Skip to main content
×
Home
    • Aa
    • Aa

A convergence result for finite volume schemes on Riemannian manifolds

  • Jan Giesselmann (a1)
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 $h^{\frac{1}{2}}.$

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 $h^{\frac{1}{2}}.$

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A convergence result for finite volume schemes on Riemannian manifolds
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      A convergence result for finite volume schemes on Riemannian manifolds
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      A convergence result for finite volume schemes on Riemannian manifolds
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

M. Ben-Artzi and P.G. LeFloch , Well-posedness theory for geometry compatible hyperbolic conservation laws on manifolds. Ann. H. Poincaré Anal. Non Linéaire 24 (2007) 9891008.

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) 723752. Available at http://www.amath.washington.edu/~rjl/pubs/circles.

J.Y.-K. Cho and L.M. Polvani , The emergence of jets and vortices in freely evolving, shallow-water turbulence on a sphere. Phys. Fluids 8 (1996) 15311552.

M. Dikpati and P.A. Gilman , A “shallow-water” theory for the sun's active longitudes. Astrophys. J. Lett. 635 (2005) L193L196.

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) 563594.

J.A. Font , Numerical hydrodynamics and magnetohydrodynamics in general relativity. Living Rev. Relativ. 11 (2008) 7. URL (cited on June 8, 2009): http://www.livingreviews.org/lrr-2008-7.

P.A. Gilman , Magnetohydrodynamic “shallow-water” equations for the solar tachocline. Astrophys. J. Lett. 544 (2000) L79L82.

F.X. Giraldo , Lagrange-Galerkin methods on spherical geodesic grids. J. Comput. Phys. 136 (1997) 197213.

F.X. Giraldo , High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere. J. Comput. Phys. 214 (2006) 447465.

R. Iacono , M.V. Struglia and C. Ronchi , Spontaneous formation of equatorial jets in freely decaying shallow water turbulence. Phys. Fluids 11 (1999) 12721274.

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) 542565.

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) 959982.

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) 93114.

J.A. Rossmanith , A wave propagation algorithm for hyperbolic systems on the sphere. J. Comput. Phys. 213 (2006) 629658.

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) 631662.

D.A. Schecter , J.F. Boyd and P.A. Gilman , “Shallow-water” magnetohydrodynamic waves in the solar tachocline. Astrophys. J. Lett. 551 (2001) L185L188.

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) 29262928.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ESAIM: Mathematical Modelling and Numerical Analysis
  • ISSN: 0764-583X
  • EISSN: 1290-3841
  • URL: /core/journals/esaim-mathematical-modelling-and-numerical-analysis
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 3 *
Loading metrics...

Abstract views

Total abstract views: 25 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd September 2017. This data will be updated every 24 hours.