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Numerical methods that approximate the solution of the Vlasov-Poisson equation by a low-rank representation have been considered recently. These methods can be extremely effective from a computational point of view, but contrary to most semi-Lagrangian or Eulerian Vlasov solvers, they do not conserve mass and momentum, neither globally nor in respecting the corresponding local conservation laws. This can be a significant limitation for intermediate and long time integration. In this paper we propose a numerical algorithm that overcomes some of these difficulties and demonstrate its utility by presenting numerical simulations. [ABSTRACT FROM AUTHOR]

Finding metastable sets as dominant structures of Markov processes has been shown to be especially useful in modeling interesting slow dynamics of various real world complex processes. Furthermore, coarse graining of such processes based on their dominant structures leads to better understanding and dimension reduction of observed systems. However, in many cases, e.g., for nonreversible Markov processes, dominant structures are often not formed by metastable sets but by important cycles or a mixture of both. This paper aims at understanding and identifying these different types of dominant structures for reversible as well as nonreversible ergodic Markov processes. Our algorithmic approach generalizes spectral-based methods for reversible processes by using Schur decomposition techniques which can also tackle nonreversible cases. We illustrate the mathematical construction of our new approach by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we discuss discrete conservation laws for diffusion equations over triangular surface meshes from the viewpoint of duality. Conservation laws are very important for us to model physical phenomenon on curved spaces. The key idea of our method is to use the concept of local dual meshes, which are determined by the centroid points of the triangles in the original meshes. The Green formula will provide us a natural way to give a discrete approximation of the Laplace-Beltrami operators on functions over curved regular surfaces. Then, we shall show that discrete conservation laws are fulfilled on the dual meshes for our algorithm. Moreover, our discrete Laplace-Beltrami operators have the local property: it only involves the information on the 1-ring of neighboring vertices. Note that this approach is quite different from the finite element/difference method, used very often in the filed of numerical analysis. Some convergence problems will also be discussed. Numerical simulations are given to support theses results. [ABSTRACT FROM AUTHOR]