We prove that a finite-difference centered approximation for the Kolmogorov equation in the whole space preserves the decay properties of continuous solutions as t -> 8, independently of the mesh-size parameters. This is a manifestation of the property of numerical hypo-coercivity, and it holds both for semi-discrete and fully discrete approximations. The method of proof is based on the energy methods developed by Herau and Villani, employing well-balanced Lyapunov functionals mixing different energies, suitably weighted and equilibrated by multiplicative powers in time. The decreasing character of this Lyapunov functional leads to the optimal decay of the L-2-norms of solutions and partial derivatives, which are of different order because of the anisotropy of the model.
Porretta, A., Zuazua, E. (2017). Numerical hypocoercivity for the Kolmogorov equation. MATHEMATICS OF COMPUTATION, 86(303), 97-119 [10.1090/mcom/3157].
Numerical hypocoercivity for the Kolmogorov equation
PORRETTA, ALESSIO;
2017-01-01
Abstract
We prove that a finite-difference centered approximation for the Kolmogorov equation in the whole space preserves the decay properties of continuous solutions as t -> 8, independently of the mesh-size parameters. This is a manifestation of the property of numerical hypo-coercivity, and it holds both for semi-discrete and fully discrete approximations. The method of proof is based on the energy methods developed by Herau and Villani, employing well-balanced Lyapunov functionals mixing different energies, suitably weighted and equilibrated by multiplicative powers in time. The decreasing character of this Lyapunov functional leads to the optimal decay of the L-2-norms of solutions and partial derivatives, which are of different order because of the anisotropy of the model.File | Dimensione | Formato | |
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