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Convergence of finite volume schemes for the coupling between the inviscid Burgers equation and a particle. (English) Zbl 1380.65192
The authors study the numerical convergence of finite volume schemes for the Cauchy problem of the following system in 1D: \(u_t +(u^2/2)_x =-\lambda (u-h'(t))\delta_{h(t)}(x)\), \(m_ph''(t)=\lambda (u(t,h(t)) -h'(t))\) with suitable initial data. This system models the behavior of a pointwise particle of position \(h\), velocity \(h'\) and acceleration \(h''\) with mass \(m_p\), immersed into a “fluid”, whose velocity \(u(t,x)\) at time \(t\) and point \(x\) is assumed to follow the inviscid Burgers equation. In this model, the particle is seen as a moving interface through which an interface condition is imposed, which links \(u(t,x)\) 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 impulsion of the system is conserved through time. The proposed finite volume schemes are consistent with a “large enough” part of the interface conditions. The proof of convergence is an extension of the one of [B. Andreianov and N. Seguin, Discrete Contin. Dyn. Syst. 32, No. 6, 1939–1964 (2012; Zbl 1246.35125)] to the case where the particle moves under the influence of the fluid. The difficulties in the proof are to treat numerically the interface conditions enclosed in the germ and the coupling between an ODE and a PDE, and are circumvented by taking into account at the numerical level the interface condition, applying schemes that preserves a “sufficiently large” part of the germ, and using a mesh that tracks the particle and updating the particle’s velocity by conservation of total impulsion.

MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35Q53 KdV equations (Korteweg-de Vries equations)
35R37 Moving boundary problems for PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L72 Second-order quasilinear hyperbolic equations
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