Porretta A., Zuazua E. Numerical hypocoercivity for the Kolmogorov equation APPL MATH COMPUT, AMS. Vol. 86. No. 303 (2017), pp. 97–119, DOI: 10.1090/mcom/3157
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 \to \infty $, 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.