Novel well-balanced continuous interior penalty stabilizations

TL;DR

This paper introduces novel high-order continuous interior penalty stabilizations for the Shallow Water equations, ensuring exact preservation of lake at rest states and capturing small perturbations with high accuracy in a fully explicit finite element framework.

Contribution

The work presents new CIP stabilizations that guarantee well-balanced properties and high-order accuracy, including for general steady states without explicit solutions.

Findings

Successfully preserves lake at rest steady state.

Achieves high-order accuracy and superconvergence.

Effective for general steady states without explicit solutions.

Abstract

In this work, the high order accuracy and the well-balanced (WB) properties of some novel continuous interior penalty (CIP) stabilizations for the Shallow Water (SW) equations are investigated. The underlying arbitrary high order numerical framework is given by a Residual Distribution (RD)/continuous Galerkin (CG) finite element method (FEM) setting for the space discretization coupled with a Deferred Correction (DeC) time integration, to have a fully-explicit scheme. If, on the one hand, the introduced CIP stabilizations are all specifically designed to guarantee the exact preservation of the lake at rest steady state, on the other hand, some of them make use of general structures to tackle the preservation of general steady states, whose explicit analytical expression is not known. Several basis functions have been considered in the numerical experiments and, in all cases, the numerical results confirm the high order accuracy and the ability of the novel stabilizations to exactly preserve the lake at rest steady state and to capture small perturbations of such equilibrium. Moreover, some of them, based on the notions of space residual and global flux, have shown very good performances and superconvergences in the context of general steady solutions not known in closed-form. Many elements introduced here can be extended to other hyperbolic systems, e.g., to the Euler equations with gravity.