On some fast well-balanced first order solvers for nonconservative systems
HTML articles powered by AMS MathViewer
- by Manuel J. Castro, Alberto Pardo, Carlos Parés and E. F. Toro;
- Math. Comp. 79 (2010), 1427-1472
- DOI: https://doi.org/10.1090/S0025-5718-09-02317-5
- Published electronically: November 23, 2009
- PDF | Request permission
Abstract:
The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good well-balanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.References
- Emmanuel Audusse, François Bouchut, Marie-Odile Bristeau, Rupert Klein, and Benoît Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput. 25 (2004), no. 6, 2050–2065. MR 2086830, DOI 10.1137/S1064827503431090
- Alfredo Bermudez and Ma. Elena Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids 23 (1994), no. 8, 1049–1071. MR 1314237, DOI 10.1016/0045-7930(94)90004-3
- François Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. MR 2128209, DOI 10.1007/b93802
- A. Canestrelli, A. Siviglia, M. Dumbser, E.F. Toro. Well-balanced high-order centered schemes for non-conservative hyperbolic systems. Applications to Shallow Water Equations with fixed and mobile bed. Adv. Water Resour, doi:10.1016/j.advwatres.2009.02.006, 2009.
- M. J. Castro, A. M. Ferreiro Ferreiro, J. A. García-Rodríguez, J. M. González-Vida, J. Macías, C. Parés, and M. Elena Vázquez-Cendón, The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems, Math. Comput. Modelling 42 (2005), no. 3-4, 419–439. MR 2163780, DOI 10.1016/j.mcm.2004.01.016
- Manuel J. Castro, Philippe G. LeFloch, María Luz Muñoz-Ruiz, and Carlos Parés, Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. Comput. Phys. 227 (2008), no. 17, 8107–8129. MR 2442446, DOI 10.1016/j.jcp.2008.05.012
- Manuel Castro, José M. Gallardo, and Carlos Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems, Math. Comp. 75 (2006), no. 255, 1103–1134. MR 2219021, DOI 10.1090/S0025-5718-06-01851-5
- Manuel Castro, Jorge Macías, and Carlos Parés, A $Q$-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system, M2AN Math. Model. Numer. Anal. 35 (2001), no. 1, 107–127. MR 1811983, DOI 10.1051/m2an:2001108
- Manuel J. Castro, Alberto Pardo Milanés, and Carlos Parés, Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Math. Models Methods Appl. Sci. 17 (2007), no. 12, 2055–2113. MR 2371563, DOI 10.1142/S021820250700256X
- Gianni Dal Maso, Philippe G. Lefloch, and François Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. MR 1365258
- Bernd Einfeldt, On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal. 25 (1988), no. 2, 294–318. MR 933726, DOI 10.1137/0725021
- José M. Gallardo, Carlos Parés, and Manuel Castro, On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. Comput. Phys. 227 (2007), no. 1, 574–601. MR 2361537, DOI 10.1016/j.jcp.2007.08.007
- Thierry Gallouët, Jean-Marc Hérard, and Nicolas Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Comput. & Fluids 32 (2003), no. 4, 479–513. MR 1966639, DOI 10.1016/S0045-7930(02)00011-7
- Ami Harten and James M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys. 50 (1983), no. 2, 235–269. MR 707200, DOI 10.1016/0021-9991(83)90066-9
- Thomas Y. Hou and Philippe G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comp. 62 (1994), no. 206, 497–530. MR 1201068, DOI 10.1090/S0025-5718-1994-1201068-0
- Philippe LeFloch and Tai-Ping Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993), no. 3, 261–280. MR 1216035, DOI 10.1515/form.1993.5.261
- Randall J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys. 146 (1998), no. 1, 346–365. MR 1650496, DOI 10.1006/jcph.1998.6058
- Antonio Marquina, Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws, SIAM J. Sci. Comput. 15 (1994), no. 4, 892–915. MR 1278006, DOI 10.1137/0915054
- María Luz Muñoz-Ruiz and Carlos Parés, Godunov method for nonconservative hyperbolic systems, M2AN Math. Model. Numer. Anal. 41 (2007), no. 1, 169–185. MR 2323696, DOI 10.1051/m2an:2007011
- Carlos Parés, Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal. 44 (2006), no. 1, 300–321. MR 2217384, DOI 10.1137/050628052
- Carlos Parés and Manuel Castro, On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems, M2AN Math. Model. Numer. Anal. 38 (2004), no. 5, 821–852. MR 2104431, DOI 10.1051/m2an:2004041
- Sebastian Noelle, Yulong Xing, and Chi-Wang Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. Comput. Phys. 226 (2007), no. 1, 29–58. MR 2356351, DOI 10.1016/j.jcp.2007.03.031
- P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981), no. 2, 357–372. MR 640362, DOI 10.1016/0021-9991(81)90128-5
- J.B. Schijf, J.C. Schonfeld. Theoretical considerations on the motion of salt and fresh water. In Proc. of the Minn. Int. Hydraulics Conv., Joint meeting IAHR and Hyd. Div. ASCE., 321–333, 1953.
- E. F. Toro and S. J. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal. 20 (2000), no. 1, 47–79. MR 1736950, DOI 10.1093/imanum/20.1.47
- E. F. Toro and A. Siviglia, PRICE: primitive centred schemes for hyperbolic systems, Internat. J. Numer. Methods Fluids 42 (2003), no. 12, 1263–1291. MR 1994077, DOI 10.1002/fld.491
- E. F. Toro and V. A. Titarev, MUSTA fluxes for systems of conservation laws, J. Comput. Phys. 216 (2006), no. 2, 403–429. MR 2235378, DOI 10.1016/j.jcp.2005.12.012
- I. Toumi, A weak formulation of Roe’s approximate Riemann solver, J. Comput. Phys. 102 (1992), no. 2, 360–373. MR 1187694, DOI 10.1016/0021-9991(92)90378-C
Bibliographic Information
- Manuel J. Castro
- Affiliation: Universidad de Málaga, Departamento Análisis Matemático, Campus de Teatinos s/n, 29071 Málaga, Spain
- Email: castro@anamat.cie.uma.es
- Alberto Pardo
- Affiliation: Universidad de Málaga, Departamento Análisis Matemático, Campus de Teatinos s/n, 29071 Málaga, Spain
- Email: pardo@anamat.cie.uma.es
- Carlos Parés
- Affiliation: Universidad de Málaga, Departamento Análisis Matemático, Campus de Teatinos s/n, 29071 Málaga, Spain
- Email: pares@anamat.cie.uma.es
- E. F. Toro
- Affiliation: University of Trento. Laboratory of Applied Mathematics. Faculty of Engineering, 38050 Mesiano di Povo, Trento, Italy
- Email: toroe@ing.unitn.it
- Received by editor(s): November 24, 2008
- Received by editor(s) in revised form: May 11, 2009
- Published electronically: November 23, 2009
- Additional Notes: This research has been partially supported by the Spanish Government Research project MTM2006-08075. The numerical computations have been performed at the Laboratory of Numerical Methods of the University of Málaga.
- © Copyright 2009 American Mathematical Society
- Journal: Math. Comp. 79 (2010), 1427-1472
- MSC (2000): Primary 74S10, 65M06, 35L60, 35L65, 35L67
- DOI: https://doi.org/10.1090/S0025-5718-09-02317-5
- MathSciNet review: 2629999