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A second order PVM flux limiter method. Application to magnetohydrodynamics and shallow stratified flows. (English) Zbl 1349.76314
Summary: In this work we propose a second order flux limiter finite volume method, named PVM-2U-FL, that only uses information of the two external waves of the hyperbolic system. This method could be seen as a natural extension of the well known WAF method introduced by E. F. Toro [Proc. R. Soc. Lond., Ser. A 423, No. 1865, 401–418 (1989; Zbl 0674.76060)]. We prove that independently of the number of unknowns of the 1D system, it recovers the second order accuracy at regular zones, while in presence of discontinuities, the scheme degenerates to PVM-2U method, which can be seen as an improvement of the HLL method (see [the first and the second author, SIAM J. Sci. Comput. 34, No. 4, A2173-A2196 (2012; Zbl 1253.65167); P. Degond et al., C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 6, 479–483 (1999; Zbl 0933.65101)]). Another interesting property of the method is that it does not need any spectral decomposition of the Jacobian or Roe matrix associated to the flux function. Therefore, it can be easily applied to systems with a large number of unknowns or in situations where no analytical expression of the eigenvalues or eigenvectors are known. In this work, we apply the proposed method to magnetohydrodynamics and to stratified multilayer flows.comparison with the two-waves WAF and HLL-MUSCL methods are also presented. The numerical results show that PVM-2U-FL is the most efficient and accurate among them.

76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76W05 Magnetohydrodynamics and electrohydrodynamics
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