Abstract
We describe a numerical implementation of a differential model for the simulation of self-organized criticality (SOC) phenomena arising from recent papers by Barbu (Annu Rev Control 34:52–61, 2010; Math Methods Appl Sci 36:1726–1733, 2013). In that singular nonlinear diffusion problem an initial supercritical state evolves in a finite time towards a given critical solution, progressively from the boundary towards the internal regions. The key elements are the Heaviside function which plays the role of a switch for the dynamics, and the initial boundary contact with the critical state.
A finite difference implicit scheme on a fixed grid is proposed for a regularized version of the problem, with the Heaviside replaced by a C 1 function, showing the same behavior of the solution: convergence in finite time toward the critical state on every single node, up to any prescribed accuracy, remaining supercritical during all the process.
The use of synchronized spatial-temporal grids with progressive refinements (in the spirit of Mosco (SIAM J Math Anal 50(3):2409–2440, 2018)) simulates the appearance of short-range interactions of an increasing number of particles, speeding up the convergence to the critical solution, and allowing a strong reduction of computational cost.
The results of some numerical simulations are discussed, in one and two dimensions.
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Acknowledgements
We wish to thank Umberto Mosco and Maria Agostina Vivaldi for their helpful support and fruitful discussions while working on this paper.
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Alberini, C., Finzi Vita, S. (2021). A Numerical Approach to a Nonlinear Diffusion Model for Self-Organized Criticality Phenomena. In: Lancia, M.R., Rozanova-Pierrat, A. (eds) Fractals in Engineering: Theoretical Aspects and Numerical Approximations. SEMA SIMAI Springer Series(), vol 8. Springer, Cham. https://doi.org/10.1007/978-3-030-61803-2_1
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DOI: https://doi.org/10.1007/978-3-030-61803-2_1
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