Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier--Stokes equations

To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements of accuracy are considered for a fully explicit particle method for incompressible Navier--Stokes equations. The explicit particle method is based on a penalty problem, which converges theoretically to the incompressible Navier--Stokes equations, and is discretized in space by generalized approximate operators defined as a wider class of approximate operators than those of the smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods. By considering an analytical derivation of the explicit particle method and truncation error estimates of the generalized approximate operators, sufficient conditions of convergence are conjectured.Under these conditions, the convergence of the explicit particle method is confirmed by numerically comparing errors between exact and approximate solutions. Moreover, by focusing on the truncation errors of the generalized approximate operators, an optimal weight function is derived by reducing the truncation errors over general particle distributions. The effectiveness of the generalized approximate operators with the optimal weight functions is confirmed using numerical results of truncation errors and driven cavity flow. As an application for flow problems with free surface effects, the explicit particle method is applied to a dam break flow.Comment: 27 pages, 13 figure

An extension of the entropic chaos degree and its positive effect

The Lyapunov exponent is used to quantify the chaos of a dynamical system, by characterizing the exponential sensitivity of an initial point on the dynamical system. However, we cannot directly compute the Lyapunov exponent for a dynamical system without its dynamical equation, although some estimation methods do exist. Information dynamics introduces the entropic chaos degree to measure the strength of chaos of the dynamical system. The entropic chaos degree can be used to compute the strength of chaos with a practical time series. It may seem like a kind of fnite space Kolmogorov-Sinai entropy, which then indicates the relation between the entropic chaos degree and the Lyapunov exponent. In this paper, we attempt to extend the defnition of the entropic chaos degree on a d-dimensional Euclidean space to improve the ability to measure the stength of chaos of the dynamical system and show several relations between the extended entropic chaos degree and the Lyapunov exponen