We develop a new moving-water equilibria preserving numerical scheme for the Saint-Venant system. The new scheme is designed in two major steps. First, the geometric source term is incorporated into the discharge flux, which results in a hyperbolic system with a global flux. Second, the discharge equation is relaxed so that the nonlinearity is moved into the stiff right-hand side of the added auxiliary equation. The main advantages of the new scheme are that (i) no special treatment of the geometric source term is required, and (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables, as it must be done in the existing alternative methods. We also develop a hybrid numerical flux, which helps to handle various flow regimes in a stable manner. Several numerical experiments are performed to verify that the proposed scheme is capable of exactly preserving general moving-water steady states and accurately capturing their small perturbations.

Wind power is an increasingly popular renewable energy. In the design process of the wind turbine blade, the accurate aerodynamic simulation is important. In the past, most of the wind turbine simulations were carried out with some low fidelity methods, such as the blade element momentum method [9]. Recently, with the rapid development of the supercomputers, high fidelity simulations based on 3D unsteady Navier-Stokes (N-S) equations become more popular. For example, Sorensen et al. studied the 3D wind turbine rotor using the Reynolds-Averaged Navier-Stokes (RANS) framework where a finite volume method and a semi-implicit method are used for the spatial and temporal discretization, respectively [17]. Bazilevs et al. investigated the aerodynamic of the NREL 5MW offshore baseline wind turbine rotor using large eddy simulation built with a deforming-spatial-domain/stabilized space-time

Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

Convergence of a second-order shock-capturing scheme for the system of isentropic gas dynamics with $ {L^\infty }$ initial data is established by analyzing the entropy dissipation measures. This scheme is modified from the classical MUSCL scheme to treat the vacuum problem in gas fluids and to capture local entropy near shock waves. Convergence of this scheme for the piston problem is also discussed.

In the paper, we consider a continuation approach for the binary quadratic program (BQP) based on a class of NCP-functions. More specifically, we recast the BQP as an equivalent minimization and then seeks its global minimizer via a global continuation method. Such approach had been considered in [11] which is based on the FischerBurmeister function. We investigate this continuation approach again by using a more general function, called the generalized FischerBurmeister function. However, the theoretical background for such extension can not be easily carried over. Indeed, it needs some subtle analysis.