Novel well-balanced continuous interior penalty stabilizations

TL;DR

The paper develops and analyzes high-order, well-balanced CIP stabilizations for the Shallow Water equations within a RD/CG finite element framework, paired with a fully explicit bDeCu time integration to yield an arbitrary-order, mass-matrix-free scheme. It introduces two well-balanced space discretizations (WB-$\text{HS}$ and WB-$\text{GF}$) and four CIP stabilizations (jt, je, jr, jg), all designed to preserve the lake-at-rest state exactly and, for jr and jg, to handle more general steady states. Numerical results in one dimension (and select 2D tests) demonstrate exact well-balancing for lake-at-rest, arbitrary high-order convergence on smooth solutions, and accurate evolution of small perturbations, with jr and jg offering superior handling of moving equilibria and general steady states. The work highlights the method’s potential for extending to other hyperbolic systems, including the Euler equations with gravity, and provides a robust, high-order, explicit framework for WB SW simulations on unstructured meshes.

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.

Figures (9)

• Figure 1: Sketch of the original DeC method on the left and of the modified one on the right
• Figure 2: Smooth steady states
• Figure 3: Convergence analysis: supercritical with B4, subcritical with PGL4 and transcritical with P3. $L^1$ error on $H$ in continuous line, on $q$ in dashed line
• Figure 4: Convergence analysis: best performing settings for basis functions with different degrees. Supercritical with B$n$, subcritical with PGL$n$ and transcritical with P$n$. $L^1$ error on $H$ in continuous line, on $q$ in dashed line
• Figure 5: Convergence analysis: comparison between B3, PGL3 and P3 with the best performing settings. $L^1$ error on $H$ in continuous line, on $q$ in dashed line

Theorems & Definitions (20)

• Remark 3.1: Generalization to a multidimensional setting
• Remark 4.1
• Proposition 4.1
• Remark 4.2: Arbitrary high order stabilizations
• Remark 4.3: On the conservation property of WB-HS
• Proposition 4.2
• Remark 4.4: Local interpolation
• Remark 4.5: Global flux quadrature
• Proposition 4.3
• proof