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Deterministic particle method approximation of a contact inhibition cross-diffusion problem

Gonzalo Galiano, Virginia Selgas

Abstract

We use a deterministic particle method to produce numerical approximations to the solutions of an evolution cross-diffusion problem for two populations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. When the matrix is definite positive, the problem is well posed and the Finite Element approximation produces convergent approximations to the exact solution. A particularly important case arises when the matrix is only positive semi-definite and the initial data are segregated: the contact inhibition problem. In this case, the solutions may be discontinuous and hence the (conforming) Finite Element approximation may exhibit instabilities in the neighborhood of the discontinuity. In this article we deduce the particle method approximation to the general cross-diffusion problem and apply it to the contact inhibition problem. We then provide some numerical experiments comparing the results produced by the Finite Element and the particle method discretizations.

Deterministic particle method approximation of a contact inhibition cross-diffusion problem

Abstract

We use a deterministic particle method to produce numerical approximations to the solutions of an evolution cross-diffusion problem for two populations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. When the matrix is definite positive, the problem is well posed and the Finite Element approximation produces convergent approximations to the exact solution. A particularly important case arises when the matrix is only positive semi-definite and the initial data are segregated: the contact inhibition problem. In this case, the solutions may be discontinuous and hence the (conforming) Finite Element approximation may exhibit instabilities in the neighborhood of the discontinuity. In this article we deduce the particle method approximation to the general cross-diffusion problem and apply it to the contact inhibition problem. We then provide some numerical experiments comparing the results produced by the Finite Element and the particle method discretizations.

Paper Structure

This paper contains 7 sections, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The particle method
  3. Implementation
  4. Numerical experiments
  5. The Finite Element Method discretization
  6. Experiments
  7. Conclusions

Figures (4)

  • Figure 1: Experiment 1. Left panel: Exact and particle method approximation. At this scale, both are indistinguishable. Right panel: A detail of the solution showing the oscillating instabilities and the need of only three particles to covering the discontinuity of the exact solution.
  • Figure 2: Experiment 1. Left panel: Exact and FEM approximation. Again, at this scale, both are indistinguishable. Right panel: A detail of the solution showing that more nodes than in the particle method are needed to cover the discontinuity of the exact solution. However, no instabilities are present.
  • Figure 3: Experiment 2. Particle (continuous line) and FEM (dotted line) approximations. Only at the discontinuity point a difference between both approximations may be observed.
  • Figure 4: Experiment 2. Detail of Fig. \ref{['exp2_1.fig']} at the contact inhibition point.