A stable and accurate control-volume technique based on integrated radial basis function networks for fluid-flow problems

Radial basis function networks (RBFNs) have been widely used in solving partial differential equations as they are able to provide fast convergence. Integrated RBFNs have the ability to avoid the problem of reduced convergence-rate caused by differentiation. This paper is concerned with the use of integrated RBFNs in the context of control-volume discretisations for the simulation of fluid-flow problems. Special attention is given to (i) the development of a stable high-order upwind scheme for the convection term and (ii) the development of a local high-order approximation scheme for the diffusion term. Benchmark problems including the lid-driven triangular-cavity flow are employed to validate the present technique. Accurate results at high values of the Reynolds number are obtained using relatively-coarse grids

Explicit symmetric Runge-Kutta-Nyström methods for parallel computers

AbstractIn this paper, we are concerned with parallel predictor-corrector (PC) iteration of Runge-Kutta-Nyström (RKN) methods in P(EC)mE mode for integrating initial value problems for the special second-order equation y″(t) = f(y(t)). We consider symmetric Runge-Kutta-Nyström (SRKN) corrector methods based on direct collocation techniques which optimize the rate of convergence of the PC iteration process. The resulting PISRKN methods (parallel iterated SRKN methods) are shown to be much more efficient when they are compared to the PC iteration process applied to the Gauss-Legendre RKN correctors

Explicit Parallel Two-Step Runge-Kutta-Nyström Methods

AbstractThe aim of this paper is to design a class of two-step Runge-Kutta-Nyström methods of arbitrarily high order for the special second-order equation y″(t) = f(y(t)), for use on parallel computers. Starting with an s-stage implicit two-step Runge-Kutta-Nyström method of order p with k = p2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P(EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta-Nyström method that has order p for all m and that requires k(m + 1) right-hand side evaluations per step of which each k evaluation can be computed in parallel. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature

QCD4_4 Glueball Masses from AdS-6 Black Hole Description

By using the generalized version of gauge/gravity correspondence, we study the mass spectra of several typical QCD$_4$ glueballs in the framework of AdS$_6$ black hole metric of Einstein gravity theory. The obtained glueball mass spectra are numerically in agreement with those from the AdS$7 \times S^4$ black hole metric of the 11-dimensional supergravity.Comment: 10 pages, references updated and minor change

Linearized Asymptotic Stability for Fractional Differential Equations

We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector \{\lambda \in \C : |\arg \lambda| > \frac{\alpha \pi}{2}\} where $\alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable