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A blob method for the aggregation equation. (English) Zbl 1339.35235

The authors start from the aggregation equation \(\rho_t+\nabla \cdot (v\rho )=0\), with \(v=-\nabla K\ast \rho \), for some kernel \(K\), with the initial condition \(\rho (x,0)=\rho_0(x)\). Considering the trajectory of a particle \[ \frac{d}{dt}X^{t}(\alpha )=-\nabla K\ast \rho (X^t(\alpha ),t),\quad X^0 (\alpha )=\alpha , \] they rewrite the aggregation equation as \[ \begin{cases} \frac{d}{dt}\rho (X^{t}(\alpha ),t)=(\Delta K\ast \rho (X^t(\alpha ),t))\rho (X^t (\alpha ),t), \\ \rho (X^{0}(\alpha ),0)=\rho _{0}(\alpha ).\end{cases} \] They introduce the approximate velocity field along the exact particle trajectories as \[ v^{h}(x,t)=-\int_{\mathbb{R}^{d}}\nabla K_{\delta }(x-X^{t}(\alpha ))\rho _{0}^{\text{particle}}(\alpha )d\alpha =-\sum_{j}\nabla K_{\delta }(x-X_{j}(t))\rho _{0_{j}}h^{d} \] where \(K_{\delta }\) is the convolution product between the kernel \(K\) and \(\psi _{\delta }\) defined through \(\psi _{\delta }(x)=\delta ^{-d}\psi (x/\delta )\) \(\psi \) being a smooth mollifier or blob function. This allows defining approximate particle trajectories and the authors build the associated blob method. The main result of the paper proves the convergence of the approximate solution built through this blob method, under assumptions on the kernel and on the mollifier. In the last part of their paper, the authors present the results of numerical simulations choosing either regular or discontinuous initial data.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q82 PDEs in connection with statistical mechanics
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
82C22 Interacting particle systems in time-dependent statistical mechanics
76M28 Particle methods and lattice-gas methods

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NumPy; Matplotlib; SciPy
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References:

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