A steady-state capturing method for hyperbolic systems with geometrical source terms
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 631-645.

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.

Classification : 35L65, 65M06, 76B15
Mots-clés : hyperbolic systems, source terms, steady state solution, shallow water equations, shock capturing methods 
@article{M2AN_2001__35_4_631_0,
     author = {Jin, Shi},
     title = {A steady-state capturing method for hyperbolic systems with geometrical source terms},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {631--645},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {4},
     year = {2001},
     mrnumber = {1862872},
     zbl = {1001.35083},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_4_631_0/}
}
TY  - JOUR
AU  - Jin, Shi
TI  - A steady-state capturing method for hyperbolic systems with geometrical source terms
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2001
SP  - 631
EP  - 645
VL  - 35
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/M2AN_2001__35_4_631_0/
LA  - en
ID  - M2AN_2001__35_4_631_0
ER  - 
%0 Journal Article
%A Jin, Shi
%T A steady-state capturing method for hyperbolic systems with geometrical source terms
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2001
%P 631-645
%V 35
%N 4
%I EDP-Sciences
%U http://www.numdam.org/item/M2AN_2001__35_4_631_0/
%G en
%F M2AN_2001__35_4_631_0
Jin, Shi. A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 631-645. http://www.numdam.org/item/M2AN_2001__35_4_631_0/

[1] A. Bernudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. & Fluids 23 (1994) 1049-1071. | Zbl

[2] R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comp. (to appear). | MR | Zbl

[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). | MR

[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. | Zbl

[7] L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms | MR | Zbl

[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 | Zbl

[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. | Zbl

[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. | Zbl

[11] S. Jin and M. Katsoulakis, Hyperbolic systems with supercharacteristic relaxations and roll waves. SIAM J. Appl. Math. 61 (2000) 271-292 (electronic). | Zbl

[12] S. Jin and Y.J. Kim, On the computation of roll waves. ESAIM: M2AN 35 (2001) 463-480. | Numdam | Zbl

[13] C. Kranenburg, On the evolution of roll waves. J. Fluid Mech. 245 (1992) 249-261. | Zbl

[14] R.J. Leveque, Numerical methods for conservation laws. Birkhäuser, Basel (1992). | MR | Zbl

[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. | Zbl

[16] P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357-372. | Zbl

[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. | Zbl

[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. | Zbl