. Duruflé, L. Gizon, and M. Leguèbe, Atmospheric radiation boundary conditions for the Helmholtz equation, ESAIM Math. Model. Numer. Anal., 52 (2018), pp. 945-964. · Zbl 1403.85001
[6] H. Barucq, F. Faucher, D. Fournier, L. Gizon, and H. Pham, Efficient Computation of the Modal Outgoing Green’s Kernel for the Scalar Wave Equation in Helioseismology, Research Report RR-9338, Inria Bordeaux Sud-Ouest; Magique 3D; Max-Planck Institute for Solar System Research, April 2020, https://hal.archives-ouvertes.fr/hal-02544701.
[7] H. Barucq, F. Faucher, and H. Pham, Outgoing Solutions to the Scalar Wave Equation in Helioseismology, Research Report RR-9280, Inria Bordeaux Sud-Ouest; Project-Team Magique3D, August 2019, https://hal.archives-ouvertes.fr/hal-02168467. · Zbl 1440.85003
[8] H. Barucq, F. Faucher, and H. Pham, Outgoing solutions and radiation boundary conditions for the ideal atmospheric scalar wave equation in helioseismology, ESAIM Math. Model. Numer. Anal., 54 (2020), pp. 1111-1138. · Zbl 1440.85003
[9] P. G. Bergmann, The wave equation in a medium with a variable index of refraction, J. Acoust. Soc. Am., 17 (1946), pp. 329-333.
[10] M. Bonnasse-Gahot, H. Calandra, J. Diaz, and S. Lanteri, Hybridizable discontinuous Galerkin method for the 2-d frequency-domain elastic wave equations, Geophys. J. Int., 213 (2017), pp. 637-659.
[11] J. Chabassier and M. Durufle, High Order Finite Element Method for solving Convected Helmholtz Equation in Radial and Axisymmetric Domains. Application to Helioseismology, Research Report RR-8893, Inria Bordeaux Sud-Ouest, Mar. 2016, https://hal.inria.fr/hal-01295077.
[12] J. Christensen-Dalsgaard, W. Däppen, S. Ajukov, E. Anderson, H. Antia, S. Basu, V. Baturin, G. Berthomieu, B. Chaboyer, S. Chitre, et al., The current state of solar modeling, Science, 272 (1996), pp. 1286-1292.
[13] B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), pp. 1319-1365, https://doi.org/10.1137/070706616. · Zbl 1205.65312
[14] E. A. Coddington, An Introduction to Ordinary Differential Equations, Dover, New York, 1961. · Zbl 0123.27301
[15] F. Faucher and O. Scherzer, Adjoint-state method for Hybridizable Discontinuous Galerkin discretization: Application to the inverse acoustic wave problem, Comput. Methods Appl. Mech. Engrg., 372 (2020), 113406, https://doi.org/10.1016/j.cma.2020.113406. · Zbl 1506.65193
[16] D. Fournier, M. Leguèbe, C. S. Hanson, L. Gizon, H. Barucq, J. Chabassier, and M. Duruflé, Atmospheric-radiation boundary conditions for high-frequency waves in time-distance helioseismology, Astron. Astrophys., 608 (2017), A109. · Zbl 1403.85001
[17] L. Gizon, H. Barucq, M. Duruflé, C. S. Hanson, M. Leguèbe, A. C. Birch, J. Chabassier, D. Fournier, T. Hohage, and E. Papini, Computational helioseismology in the frequency domain: Acoustic waves in axisymmetric solar models with flows, Astron. Astrophys., 600 (2017), A35.
[18] L. Gizon and A. Birch, Time-distance helioseismology: The forward problem for random distributed sources, Astrophys. J., 571 (2002), pp. 966-986.
[19] L. Gizon, R. H. Cameron, M. Pourabdian, Z.-C. Liang, D. Fournier, A. C. Birch, and C. S. Hanson, Meridional flow in the Sun’s convection zone is a single cell in each hemisphere, Science, 368 (2020), pp. 1469-1472.
[20] R. Griesmaier and P. Monk, Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Sci. Comput., 49 (2011), pp. 291-310. · Zbl 1246.65211
[21] F. Johansson, Arb: Efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput., 66 (2017), pp. 1281-1292. · Zbl 1388.65037
[22] R. M. Kirby, S. J. Sherwin, and B. Cockburn, To CG or to HDG: A comparative study, J. Sci. Comput., 51 (2012), pp. 183-212. · Zbl 1244.65174
[23] D. Lynden-Bell and J. Ostriker, On the stability of differentially rotating bodies, Mon. Not. R. Astron. Soc., 136 (1967), pp. 293-310. · Zbl 0168.23205
[24] P. Martin, Acoustic scattering by inhomogeneous spheres, J. Acoust. Soc. Am., 111 (2002), pp. 2013-2018.
[25] P. A. Martin, Acoustic scattering by inhomogeneous obstacles, SIAM J. Appl. Math., 64 (2003), pp. 297-308, http://doi.org/10.1137/S0036139902414379. · Zbl 1063.76091
[26] K. Nagashima, B. Löptien, L. Gizon, A. C. Birch, R. Cameron, S. Couvidat, S. Danilovic, B. Fleck, and R. Stein, Interpreting the helioseismic and magnetic imager (HMI) multi-height velocity measurements, Sol. Phys., 289 (2014), pp. 3457-3481.
[27] R. Snieder, M. Miyazawa, E. Slob, I. Vasconcelos, and K. Wapenaar, A comparison of strategies for seismic interferometry, Surv. Geophys., 30 (2009), pp. 503-523.
[28] J. E. Vernazza, E. H. Avrett, and R. Loeser, Structure of the solar chromosphere. III-Models of the EUV brightness components of the quiet-sun, Astrophys. J. Suppl. Ser., 45 (1981), pp. 635-725.
[29] D. Yang, Modeling Experiments in Helioseismic Holography, Ph.D. thesis, The Georg-August-Universität Göttingen, Göttingen, Germany, 2018.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.