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On some fast well-balanced first order solvers for nonconservative systems. (English) Zbl 1369.65107
Summary: 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 I. Toumi [J. Comput. Phys. 102, No. 2, 360–373 (1992; Zbl 0783.65068)] 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.

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M20 Finite difference methods applied to problems in fluid mechanics
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[1] 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. · Zbl 1133.65308
[2] Alfredo Bermudez and Ma. Elena Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids 23 (1994), no. 8, 1049 – 1071. · Zbl 0816.76052
[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. · Zbl 1086.65091
[4] 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.
[5] 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. · Zbl 1121.76008
[6] 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. · Zbl 1176.76084
[7] 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. · Zbl 1096.65082
[8] Manuel Castro, Jorge Macías, and Carlos Parés, A \?-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. · Zbl 1094.76046
[9] 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. · Zbl 1137.76038
[10] 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. · Zbl 0853.35068
[11] Bernd Einfeldt, On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal. 25 (1988), no. 2, 294 – 318. · Zbl 0642.76088
[12] 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. · Zbl 1126.76036
[13] 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. · Zbl 1084.76540
[14] 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. · Zbl 0565.65049
[15] Thomas Y. Hou and Philippe G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comp. 62 (1994), no. 206, 497 – 530. · Zbl 0809.65102
[16] Philippe LeFloch and Tai-Ping Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993), no. 3, 261 – 280. · Zbl 0804.35086
[17] 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. · Zbl 0931.76059
[18] Antonio Marquina, Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws, SIAM J. Sci. Comput. 15 (1994), no. 4, 892 – 915. · Zbl 0805.65088
[19] María Luz Muñoz-Ruiz and Carlos Parés, Godunov method for nonconservative hyperbolic systems, M2AN Math. Model. Numer. Anal. 41 (2007), no. 1, 169 – 185. · Zbl 1124.65077
[20] Carlos Parés, Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal. 44 (2006), no. 1, 300 – 321. · Zbl 1130.65089
[21] 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. · Zbl 1130.76325
[22] 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. · Zbl 1120.76046
[23] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981), no. 2, 357 – 372. · Zbl 0474.65066
[24] 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.
[25] E. F. Toro and S. J. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal. 20 (2000), no. 1, 47 – 79. · Zbl 0943.65100
[26] E. F. Toro and A. Siviglia, PRICE: primitive centred schemes for hyperbolic systems, Internat. J. Numer. Methods Fluids 42 (2003), no. 12, 1263 – 1291. · Zbl 1078.76566
[27] E. F. Toro and V. A. Titarev, MUSTA fluxes for systems of conservation laws, J. Comput. Phys. 216 (2006), no. 2, 403 – 429. · Zbl 1097.65091
[28] I. Toumi, A weak formulation of Roe’s approximate Riemann solver, J. Comput. Phys. 102 (1992), no. 2, 360 – 373. · Zbl 0783.65068
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