Versatile anisotropic mesh adaptation methodology applied to pure quantity of interest error estimator. Steady,laminar incompressible flow

We introduce a new flexible mesh adaptation approach to efficiently compute a quantity of interest by the finite element method. Efficiently, we mean that the method provides an evaluation of that quantity up to a predetermined accuracy at a lower computational cost than other classical methods. The central pillar of the method is our scalar error estimator based on sensitivities of the quantity of interest to the residuals. These sensitivities result from the computation of a continuous adjoint problem. The mesh adaptation strategy can drive anisotropic mesh adaptation from a general scalar error contribution of each element. The full potential of our error estimator is then reached. The proposed method is validated by evaluating the lift, the drag, and the hydraulic losses on a 2D benchmark case: the flow around a cylinder at a Reynolds number of 20.     

An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection-diffusion equations

In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second-order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.     

An improved depth-averaged nonhydrostatic shallow water model with quadratic pressure approximation

Phase-resolved information is necessary for many coastal wave problems, for example, for the wave conditions in the vicinity of harbor structures. Two-dimensional (2D) depth-averaging shallow water models are commonly used to obtain a phase-resolved solution near the coast. These models are in general more computationally effective compared with computational fluid dynamics software and will be even more capable if equipped with a parallelized code. In the current article, a 2D wave model solving the depth-averaged continuity equation and the Euler equations is implemented in the open-source hydrodynamic code REEF3D. The model is based on a nonhydrostatic extension and a quadratic vertical pressure profile assumption, which provides a better approximation of the frequency dispersion. It is the first model of its kind to employ high-order discretization schemes and to be fully parallelized following the domain decomposition strategy. Wave generation and absorption are achieved with a relaxation method. The simulations of nonlinear long wave propagations and transformations over nonconstant bathymetries are presented. The results are compared with benchmark wave propagation cases. A large-scale wave propagation simulation over realistic irregular topography is shown to demonstrate the model's capability of solving operational large-scale problems.     

Existence of multiple and sign-changing solutions for a second-order nonlinear functional difference equation with periodic coefficients

In the present paper, we apply the method of invariant sets of descending flow to establish a series of criteria to ensure that a second-order nonlinear functional difference equation with periodic boundary conditions possesses at least one trivial solution and three nontrivial solutions. These nontrivial solutions consist of sign-changing solutions, positive solutions and negative solutions. Moreover, as an application of our theoretical results, an example is elaborated. Our results generalize and improve some existing ones.     

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.