A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows. (English) Zbl 1390.76760
Summary: The present paper is concerned with the development of a new path-conservative meshless Lagrangian particle method (SPH), for the solution of non-conservative systems of hyperbolic partial differential equations, with applications to the shallow water equations, compressible and incompressible multi-phase flows. For shallow water flows, the proposed method is well-balanced. The starting point of our work is the SPH formulation of B. Ben Moussa et al. [ISNM, Int. Ser. Numer. Math. 129, 31–40 (1999; Zbl 0930.65102)]. The method is based on arbitrary-Lagrangian-Eulerian (ALE) numerical flux functions (Riemann solvers) and the scheme is rewritten in flux-difference form, thus ensuring at least zeroth order consistency for constant solutions at the discrete level. Furthermore, a smoothed velocity field is used, in order to obtain a final particle velocity that is consistent with the chosen ALE interface velocity. Starting from the formulation previously described, we then use a path-conservative discretization of the non-conservative terms, following the pioneering work of Castro and Parés. The scheme uses a generalized Roe-matrix that is computed as the path integral of the non-conservative terms along a prescribed integration path, which in this paper is chosen to be a simple straight-line segment. For the shallow water systems under consideration, the segment path leads to well-balanced schemes that exactly preserve the water at rest solution for arbitrary bottom topography. To the knowledge of the authors, this is the first well-balanced SPH scheme based on approximate Riemann solvers and the framework of path-conservative schemes for non-conservative hyperbolic systems. Three different approximate Riemann solvers have been investigated and compared in this paper: the Rusanov scheme, the Osher-type DOT scheme and the HLLEM method. A detailed comparison concerning the accuracy and computational efficiency of each Riemann solver is provided. Several test problems (both in 1D and 2D) are presented in the final part of the paper for different systems of non-conservative hyperbolic equations: the Baer Nunziato model of compressible multi-phase flows, the one- and two-layer shallow water equations and the E. B. Pitman and L. Le two-phase debris flow model [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 363, No. 1832, 1573–1601 (2005; Zbl 1152.86302)]. The numerical results for each system of equations have been compared with available reference solutions to verify the accuracy of the proposed scheme. For the shallow water type models under consideration the method proposed in this paper has furthermore been shown to be well-balanced up to machine precision.
MSC:
| 76M28 | Particle methods and lattice-gas methods |
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
Keywords:
smoothed particle hydrodynamics (SPH) based on Riemann solvers; path-conservative SPH schemes for non-conservative hyperbolic PDE; well-balanced SPH schemes; comparison of different approximate Riemann solvers: Rusanov; Osher (DOT); and HLLEM; Baer Nunziato model of compressible multiphase flow; single and two-layer shallow water equationsSoftware:
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