Convergence of a linearly transformed particle method for aggregation equations

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in $L^1$ and $L^\infty$ norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in $L^1 \cap L^p$ norm

A blob method for diffusion

As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocity field that ensures particles do remain particles, and we apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient flow type, including the heat equation, the porous medium equation, the Fokker-Planck equation, the Keller-Segel equation, and its variants. Our choice of regularization is guided by the Wasserstein gradient flow structure, and the corresponding energy has a novel form, combining aspects of the well-known interaction and potential energies. In the presence of a confining drift or interaction potential, we prove that minimizers of the regularized energy exist and, as the regularization is removed, converge to the minimizers of the unregularized energy. We then restrict our attention to nonlinear diffusion of porous medium type with at least quadratic exponent. Under sufficient regularity assumptions, we prove that gradient flows of the regularized energies converge to solutions of the porous medium equation. As a corollary, we obtain convergence of our numerical blob method, again under sufficient regularity assumptions. We conclude by considering a range of numerical examples to demonstrate our method's rate of convergence to exact solutions and to illustrate key qualitative properties preserved by the method, including asymptotic behavior of the Fokker-Planck equation and critical mass of the two-dimensional Keller-Segel equation

A particle method for the homogeneous Landau equation

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains all the important properties of the Landau operator, namely the conservation of mass, momentum and energy, and the decay of entropy. We illustrate our new method by showing its performance in several test cases including the physically relevant case of the Coulomb interaction. The comparison to the exact solution and the spectral method is strikingly good maintaining 2nd order accuracy. Moreover, an efficient implementation of the method via the treecode is explored. This gives a proof of concept for the practical use of our method when coupled with the classical PIC method for the Vlasov equation.Comment: 27 pages, 14 figures, debloated some figures, improved explanations in sections 2, 3, and

Multivariate calibration of a water and energy balance model in the spectral domain

The objective of this paper is to explore the possibility of using multiple variables in the calibration of hydrologic models in the spectral domain. A simple water and energy balance model was used, combined with observations of the energy balance and the soil moisture profile. The correlation functions of the model outputs and the observations for the different variables have been calculated after the removal of the diurnal cycle of the energy balance variables. These were transformed to the frequency domain to obtain spectral density functions, which were combined in the calibration algorithm. It has been found that it is best to use the square root of the spectral densities in the parameter estimation. Under these conditions, spectral calibration performs almost equally as well as time domain calibration using least squares differences between observed and simulated time series. Incorporation of the spectral coefficients of the cross-correlation functions did not improve the results of the calibration. Calibration on the correlation functions in the time domain led to worse model performance. When the meteorological forcing and model calibration data are not overlapping in time, spectral calibration has been shown to lead to an acceptable model performance. Overall, the results in this paper suggest that, in case of data scarcity, multivariate spectral calibration can be an attractive tool to estimate model parameters

A Boltzmann model for rod alignment and schooling fish

We consider a Boltzmann model introduced by Bertin, Droz and Greegoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the equilibria of this Boltzmann model and we rigorously show the existence of a pitchfork bifurcation when a parameter measuring the inverse of the noise intensity crosses a critical threshold. The analysis is carried over rigorously when there are only finitely many non-zero Fourier modes of the noise distribution. In this case, we can show that the critical exponent of the bifurcation is exactly 1/2. In the case of an infinite number of non-zero Fourier modes, a similar behavior can be formally obtained thanks to a method relying on integer partitions first proposed by Ben-Naim and Krapivsky.Comment: 22 pages, 3 figure