A Very High-Order Accurate Staggered Finite Volume Scheme for the Stationary Incompressible Navier–Stokes and Euler Equations on Unstructured Meshes

International audienceWe propose a sixth-order staggered finite volume scheme based on polynomial reconstructions to achieve high accurate numerical solutions for the incompressible Navier-Stokes and Euler equations. The scheme is equipped with a fixed-point algorithm with solution relaxation to speed-up the convergence and reduce the computation time. Numerical tests are provided to assess the effectiveness of the method to achieve up to sixth-order con-2 Ricardo Costa et al. vergence rates. Simulations for the benchmark lid-driven cavity problem are also provided to highlight the benefit of the proposed high-order scheme

Finite element methods for the parabolic equation with interfaces

We consider the standard parabolic approximation of underwater acoustics in a medium consisting of fluid layers with depth- and range-dependent speed of sound and dissipation coefficients. In the case of horizontal layers we construct two types of finite element discretizations in the depth variable which treat the interfaces in different ways. We couple these methods with conservative Crank-Nicolson and implicit Runge-Kutta schemes in the range variable and analyze the stability and convergence of the resulting fully discrete methods. Using a change of variable technique, we show how these schemes may be extended to treat interfaces with range-dependent topography; we also couple them with a nonlocal (’impedance’) bottom boundary condition. The methods are applied to several benchmark problems in the literature and their results are compared to those obtained from standard numerical codes

A finite element code for the numerical solution of the Helmholtz equation in axially symmetric waveguides with interfaces

We consider the Helmholtz equation in an axisymmetric cylindrical waveguide consisting of fluid layers overlying a rigid bottom. The medium may have range-dependent speed of sound and interface and bottom topography in the interior nonhomogeneous part of the waveguide, while in the far-field the interfaces and bottom are assumed to be horizontal and the problem separable. A nonlocal boundary condition based on the DtN map of the exterior problem is posed at the far-field artificial boundary. The problem is discretized by a standard Galerkin/finite element method and the resulting numerical scheme is implemented in a Fortran code that is interfaced with general mesh generation programs from the MODULEF finite element library and iterative linear solvers from QMRPACK. The code is tested on several small scale examples of acoustic propagation and scattering in the sea and its results are found to compare well with those of COUPLE