A mass conservative scheme for fluid-structure interaction problems by the staggered discontinuous Galerkin method

TL;DR

This paper introduces a new mass conservative numerical scheme for fluid-structure interaction problems using the immersed boundary method combined with a staggered discontinuous Galerkin discretization to ensure divergence-free velocity and improved energy stability.

Contribution

The paper develops a novel mass conservative scheme employing staggered discontinuous Galerkin methods with divergence-free postprocessing for fluid-structure interaction simulations.

Findings

The scheme preserves mass conservation and divergence-free velocity.

Numerical results demonstrate improved energy stability.

The method achieves superconvergence in velocity computation.

Abstract

In this paper, we develop a new mass conservative numerical scheme for the simulations of a class of fluid-structure interaction problems. We will use the immersed boundary method to model the fluid-structure interaction, while the fluid flow is governed by the incompressible Navier-Stokes equations. The immersed boundary method is proven to be a successful scheme to model fluid-structure interactions. To ensure mass conservation, we will use the staggered discontinuous Galerkin method to discretize the incompressible Navier-Stokes equations. The staggered discontinuous Galerkin method is able to preserve the skew-symmetry of the convection term. In addition, by using a local postprocessing technique, the weakly divergence free velocity can be used to compute a new postprocessed velocity, which is exactly divergence free and has a superconvergence property. This strongly divergence free velocity field is the key to the mass conservation. Furthermore, energy stability is improved by the skew-symmetric discretization of the convection term. We will present several numerical results to show the performance of the method.