On some fast well-balanced first order solvers for nonconservative systems.

*(English)*Zbl 1369.65107Summary: 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.

##### MSC:

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 |

##### Keywords:

nonconservative hyperbolic systems; finite volume method; approximate Riemann solvers; coefficient-splitting schemes; GFORCE method; well-balanced schemes; high order methods##### Software:

GFORCE
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\textit{M. J. Castro} et al., Math. Comput. 79, No. 271, 1427--1472 (2010; Zbl 1369.65107)

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##### References:

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