Convergence of finite volume schemes for the coupling between the inviscid Burgers equation and a particle

Aguillon, Nina; Lagoutière, Frédéric; Seguin, Nicolas

doi:10.1090/mcom/3082

Convergence of finite volume schemes for the coupling between the inviscid Burgers equation and a particle
HTML articles powered by AMS MathViewer

by Nina Aguillon, Frédéric Lagoutière and Nicolas Seguin PDF

Math. Comp. 86 (2017), 157-196 Request permission

Abstract:

The convergence of a class of finite volume schemes for a model of coupling between a Burgers fluid and a pointwise particle is proved. In this model, introduced by Lagoutière, Seguin and Takahashi in 2008, the particle is seen as a moving point through which an interface condition is imposed, which links the velocity of the fluid on the left and on the right of the particle and the velocity of the particle (the three quantities are all not equal in general). The total momentum of the system is conserved through time.

The proposed schemes are consistent with a “large enough” part of the interface conditions. The proof of convergence is an extension of the one of Andreianov and Seguin (2012) to the case where the particle moves under the influence of the fluid (two-way coupling). This extension contains two new main difficulties: first, the fluxes and interface conditions are time-dependent, and second, the coupling between an ODE and a PDE.

References

Nina Aguillon, Riemann problem for a particle–fluid coupling, Math. Models Methods Appl. Sci. 25 (2015), no. 1, 39–78. MR 3277284, DOI 10.1142/S0218202515500025
Boris Andreianov, Kenneth Hvistendahl Karlsen, and Nils Henrik Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal. 201 (2011), no. 1, 27–86. MR 2807133, DOI 10.1007/s00205-010-0389-4
Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, and Takéo Takahashi, Small solids in an inviscid fluid, Netw. Heterog. Media 5 (2010), no. 3, 385–404. MR 2670647, DOI 10.3934/nhm.2010.5.385
Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, and Takéo Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model, SIAM J. Math. Anal. 46 (2014), no. 2, 1030–1052. MR 3174172, DOI 10.1137/130907963
Boris Andreianov and Nicolas Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst. 32 (2012), no. 6, 1939–1964. MR 2885792, DOI 10.3934/dcds.2012.32.1939
Raul Borsche, Rinaldo M. Colombo, and Mauro Garavello, On the interactions between a solid body and a compressible inviscid fluid, Interfaces Free Bound. 15 (2013), no. 3, 381–403. MR 3148597, DOI 10.4171/IFB/307
C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017–1034. MR 542510, DOI 10.1080/03605307908820117
Clément Cancès and Nicolas Seguin, Error estimate for Godunov approximation of locally constrained conservation laws, SIAM J. Numer. Anal. 50 (2012), no. 6, 3036–3060. MR 3022253, DOI 10.1137/110836912
Michael G. Crandall and Luc Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78 (1980), no. 3, 385–390. MR 553381, DOI 10.1090/S0002-9939-1980-0553381-X
Ami Harten, On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. Numer. Anal. 21 (1984), no. 1, 1–23. With an appendix by Peter D. Lax. MR 731210, DOI 10.1137/0721001
M. Hillairet, Asymptotic collisions between solid particles in a Burgers-Hopf fluid, Asymptot. Anal. 43 (2005), no. 4, 323–338. MR 2160703
A. Y. le Roux, A numerical conception of entropy for quasi-linear equations, Math. Comp. 31 (1977), no. 140, 848–872. MR 478651, DOI 10.1090/S0025-5718-1977-0478651-3
Frédéric Lagoutière, Nicolas Seguin, and Takéo Takahashi, A simple 1D model of inviscid fluid-solid interaction, J. Differential Equations 245 (2008), no. 11, 3503–3544. MR 2460032, DOI 10.1016/j.jde.2008.03.011
E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ. 4 (2007), no. 4, 729–770. MR 2374223, DOI 10.1142/S0219891607001343
John Towers, A fixed grid, shifted stencil scheme for inviscid fluide-particle interaction, preprint (2015).
Alexis Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal. 160 (2001), no. 3, 181–193. MR 1869441, DOI 10.1007/s002050100157
Julien Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains, Numer. Math. 90 (2002), no. 3, 563–596. MR 1884231, DOI 10.1007/s002110100307
Juan Luis Vázquez and Enrique Zuazua, Large time behavior for a simplified 1D model of fluid-solid interaction, Comm. Partial Differential Equations 28 (2003), no. 9-10, 1705–1738. MR 2001181, DOI 10.1081/PDE-120024530
Juan Luis Vázquez and Enrique Zuazua, Lack of collision in a simplified 1D model for fluid-solid interaction, Math. Models Methods Appl. Sci. 16 (2006), no. 5, 637–678. MR 2226121, DOI 10.1142/S0218202506001303