The Hybrid Discontinuous Galerkin method for elliptic problems and applications in vertical ocean-slice modeling

TL;DR

This paper introduces a Hybrid Discontinuous Galerkin method tailored for elliptic problems, demonstrating its efficiency and accuracy in vertical ocean-slice modeling with density effects, offering a promising approach for high-order coastal ocean simulations.

Contribution

It develops a hybridized DG method for elliptic problems, reducing computational costs and enabling high-order accurate solutions in ocean modeling applications.

Findings

High-order implementation yields fast, accurate approximations on coarse meshes.

The method effectively solves Poisson problems in ocean-slice models.

Hybridization reduces computational costs compared to classical DG methods.

Abstract

Element Method. The Finite Volume Method guarantees local and global mass conservation. A property not satisfied by the Finite Volume Method. On the down side, the Finite Volume Method requires non trivial modifications to attain high order approximations unlike the Finite Volume Method. It has been contended that the Discontinuous Galerkin Method, locally conservative and high order, is a natural progression for Coastal Ocean Modeling. Consequently, as a primer we consider the vertical ocean-slice model with the inclusion of density effects. To solve these non steady Partial Differential Equations, we develop a pressure projection method for solution. We propose a Hybridized Discontinuous Galerkin solution for the required Poisson Problem in each time step. The purpose, is to reduce the computational cost of classical applications of the Discontinuous Galerkin method. The Hybridized Discontinuous Galerkin method is first presented as a general elliptic problem solver. It is shown that a high order implementation yields fast and accurate approximations on coarse meshes.