Towers, John D. A fixed grid, shifted stencil scheme for inviscid fluid-particle interaction. (English) Zbl 06638248 Appl. Numer. Math. 110, 26-40 (2016). Summary: This paper presents a finite volume scheme for a scalar one-dimensional fluid-particle interaction model. When devising a finite volume scheme for this model, one difficulty that arises is how to deal with the moving source term in the PDE while maintaining a fixed grid. The fixed grid requirement comes from the ultimate goal of accommodating two or more particles. The finite volume scheme that we propose addresses the moving source term in a novel way. We use a modified computational stencil, with the lower part of the stencil shifted during those time steps when the particle crosses a mesh point. We then employ an altered convective flux to compensate the stencil shifts. The resulting scheme uses a fixed grid, preserves total momentum, and enforces several stability properties in the single-particle case. The single-particle scheme is easily extended to multiple particles by a splitting method. Cited in 3 Documents MSC: 65 Numerical analysis Keywords:solid-fluid interaction; Burgers equation; finite volume scheme; singular source term; moving mesh scheme; well-balanced scheme; moving interface; PDE-ODE coupling PDF BibTeX XML Cite \textit{J. D. Towers}, Appl. Numer. Math. 110, 26--40 (2016; Zbl 06638248) Full Text: DOI References: [1] Aguillon, N., Numerical simulations of a fluid-particle coupling, (Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Springer Proc. Math. Stat., vol. 78, (2014)), 759-767 · Zbl 1426.76328 [2] Aguillon, N., Riemann problem for a particle-fluid coupling, Math. Models Methods Appl. Sci., 25, 39-78, (2015) · Zbl 1314.35089 [3] Aguillon, N.; Lagoutière, F.; Seguin, N., Convergence of finite volume schemes for coupling between the inviscid Burgers equation and a particle, preprint accessed at · Zbl 1380.65192 [4] Andreianov, B.; Karlsen, K.; Risebro, N., A theory of \(L^1\)-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201, 27-86, (2011) · Zbl 1261.35088 [5] Andreianov, B.; Lagoutière, F.; Seguin, N.; Takahashi, T., Small solids in an inviscid fluid, Netw. Heterog. Media, 5, 3, 385-404, (2010) · Zbl 1262.35180 [6] Andreianov, B.; Lagoutière, F.; Seguin, N.; Takahashi, T., Well-posedness for a one-dimensional fluid-particle interaction model, SIAM J. Math. Anal., 46, 1030-1052, (2014) · Zbl 1302.35263 [7] Andreianov, B.; Seguin, N., Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst., 32, 6, 1939-1964, (2012) · Zbl 1246.35125 [8] Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49, 357-393, (1983) · Zbl 0565.65050 [9] Holden, H.; Karlsen, K. H.; Lie, K.-A.; Risebro, N. H., Splitting for partial differential equations with rough solutions, (2010), European Math. Soc. Publishing House Zurich [10] Lagoutière, F.; Seguin, N.; Takahashi, T., A simple 1D model of inviscid fluid-solid interaction, J. Differ. Equ., 245, 3503-3544, (2008) · Zbl 1151.76033 [11] LeRoux, A., A numerical conception of entropy for quasi-linear equations, Math. Comput., 31, 848-872, (1977) · Zbl 0378.65053 [12] Leveque, R. J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press Cambridge, UK · Zbl 1010.65040 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.