A blob method for diffusion
| Autor: | José A. Carrillo, Katy Craig, Francesco S. Patacchini |
|---|---|
| Přispěvatelé: | Engineering & Physical Science Research Council (EPSRC) |
| Rok vydání: | 2019 |
| Předmět: |
NONLINEAR CONTINUITY EQUATIONS
General Mathematics Mathematics Applied APPROXIMATIONS 010103 numerical & computational mathematics 35Q35 35Q82 65M12 82C22 01 natural sciences 0101 Pure Mathematics GRADIENT FLOW Quadratic equation Mathematics - Analysis of PDEs CRITICAL MASS 0102 Applied Mathematics FOS: Mathematics Mathematics - Numerical Analysis 0101 mathematics Mathematics Partial differential equation Science & Technology Applied Mathematics SCHEME Mathematical analysis Numerical Analysis (math.NA) AGGREGATION 010101 applied mathematics GAMMA-CONVERGENCE Rate of convergence Regularization (physics) Physical Sciences Vector field Heat equation STEEPEST DESCENT Balanced flow NUMERICAL-SIMULATION Gradient descent WEIGHTED PARTICLE METHOD Analysis Analysis of PDEs (math.AP) |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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