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.
翻译:在本文中,我们开发了一种平衡的无振动不连续的Galerkin(OFDG)方法,用非倾斜的底部地形来解决浅水方程,所建办法的一个显著特征是完全平衡的财产,它完全保持了静水平衡的解决方案,直至机器错误。另一个特征是非振动性财产,在数字模拟中,当出现某种震动不连续时,这一点非常重要。为了控制虚假的振动,我们构建了一种OFDG方法,对[Y. Xing和C. W. Shu、CICP、1,2006年1,100-134.]中提议的现有平衡的DG方案作了额外的分隔期。在仔细构建了悬浮时,拟议的方法既实现了平衡性财产,又实现了非凝固性财产,同时没有损害任何准确性的顺序。我们还为维护平衡性财产提出了详细的构建程序和理论分析。广泛的数字实验包括一维和二维空间,表明拟议的方法拥有理想的准确性能。