Finite element approximation of the modified Boussinesq equations using a stabilized formulation

Abstract

SUMMARYIn this work, we present a finite element model to approximate the modified Boussinesq equations. The objective is to deal with the major problem associated with this system of equations, namely, the need to use stable velocity-depth interpolations, which can be overcome by the use of a stabilization technique. The one described in this paper is based on the splitting of the unknowns into their finite element component and the remainder, which we call the subgrid scale. We also discuss the treatment of high-order derivatives of the mathematical model and describe the time integration scheme.

A Stabilized Finite Element Method for the Mixed Wave Equation in an ALE Framework With Application to Diphthong Production

“…The Galerkin FEM solution for the latter is known to suffer from strong high-frequency oscillations [29,30] when using equal interpolations for the gravity waves depth and velocity, but these can be overcome if one relies on stabilization strategies [31,32]. These stabilization methods also serve to deal with the convective nature of the terms arising from the ALE formulation.…”

“…This will avoid locking effects when using the same interpolations for the acoustic pressure and velocity, as well as accounting for the treatment of the convective terms arising from the ALE formulation. To do so we will closely follow the main lines in previous works on stabilization of the wave equation in mixed form [4] and on the modified Boussinesq equations [32].…”

“…Numerical experiments show that the Galerkin FEM solution for the latter exhibits strong high-frequency oscillations if equal interpolations are used for the gravity waves depth and velocity [29,30]. Again these oscillations can be overcome if one draws on stabilization strategies [31,32]. In this work we will present a subgrid scale stabilization FEM approach tailored to the ALE wave equation in mixed form, following the main lines in [32].…”

“…Again these oscillations can be overcome if one draws on stabilization strategies [31,32]. In this work we will present a subgrid scale stabilization FEM approach tailored to the ALE wave equation in mixed form, following the main lines in [32]. It will be shown that the proposed strategy will allow us to use equal interpolations for the acoustic pressure and velocity, and prevent the FEM solution to blow up.…”

Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling

“…In the following subsection, this approximation will be used for the spatial operator arising from the linearization of the left-hand-side in (15)- (17), now u being composed of the velocity components, the pressure and the temperature. In order to obtain (19), we use a heuristic Fourier analysis, introduced in [6] and extended in [8], for example. Let us denote the Fourier transform by .…”