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Numerical study of a particle method for gradient flows

  • * Corresponding author: J. A. Carrillo

    * Corresponding author: J. A. Carrillo 

The first, second and third authors are supported by Engineering and Physical Sciences Research Council grant EP/K008404/1. The first author is also supported by the Royal Society through a Wolfson Research Merit Award. The last author is supported by ISF grant 998/5

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  • We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

    Mathematics Subject Classification: Primary: 65M12; Secondary: 35K05.


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  • Figure 1.  The heat equation

    Figure 2.  The porous medium equation with $m=2$

    Figure 3.  The linear Fokker-Planck equation with $\Delta t = 10^{-5}$ --Stabilisation in time of the scheme (rate of convergence to the discrete steady state)

    Figure 4.  The nonlinear Fokker-Planck equation with $m=2$ and $\Delta t = \frac{0.1}{N^2}$ -Stabilisation in time of the scheme (rate of convergence to the discrete steady state)

    Figure 5.  The particles' positions for the two-dimensional heat equation for $N=100$ with $\Delta t = 10^{-4}$

    Figure 6.  Accuracy for the two-dimensional heat equation with $\Delta t = 10^{-4}$

    Figure 7.  The modified Keller-Segel equation with $\chi = 1.5$ for $N=50$

    Figure 8.  The blow-up of the modified Keller-Segel equation with $\chi = 1.5$

    Figure 9.  Blow-up formation for the modified Keller-Segel equation with two initial Gaussian bumps

    Figure 10.  The modified nonlinear Keller-Segel equation with $\chi = 1.4$ for different choices of $m$, for $N=50$ at $T=4$ with $\Delta t = 10^{-5}$

    Figure 11.  Compactly supported potential $W(x) = -c\max(1-|x|,0) + c$ with nonlinear diffusion with $c = 8$ and $m = 3$, for $N=80$ with $\Delta t = 10^{-5}$

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