A convergence result for finite volume schemes on Riemannian manifolds

Jan Giesselmann

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 5, page 929-955
  • ISSN: 0764-583X

Abstract

top
This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law u t + g · f ( x , u ) = 0 on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a h 1 4 convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to h 1 2 .

How to cite

top

Giesselmann, Jan. "A convergence result for finite volume schemes on Riemannian manifolds." ESAIM: Mathematical Modelling and Numerical Analysis 43.5 (2009): 929-955. <http://eudml.org/doc/250606>.

@article{Giesselmann2009,
abstract = { This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +\nabla_g \cdot f(x,u)=0$ on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a $h^\{\frac\{1\}\{4\}\} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^\{\frac\{1\}\{2\}\}.$},
author = {Giesselmann, Jan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite volume method; conservation law; curved manifold; finite volume method},
language = {eng},
month = {6},
number = {5},
pages = {929-955},
publisher = {EDP Sciences},
title = {A convergence result for finite volume schemes on Riemannian manifolds},
url = {http://eudml.org/doc/250606},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Giesselmann, Jan
TI - A convergence result for finite volume schemes on Riemannian manifolds
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/6//
PB - EDP Sciences
VL - 43
IS - 5
SP - 929
EP - 955
AB - This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +\nabla_g \cdot f(x,u)=0$ on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$
LA - eng
KW - Finite volume method; conservation law; curved manifold; finite volume method
UR - http://eudml.org/doc/250606
ER -

References

top
  1. P. Amorim, M. Ben-Artzi and P.G. LeFloch, Hyperbolic conservation laws on manifolds: total variation estimates and the finite volume method. Methods Appl. Anal.12 (2005) 291–323.  
  2. M. Ben-Artzi and P.G. LeFloch, Well-posedness theory for geometry compatible hyperbolic conservation laws on manifolds. Ann. H. Poincaré Anal. Non Linéaire24 (2007) 989–1008.  
  3. D.A. Calhoun, C. Helzel and R.J. LeVeque, Logically rectangular grids and finite volume methods for PDEs in circular and spherical domains. SIAM Rev.50 (2008) 723–752. Available at .  URIhttp://www.amath.washington.edu/~rjl/pubs/circles
  4. J.Y.-K. Cho and L.M. Polvani, The emergence of jets and vortices in freely evolving, shallow-water turbulence on a sphere. Phys. Fluids8 (1996) 1531–1552.  
  5. M. Dikpati and P.A. Gilman, A “shallow-water” theory for the sun's active longitudes. Astrophys. J. Lett.635 (2005) L193–L196.  
  6. M.P. do Carmo, Riemannian geometry, Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, USA (1992).  
  7. R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal.18 (1998) 563–594.  
  8. J.A. Font, Numerical hydrodynamics and magnetohydrodynamics in general relativity. Living Rev. Relativ.11 (2008) 7. URL (cited on June 8, 2009): .  URIhttp://www.livingreviews.org/lrr-2008-7
  9. P.A. Gilman, Magnetohydrodynamic “shallow-water” equations for the solar tachocline. Astrophys. J. Lett.544 (2000) L79–L82.  
  10. F.X. Giraldo, Lagrange-Galerkin methods on spherical geodesic grids. J. Comput. Phys.136 (1997) 197–213.  
  11. F.X. Giraldo, High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere. J. Comput. Phys.214 (2006) 447–465.  
  12. R. Iacono, M.V. Struglia and C. Ronchi, Spontaneous formation of equatorial jets in freely decaying shallow water turbulence. Phys. Fluids11 (1999) 1272–1274.  
  13. J. Jost, Riemannian Geometry and Geometric Analysis. Springer Universitext, Springer (2002).  
  14. D. Lanser, J.G. Blom and J.G. Verwer, Spatial discretization of the shallow water equations in spherical geometry using osher's scheme. J. Comput. Phys.165 (2000) 542–565.  
  15. J.M. Martíand E. Müller, Numerical hydrodynamics in special relativity. Living Rev. Relativ.6 (2003) 7. URL (cited on June 8, 2009): .  URIhttp://www.livingreviews.org/lrr-2003-7
  16. M.J. Miranda, D. Pallara, F. Paronetto and M. Preunkert, Heat semigroup and functions of bounded variation on Riemannian manifolds. J. reine angew. Math.613 (2007) 99–119.  
  17. M. Rancic, R.J. Purser and F. Mesinger, A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates. Q. J. R. Meteorolog. Soc.122 (1996) 959–982.  
  18. C. Ronchi, R. Iacono and P.S. Paolucci, The cubed sphere: A new method for the solution of partial differential equations in spherical geometry. J. Comput. Phys.124 (1996) 93–114.  
  19. J.A. Rossmanith, A wave propagation algorithm for hyperbolic systems on the sphere. J. Comput. Phys.213 (2006) 629–658.  
  20. J.A. Rossmanith, D.S. Bale and R.J. LeVeque, A wave propagation algorithm for hyperbolic systems on curved manifolds. J. Comput. Phys.199 (2004) 631–662.  
  21. D.A. Schecter, J.F. Boyd and P.A. Gilman, “Shallow-water” magnetohydrodynamic waves in the solar tachocline. Astrophys. J. Lett.551 (2001) L185–L188.  
  22. Y. Tsukahara, N. Nakaso, H. Cho and K. Yamanaka, Observation of diffraction-free propagation of surface acoustic waves around a homogeneous isotropic solid sphere. Appl. Phys. Lett.77 (2000) 2926–2928.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.