# American Institute of Mathematical Sciences

March  2020, 15(1): 143-169. doi: 10.3934/nhm.2020007

## The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles

 MiraCosta College, 3333 Manchester Avenue, Cardiff-by-the-Sea, CA 92007-1516, USA

Received  July 2019 Revised  October 2019 Published  December 2019

We propose a finite difference method based on the Lax-Friedrichs scheme for a model of interaction between multiple solid particles and an inviscid fluid. The single-particle version has been studied extensively during the past decade. The model studied here consists of the inviscid Burgers equation with multiple nonconservative moving source terms that are singular and account for drag force interaction between the fluid and the particles. Each particle trajectory satisfies a differential equation that ensures conservation of momentum of the entire system. To deal with the singular source terms we discretize a model that associates with each particle an advection PDE whose solution is a shifted Heaviside function. This alternative model is well known but has not previously been used in numerical methods. We propose a definition of entropy solution which directly generalizes the previously defined single-particle notion of entropy solution. We prove convergence (along a subsequence) of the Lax-Friedrichs approximations, and also prove that if the set of times where the particle paths intersect has Lebesgue measure zero, then the limit is an entropy solution. We also propose a higher resolution version of the scheme, based on MUSCL processing, and present the results of numerical experiments.

Citation: John D. Towers. The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles. Networks and Heterogeneous Media, 2020, 15 (1) : 143-169. doi: 10.3934/nhm.2020007
##### References:
 [1] N. Aguillon, F. Lagoutière and N. Seguin, Convergence of finite volume schemes for the coupling between the inviscid Burgers equation and a particle, Math. Comp., 86 (2017), 157-196.  doi: 10.1090/mcom/3082. [2] N. Aguillon, Riemann problem for a particle-fluid coupling, Math. Models Methods Appl. Sci., 25 (2015), 39-78.  doi: 10.1142/S0218202515500025. [3] N. Aguillon, Numerical simulations of a fluid-particle coupling, in Finite Volumes for Complex Applications Ⅶ-Elliptic, Parabolic and Hyperbolic Problems. Springer Proceedings in Mathematics & Statistics (eds. J. Fuhrmann, M. Ohlberger M. and C. Rohde), Springer, 78 (2014), 759–767. doi: 10.1007/978-3-319-05591-6_76. [4] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4. [5] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Small solids in an inviscid fluid, Netw. Heterog. Media, 5 (2010), 385-404.  doi: 10.3934/nhm.2010.5.385. [6] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model, SIAM J. Math. Anal., 46 (2014), 1030-1052.  doi: 10.1137/130907963. [7] B. Andreianov and N. Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst., 32 (2012), 1939-1964.  doi: 10.3934/dcds.2012.32.1939. [8] M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21. [9] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, New York, 2002. doi: 10.1007/978-3-642-56139-9. [10] C. Klingenberg, J. Klotzky and N. Seguin, On well-posedness for a multi-particle fluid model, in Theory, Numerics and Applications of Hyperbolic Problems Ⅱ. HYP 2016. Springer Proceedings in Mathematics & Statistics, (eds. C. Klingenberg and M. Westdickenberg), Springer, 237 (2018), 167–177. [11] F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction, J. Differ. Equ., 245 (2008), 3503-3544.  doi: 10.1016/j.jde.2008.03.011. [12] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, UK, 2002.  doi: 10.1017/CBO9780511791253. [13] E. Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770.  doi: 10.1142/S0219891607001343. [14] J. Towers, A fixed grid, shifted stencil scheme for inviscid fluid-particle interaction, Appl. Numer. Math., 110 (2016), 26-40.  doi: 10.1016/j.apnum.2016.08.002.

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##### References:
 [1] N. Aguillon, F. Lagoutière and N. Seguin, Convergence of finite volume schemes for the coupling between the inviscid Burgers equation and a particle, Math. Comp., 86 (2017), 157-196.  doi: 10.1090/mcom/3082. [2] N. Aguillon, Riemann problem for a particle-fluid coupling, Math. Models Methods Appl. Sci., 25 (2015), 39-78.  doi: 10.1142/S0218202515500025. [3] N. Aguillon, Numerical simulations of a fluid-particle coupling, in Finite Volumes for Complex Applications Ⅶ-Elliptic, Parabolic and Hyperbolic Problems. Springer Proceedings in Mathematics & Statistics (eds. J. Fuhrmann, M. Ohlberger M. and C. Rohde), Springer, 78 (2014), 759–767. doi: 10.1007/978-3-319-05591-6_76. [4] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4. [5] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Small solids in an inviscid fluid, Netw. Heterog. Media, 5 (2010), 385-404.  doi: 10.3934/nhm.2010.5.385. [6] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model, SIAM J. Math. Anal., 46 (2014), 1030-1052.  doi: 10.1137/130907963. [7] B. Andreianov and N. Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst., 32 (2012), 1939-1964.  doi: 10.3934/dcds.2012.32.1939. [8] M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21. [9] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, New York, 2002. doi: 10.1007/978-3-642-56139-9. [10] C. Klingenberg, J. Klotzky and N. Seguin, On well-posedness for a multi-particle fluid model, in Theory, Numerics and Applications of Hyperbolic Problems Ⅱ. HYP 2016. Springer Proceedings in Mathematics & Statistics, (eds. C. Klingenberg and M. Westdickenberg), Springer, 237 (2018), 167–177. [11] F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction, J. Differ. Equ., 245 (2008), 3503-3544.  doi: 10.1016/j.jde.2008.03.011. [12] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, UK, 2002.  doi: 10.1017/CBO9780511791253. [13] E. Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770.  doi: 10.1142/S0219891607001343. [14] J. Towers, A fixed grid, shifted stencil scheme for inviscid fluid-particle interaction, Appl. Numer. Math., 110 (2016), 26-40.  doi: 10.1016/j.apnum.2016.08.002.
Example 8.3. Particle trajectories using basic scheme (upper plot) and MUSCL (lower plot). Both the true (thick line) and approximate (thin line) trajectories are plotted. For the MUSCL scheme (lower plot) the true and approximate trajectories are visually indistinguishable at this level of discretization. $\Delta x = 1.95 \times 10^{-5}$, $\mu = .25$, 102401 time steps
Example 8.4. Basic scheme (left) and MUSCL (right). The horizontal axis represents $x$, and the vertical axis represents $t$. Top level plots: $\Delta x_1 = 3.75 \times 10^{-4}$. Middle level plots: $\Delta x_2 = {1\over 2} \Delta x_1$. Bottom level plots: $\Delta x_3 = {1\over 4} \Delta x_1$. $\mu = .125$ for all plots
Example 8.1. Top: Fluid velocity $u$ at $t = 1$. Exact solution is solid line, with sharp corners. Bottom: Particle position error vs. time. Basic scheme (left plots) and MUSCL scheme (right plots). $\Delta x = .0025$ (dashed line), and $\Delta x = .00125$ (solid line). Both approximations used $\mu = .25$
Example 8.2. Fluid velocity $u$ at $t = 1$. Basic scheme (left plots) and MUSCL scheme (right plots). Exact solution (dashed line) and approximate solution (solid line). Top plots used $\Delta x = .005$, bottom plots used $\Delta x = .000625$. All approximations used $\mu = .25$. A spurious kink is visible. Its magnitude diminishes with grid refinement
Example 8.3. Solution $u$ using basic scheme at $t = .125$ (upper left), and using MUSCL (upper right). True solution (dashed line) and approximate solution (solid line). Both upper plots computed with $\Delta x = .00325$, $\mu = .25$. The lower plots show the error in $u$ in discrete $L^1$ norm as a function of time using the basic scheme (lower left) and MUSCL scheme (lower right). Uses $\Delta x = .00325$ and $\Delta x = .001625$, $\mu = .25$
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