A well-balanced oscillation-free discontinuous Galerkin method for shallow water equations

TL;DR

This paper introduces a novel oscillation-free discontinuous Galerkin method for shallow water equations that maintains hydrostatic equilibrium and prevents spurious oscillations, ensuring high accuracy in complex topographies.

Contribution

It presents a new well-balanced, oscillation-free DG scheme with an added damping term, achieving both properties without losing accuracy.

Findings

Preserves hydrostatic equilibrium exactly.

Effectively suppresses spurious oscillations near shocks.

Maintains high-order accuracy in numerical simulations.

Abstract

In this paper, we develop a well-balanced oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. One notable feature of the constructed scheme is the well-balanced property, which preserves exactly the hydrostatic equilibrium solutions up to machine error. Another feature is the non-oscillatory property, which is very important in the numerical simulation when there exist some shock discontinuities. To control the spurious oscillations, we construct an OFDG method with an extra damping term to the existing well-balanced DG schemes proposed in [Y. Xing and C.-W. Shu, CICP, 1(2006), 100-134.]. With a careful construction of the damping term, the proposed method achieves both the well-balanced property and non-oscillatory property simultaneously without compromising any order of accuracy. We also present a detailed procedure for the construction and a theoretical analysis for the preservation of the well-balancedness property. Extensive numerical experiments including one- and two-dimensional space demonstrate that the proposed methods possess the desired properties without sacrificing any order of accuracy.