论文标题
确保通过不连续的Galerkin有限元方法确保“均衡”的浅水流：以最低顺序发行
Ensuring 'well-balanced' shallow water flows via a discontinuous Galerkin finite element method: issues at lowest order
论文作者
论文摘要
Rhebergen等人开发的不连续的Galerkin有限元法（DGFEM）。 （2008年）提供了一种可靠的方法来求解非保守双曲线部分偏微分方程的系统，但是，正如我们在这里所显示的那样，并未令人满意地处理以最低顺序（所谓的DG0或等效有限量）中浅水流中的地形处理。特别是，在不同地形上离散的一维浅水方程的太空-DG0的数值解决方案并不是真正的“均衡”。如果在数值解决方案中满足微不足道的稳态，则数值方案是合理的。在浅水方程的情况下，初始化的休息流程应始终保持静止。虽然自由表面的高度和动量分别保持恒定和零，这表明该方案确实是平衡的，但流体深度和地形随时间而发展。这既是不良的且不理想的，导致对流体深度的数值解，因此从预测建模的角度来看是一个关注的问题。我们在分析和数值上暴露了这个不令人满意的问题，并指出了一种解决方案，该解决方案将非保守产品的DGFEM公式与快速稳定的有限量有限体积方法结合在一起。该联合方案绕过了有问题的问题，并成功地集成了非保守双曲线浅水型模型，并以最低顺序的地形变化。我们简要讨论了对均衡方案的定义的含义，并突出了可能不需要高阶方案时的应用，这为我们的发现提供了进一步的价值。
The discontinuous Galerkin finite element method (DGFEM) developed by Rhebergen et al. (2008) offers a robust method for solving systems of nonconservative hyperbolic partial differential equations but, as we show here, does not satisfactorily deal with topography in shallow water flows at lowest order (so-called DG0, or equivalently finite volume). In particular, numerical solutions of the space-DG0 discretised one-dimensional shallow water equations over varying topography are not truly `well-balanced'. A numerical scheme is well-balanced if trivial steady states are satisfied in the numerical solution; in the case of the shallow water equations, initialised rest flow should remain at rest for all times. Whilst the free-surface height and momentum remain constant and zero, respectively, suggesting that the scheme is indeed well-balanced, the fluid depth and topography evolve in time. This is both undesirable and unphysical, leading to incorrect numerical solutions for the fluid depth, and is thus a concern from a predictive modelling perspective. We expose this unsatisfactory issue, both analytically and numerically, and indicate a solution that combines the DGFEM formulation for nonconservative products with a fast and stable well-balanced finite-volume method. This combined scheme bypasses the offending issue and successfully integrates nonconservative hyperbolic shallow water-type models with varying topography at lowest order. We briefly discuss implications for the definition of a well-balanced scheme, and highlight applications when higher-order schemes may not be desired, which give further value to our finding beyond its exposure alone.
