54 research outputs found
Numerical simulation of continuity equations by evolving diffeomorphisms
In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on its variational formulation as a gradient flow with respect to the Wasserstein distance. Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this properties with various examples in spatial dimension one and two
Numerical Study of a Particle Method for Gradient Flows
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.Comment: 27 pages, 21 figure
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
Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons
Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with
quadratic drift as a PDE model for the dynamics of bosons in the spatially
homogeneous setting. It is an open question whether this equation has solutions
exhibiting condensates in finite time. The main analytical challenge lies in
the continuation of exploding solutions beyond their first blow-up time while
having a linear diffusion term. We present a thoroughly validated time-implicit
numerical scheme capable of simulating solutions for arbitrarily large time,
and thus enabling a numerical study of the condensation process in the
Kaniadakis--Quarati model. We show strong numerical evidence that above the
critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model
in 3D form a condensate in finite time and converge in entropy to the unique
minimiser of the natural entropy functional at an exponential rate. Our
simulations further indicate that the spatial blow-up profile near the origin
follows a universal power law and that transient condensates can occur for
sufficiently concentrated initial data.Comment: To appear in Kinet. Relat. Model
A hybrid mass transport finite element method for Keller--Segel type systems
We propose a new splitting scheme for general reaction–taxis–diffusion systems in one spatial dimension capable to deal with simultaneous concentrated and diffusive regions as well as travelling waves and merging phenomena. The splitting scheme is based on a mass transport strategy for the cell density coupled with classical finite element approximations for the rest of the system. The built-in mass adaption of the scheme allows for an excellent performance even with respect to dedicated mesh-adapted AMR schemes in original variables
- …