A steady-state capturing method for hyperbolic systems with geometrical source terms

Shi Jin

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 4, page 631-645
  • ISSN: 0764-583X

Abstract

top
We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar conservation laws and the one dimensional shallow water equations show much better resolution of the steady state than the conventional method, with almost no new numerical complexity.

How to cite

top

Jin, Shi. "A steady-state capturing method for hyperbolic systems with geometrical source terms." ESAIM: Mathematical Modelling and Numerical Analysis 35.4 (2010): 631-645. <http://eudml.org/doc/197530>.

@article{Jin2010,
abstract = { We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar conservation laws and the one dimensional shallow water equations show much better resolution of the steady state than the conventional method, with almost no new numerical complexity. },
author = {Jin, Shi},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Hyperbolic systems; source terms; steady state solution; shallow water equations; shock capturing methods.; steady state solution; shock capturing methods; Godunov or Roe-type upwind methods},
language = {eng},
month = {3},
number = {4},
pages = {631-645},
publisher = {EDP Sciences},
title = {A steady-state capturing method for hyperbolic systems with geometrical source terms},
url = {http://eudml.org/doc/197530},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Jin, Shi
TI - A steady-state capturing method for hyperbolic systems with geometrical source terms
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 4
SP - 631
EP - 645
AB - We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar conservation laws and the one dimensional shallow water equations show much better resolution of the steady state than the conventional method, with almost no new numerical complexity.
LA - eng
KW - Hyperbolic systems; source terms; steady state solution; shallow water equations; shock capturing methods.; steady state solution; shock capturing methods; Godunov or Roe-type upwind methods
UR - http://eudml.org/doc/197530
ER -

References

top
  1. A. Bernudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. & Fluids23 (1994) 1049-1071.  
  2. R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comp. (to appear).  
  3. A. Chinnayya and A.Y. Le Roux, A new general Riemann solver for the shallow-water equations with friction and topography. Preprint (1999).  
  4. T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. AIAA J. (to appear 2001).  
  5. S.K. Godunov, Finite difference schemes for numerical computation of solutions of the equations of fluid dynamics. Math. USSR-Sb.47 (1959) 271-306.  
  6. L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl.39 (2000) 135-159.  
  7. L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. M 3AS (to appear).  
  8. L. Gosse and A.-Y. Le Roux, A well-balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Sér. I Math.323 (1996). 543-546  
  9. J.M. Greenberg and A.-Y. Le Roux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal.33 1-16 1996.  
  10. J.M. Greenberg, A.-Y. Le Roux, R. Baraille and A. Noussair, Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal.34 (1997) 1980-2007.  
  11. S. Jin and M. Katsoulakis, Hyperbolic systems with supercharacteristic relaxations and roll waves. SIAM J. Appl. Math.61 (2000) 271-292 (electronic).  
  12. S. Jin and Y.J. Kim, On the computation of roll waves. ESAIM: M2AN 35 (2001) 463-480.  
  13. C. Kranenburg, On the evolution of roll waves. J. Fluid Mech.245 (1992) 249-261.  
  14. R.J. LeVeque, Numerical methods for conservation laws. Birkhäuser, Basel (1992).  
  15. R.J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys.146 (1998) 346-365.  
  16. P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys.43 (1981) 357-372.  
  17. P.L. Roe, Upwind differenced schemes for hyperbolic conservation laws with source terms, in Nonlinear Hyperbolic Problems, Proc. Adv. Res. Workshop, St. Étienne, 1986, Lect. Notes Math. Springer, Berlin, 1270 (1987) 41-45.  
  18. M.E. Vazquez-Cendon, Improved treatment of source terms in upwind schemes for shallow water equations in channels with irregular geometry. J. Comput. Phys.148 (1999) 497-526.  

Citations in EuDML Documents

top
  1. Emmanuel Audusse, Marie-Odile Bristeau, Transport of pollutant in shallow water : a two time steps kinetic method
  2. Emmanuel Audusse, Marie-Odile Bristeau, Transport of Pollutant in Shallow Water A Two Time Steps Kinetic Method
  3. Steve Bryson, Yekaterina Epshteyn, Alexander Kurganov, Guergana Petrova, Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
  4. Tomás Chacón Rebollo, Antonio Domínguez Delgado, Enrique D. Fernández Nieto, An entropy-correction free solver for non-homogeneous shallow water equations
  5. Stefania Ferrari, Fausto Saleri, A new two-dimensional shallow water model including pressure effects and slow varying bottom topography
  6. Tomás Chacón Rebollo, Antonio Domínguez Delgado, Enrique D. Fernández Nieto, An entropy-correction free solver for non-homogeneous shallow water equations
  7. Stefania Ferrari, Fausto Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography
  8. Steve Bryson, Yekaterina Epshteyn, Alexander Kurganov, Guergana Petrova, Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
  9. Alexander Kurganov, Doron Levy, Central-upwind schemes for the Saint-Venant system
  10. Alexander Kurganov, Doron Levy, Central-Upwind Schemes for the Saint-Venant System

NotesEmbed ?

top

You must be logged in to post comments.