A Discontinuous Galerkin and Semismooth Newton Approach for the Numerical Solution of Bingham Flow with Variable Density

TL;DR

This paper introduces a novel numerical method combining discontinuous Galerkin and semismooth Newton techniques to effectively solve Bingham flow problems with variable density, ensuring stability and accuracy.

Contribution

It develops a new regularization and discretization approach for Bingham flow with variable density, integrating divergence-conforming and discontinuous Galerkin methods with a semismooth Newton solver.

Findings

The scheme is stable and convergent.

Numerical examples demonstrate effectiveness.

The method accurately captures flow features.

Abstract

This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of the divergence conforming and discontinuous Galerkin formulations to incorporate upwind discretizations to stabilize the formulation. The stability of the continuous problem and the full-discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully-discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented.