Skip to main content
SpringerLink
Log in

Wave Equation Solver

  • Chapter
  • First Online:
Imaging Convection and Magnetism in the Sun

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

  • 540 Accesses

Abstract

The fast computation of solutions to the helioseismic wave propagation problem can help propel us closer to the goal of accurately inverting for the sub-surface structure and dynamics of the Sun. It also improves our understanding of wave physics, assists us in interpreting the helioseismic measurement response to the presence of anomalies in the interior and can play a central role in obtaining solutions of the inverse problem (adjoint method; Hanasoge et al 2011; also chapter 4 of this monograph).

The content of this chapter is taken from Hanasoge et al (2006), Hanasoge and Duvall (2007), Hanasoge (2008), Hanasoge et al (2008), Hanasoge et al (2010b). The spherical solver is available on request and the Cartesian code can be downloaded from http://www.tifr.res.in/~hanasoge/sparc.html

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Appelö D, Hagstrom T, Kreiss G (2006) Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability. SIAM J Appl Math 67(1):1–23

    Article  MathSciNet  MATH  Google Scholar 

  • Bécache E, Fauqueux S, Joly P (2003) Stability of perfectly matched layers, group velocities and anisotropic waves. J Comput Phys 188(2):399–433

    Article  MathSciNet  MATH  Google Scholar 

  • Belousov SL (1962) Tables of normalized associated Legendre polynomials, vol 18. Pergamon Press, New York

    MATH  Google Scholar 

  • Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114(2):185–200. DOI 10.1006/jcph.1994.1159. http://www.sciencedirect.com/science/article/B6WHY-45P0TJR-1P/2/9aa5d47e587908f2e75691914c79ac80

  • Bérenger JP (2002) Application of the CFS PML to the absorption of evanescent waves in waveguides. IEEE Microwave Wireless Compon Lett 12(6):218–220. DOI 10.1109/LMWC.2002.1010000

    Article  Google Scholar 

  • Berland J, Bogey C, Bailly C (2006) Low-dissipation and low-dispersion fourth-order Runge–Kutta algorithm. Comput Fluids 35(10):1459

    Article  MathSciNet  MATH  Google Scholar 

  • Birch AC, Braun DC, Hanasoge SM, Cameron R (2009) Surface-focused seismic holography of sunspots: II. Expectations from numerical simulations using sound-speed perturbations. Sol Phys 254:17–27. DOI 10. 1007/s11207-008-9282-9

    Google Scholar 

  • Bogdan TJ, Hindman BW, Cally PS, Charbonneau P (1996) Absorption of p-modes by slender magnetic flux tubes and p-mode lifetimes. Astrophys J 465:406. DOI 10.1086/177429

    Article  Google Scholar 

  • Bray RJ, Loughhead RE (1974) The solar chromosphere. The international astrophysics series. Chapman and Hall, London

    Google Scholar 

  • Cameron R, Gizon L, Daiffallah K (2007) SLiM: a code for the simulation of wave propagation through an inhomogeneous, magnetised solar atmosphere. Astron Nachr 328:313. DOI 10.1002/asna.200610736

    Article  MATH  Google Scholar 

  • Cameron R, Gizon L, Duvall TL Jr (2008) Helioseismology of sunspots: confronting observations with three-dimensional MHD simulations of wave propagation. Sol Phys 251:291–308. DOI 10.1007/s11207-008-9148-1. 0802.1603

    Article  Google Scholar 

  • Cameron R, Gizon L, Schunker H, Pietarila A (2010) Constructing semi-empirical sunspot models for helioseismology. arXiv e-prints 1003.0528

    Google Scholar 

  • Christensen–Dalsgaard J (2003) Lecture notes on stellar oscillations, 5th edn. http://astro.phys.au.dk/~jcd/oscilnotes/

    Google Scholar 

  • Christensen-Dalsgaard J, Dappen W, Ajukov SV, Anderson ER, Antia HM, Basu S, Baturin VA, Berthomieu G, Chaboyer B, Chitre SM, Cox AN, Demarque P, Donatowicz J, Dziembowski WA, Gabriel M, Gough DO, Guenther DB, Guzik JA, Harvey JW, Hill F, Houdek G, Iglesias CA, Kosovichev AG, Leibacher JW, Morel P, Proffitt CR, Provost J, Reiter J, Rhodes EJ Jr, Rogers FJ, Roxburgh IW, Thompson MJ, Ulrich RK (1996) The current state of solar modeling. Science 272:1286

    Article  Google Scholar 

  • Collino F, Monk P (1998) Optimizing the perfectly matched layer. Comput Methods Appl Mech Eng 164:157–171

    Article  MathSciNet  MATH  Google Scholar 

  • Colonius T (2004) Modeling artificial boundary conditions for compressible flow. Annu Rev Fluid Mech 36:315–345

    Article  MathSciNet  MATH  Google Scholar 

  • Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex fourier series. Math Comput 19(90):297–301

    Article  MathSciNet  MATH  Google Scholar 

  • Felipe T, Khomenko E, Collados M (2010) Magneto-acoustic waves in sunspots: first results from a new three-dimensional nonlinear magnetohydrodynamic code. Astrophys J 719:357–377. DOI 10.1088/ 0004-637X/719/1/357. 1006.2998

    Article  Google Scholar 

  • Felipe T, Crouch AD, Birch AC (2014) Evaluation of the capability of local helioseismology to discern between monolithic and spaghetti sunspot models. Astrophys J 788:136. DOI 10.1088/0004-637X/788/2/136. 1405.0036

    Article  Google Scholar 

  • Festa G, Vilotte JP (2005) The Newmark scheme as velocity-stress time-staggering: an efficient PML implementation for spectral-element simulations of elastodynamics. Geophys J Int 161:789–812. DOI 10.1111/ j.1365-246X.2005.02601.x

    Article  Google Scholar 

  • Goedbloed JPH, Poedts S (2004) Principles of magnetohydrodynamics: with applications to laboratory and astrophysical plasmas. Cambridge University Press, Cambridge. DOI 10.2277/0521626072

    Google Scholar 

  • Hanasoge SM (2007) Theoretical studies of wave interactions in the sun. Ph.D. thesis, Stanford University, California, USA

    Google Scholar 

  • Hanasoge SM (2008) Seismic halos around active regions: a magnetohydrodynamic theory. Astrophys J 680:1457–1466. DOI 10.1086/587934. 0712.3578

    Article  Google Scholar 

  • Hanasoge SM (2009) A wave scattering theory of solar seismic power haloes. Astron Astrophys 503:595–599. DOI 10.1051/0004-6361/200912449. 0906.4671

    Article  Google Scholar 

  • Hanasoge SM, Duvall TL Jr (2007) The solar acoustic simulator: % applications and results. Astron Nachr 328:319. DOI 10.1002/asna. 200610737

    Article  MATH  Google Scholar 

  • Hanasoge SM, Duvall TL (2009) Subwavelength resolution imaging of the solar deep interior. Astrophys J 693:1678–1685. DOI 10.1088/0004-637X/ 693/2/1678. 0812.0119

    Article  Google Scholar 

  • Hanasoge SM, Larson TP (2008) Global effects of local sound-speed perturbations in the sun: a theoretical study. Sol Phys 251:91–100. DOI 10. 1007/s11207-008-9208-6. 0711.1877

    Google Scholar 

  • Hanasoge SM, Larsen RM, Duvall TL Jr, DeRosa ML, Hurlburt NE, Schou J, Roth M, Christensen-Dalsgaard J, Lele SK (2006) Computational acoustics in spherical geometry: steps toward validating helioseismology. Astrophys J 648: 1268–1275. DOI 10.1086/505927

    Article  Google Scholar 

  • Hanasoge SM, Duvall TL Jr, Couvidat S (2007) Validation of helioseismology through forward modeling: realization noise subtraction and kernels. Astrophys J 664:1234–1243. DOI 10.1086/519070

    Article  Google Scholar 

  • Hanasoge SM, Couvidat S, Rajaguru SP, Birch AC (2008) Impact of locally suppressed wave sources on helioseismic traveltimes. Mon Not R Astron Soc 391:1931–1939. DOI 10.1111/j.1365-2966.2008.14013.x. 0707.1369

    Article  Google Scholar 

  • Hanasoge SM, Duvall TL, DeRosa ML (2010a) Seismic constraints on interior solar convection. Astrophys J Lett 712:L98–L102. DOI 10.1088/2041-8205/ 712/1/L98. 1001.4508

    Article  Google Scholar 

  • Hanasoge SM, Komatitsch D, Gizon L (2010b) An absorbing boundary formulation for the stratified, linearized, ideal MHD equations based on an unsplit, convolutional perfectly matched layer. Astron Astrophys 522:A87. DOI 10.1051/0004-6361/201014345. 1003.0725

    Article  Google Scholar 

  • Hanasoge SM, Birch A, Gizon L, Tromp J (2011) The adjoint method applied to time-distance helioseismology. Astrophys J 738:100. DOI 10. 1088/0004-637X/738/1/100. 1105.4263

    Google Scholar 

  • Hartlep T, Miesch MS, Mansour NN (2008a) Wave propagation in the magnetic sun. arXiv e-prints 0805.0333

    Google Scholar 

  • Hartlep T, Zhao J, Mansour NN, Kosovichev AG (2008b) Validating time-distance far-side imaging of solar active regions through numerical simulations. Astrophys J 689:1373–1378. DOI 10.1086/592721. 0805.0472

    Article  Google Scholar 

  • Hu FQ (2001) A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables. J Comput Phys 173(2):455–480. DOI 10.1006/jcph.2001.6887. http://www.sciencedirect.com/science/article/B6WHY-45BC24W-1M/2/948d733fe6a2ba168e816194e9b481da

  • Hu FQ, Hussaini MY, Manthey JL (1996) Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics. J Comput Phys 124(1): 177–191. http://www.science-direct.com/science/article/B6WHY-45MGWBD-5S/1/8c70a06b8c29b3e947e1f0d1f2a65fc4

  • Hurlburt NE, Rucklidge AM (2000) Development of structure in pores and sunspots: flows around axisymmetric magnetic flux tubes. Mon Not R Astron Soc 314: 793–806

    Article  Google Scholar 

  • Khomenko E, Collados M (2006) Numerical modeling of magnetohydrodynamic wave propagation and refraction in sunspots. Astrophys J 653:739–755. DOI 10.1086/507760

    Article  Google Scholar 

  • Khomenko E, Collados M (2008) Magnetohydrostatic sunspot models from deep subphotospheric to chromospheric layers. Astrophys J 689:1379–1387. DOI 10.1086/592681. 0808.3571

    Article  Google Scholar 

  • Khomenko E, Collados M (2009) Sunspot seismic halos generated by fast MHD wave refraction. arXiv e-prints 0905.3060

    Google Scholar 

  • Komatitsch D, Martin R (2007) An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation. Geophysics 72(5):SM155–SM167. DOI 10.1190/1.2757586

    Article  Google Scholar 

  • Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16–42

    Article  MathSciNet  MATH  Google Scholar 

  • Leroy B (1985) On the derivation of the energy flux of linear magnetohydrodynamic waves. Geophys Astrophys Fluid Dyn 32:123–133. DOI 10.1080/03091928508208781

    Article  Google Scholar 

  • Lui C (2003) A numerical investigation of shock-associated noise. Ph.D. thesis, Stanford University

    Google Scholar 

  • Lynden-Bell D, Ostriker JP (1967) On the stability of differentially rotating bodies. Mon Not R Astron Soc 136:293

    Article  MATH  Google Scholar 

  • Meza-Fajardo KC, Papageorgiou AS (2008) A nonconvolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: stability analysis. Bull Seismol Soc Am 98(4):1811–1836. http://www.bssaonline.org/cgi/content/abstract/98/4/1811

  • Moradi H, Cally PS (2014) Sensitivity of helioseismic travel times to the imposition of a Lorentz force limiter in computational helioseismology. Astrophys J Lett 782:L26. DOI 10.1088/2041-8205/782/ 2/L26. 1401.5518

    Article  Google Scholar 

  • Moradi H, Hanasoge SM, Cally PS (2009) Numerical models of travel-time inhomogeneities in sunspots. Astrophys J Lett 690:L72–L75. DOI 10.1088/ 0004-637X/690/1/L72. 0808.3628

    Article  Google Scholar 

  • Moradi H, Baldner C, Birch AC, Braun DC, Cameron RH, Duvall TL, Gizon L, Haber D, Hanasoge SM, Hindman BW, Jackiewicz J, Khomenko E, Komm R, Rajaguru P, Rempel M, Roth M, Schlichenmaier R, Schunker H, Spruit HC, Strassmeier KG, Thompson MJ, Zharkov S (2010) Modeling the subsurface structure of sunspots. Sol Phys 267:1–62. DOI 10.1007/ s11207-010-9630-4. 0912.4982

    Article  Google Scholar 

  • Nordlund Ã…, Stein RF, Asplund M (2009) Solar surface convection. Living Rev Sol Phys 6:2

    Article  Google Scholar 

  • Orszag SA (1971) On the elimination of aliasing in finite difference schemes by filtering high-wavenumber components. J Atmos Sci 28:1074

    Article  Google Scholar 

  • Parchevsky KV, Kosovichev AG (2007) Three-dimensional numerical simulations of the acoustic wave field in the upper convection zone of the sun. Astrophys J 666:547–558. DOI 10.1086/520108. arXiv:astro-ph/0612364

    Google Scholar 

  • Parchevsky KV, Zhao J, Kosovichev AG (2008) Influence of nonuniform distribution of acoustic wavefield strength on time-distance helioseismology measurements. Astrophys J 678:1498–1504. DOI 10.1086/533495. arXiv:0802.3866

    Google Scholar 

  • Parker EN (1979) Cosmical magnetic fields: their origin and their activity. Clarendon Press, Oxford; Oxford University Press, New York, 858 p.

    Google Scholar 

  • Rempel M, Schüssler M, Knölker M (2009) Radiative magnetohydrodynamic simulation of sunspot structure. Astrophys J 691:640–649. DOI 10.1088/0004-637X/691/1/640. 0808.3294

    Article  Google Scholar 

  • Roden JA, Gedney SD (2000) Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media. Microw Opt Technol Lett 27(5):334–339

    Article  Google Scholar 

  • Schlüter A, Temesváry S (1958) The internal constitution of sunspots. In: Lehnert B (ed) Electromagnetic phenomena in cosmical physics, IAU symposium, vol 6, p 263

    Google Scholar 

  • Schunker H, Cameron RH, Gizon L, Moradi H (2011) Constructing and characterising solar structure models for computational helioseismology. Sol Phys 271:1–26. DOI 10.1007/s11207-011-9790-x. 1105.0219

    Article  Google Scholar 

  • Shelyag S, Erdélyi R, Thompson MJ (2006) Forward modeling of acoustic wave propagation in the quiet solar subphotosphere. Astrophys J 651:576–583. DOI 10.1086/507463

    Article  Google Scholar 

  • Shelyag S, Zharkov S, Fedun V, Erdélyi R, Thompson MJ (2009) Acoustic wave propagation in the solar sub-photosphere with localised magnetic field concentration: effect of magnetic tension. Astron Astrophys 501:735–743. DOI 10.1051/0004-6361/200911709. 0901.3680

    Article  Google Scholar 

  • Stein RF, Nordlund Ã… (2000) Realistic solar convection simulations. Sol Phys 192:91–108. DOI 10.1023/A:1005260918443

    Article  Google Scholar 

  • Thompson KW (1990) Time-dependent boundary conditions for hyperbolic systems. J Comput Phys 89(2):439–461

    Article  MathSciNet  MATH  Google Scholar 

  • Tóth G (2000) The divb=0 constraint in shock-capturing magnetohydrodynamics codes. J Comput Phys 161(2):605–652 http://www.sciencedirect.com/science/article/B6WHY-45FC8HX-47/2/61db392c9a72d394a62a02958881b046

  • Vichnevetsky R, Bowles JB (1982) Fourier analysis of numerical approximations of hyperbolic equations, vol 5. Society for Industrial and Applied Mathematics, Philadelphia

    Book  MATH  Google Scholar 

  • Winton SC, Rappaport CM (2000) Specifying PML conductivities by considering numerical reflection dependencies. IEEE Trans Antennas Propag 48(7): 1055–1063

    Article  Google Scholar 

  • Zhao J, Hartlep T, Kosovichev AG, Mansour NN (2009) Imaging the solar tachocline by time-distance helioseismology. Astrophys J 702:1150–1156. DOI 10.1088/0004-637X/702/2/1150. 0907.2118

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai, India

    Shravan Hanasoge

Authors
  1. Shravan Hanasoge
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 The Author

About this chapter

Cite this chapter

Hanasoge, S. (2015). Wave Equation Solver. In: Imaging Convection and Magnetism in the Sun. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-27330-3_2

Download citation

Publish with us

Policies and ethics

Search

Navigation