We present a novel approach for solving the shallow water equations using a discontinuous Galerkin spectral element method. The method we propose has three main features. First, it enjoys a discrete well-balanced property, in a spirit similar to the one of e.g. [20]. As in the reference, our scheme does not require any a-priori knowledge of the steady equilibrium, moreover it does not involve the explicit solution of any local auxiliary problem to approximate such equilibrium. The scheme is also arbitrarily high order, and verifies a continuous in time cell entropy equality. The latter becomes an inequality as soon as additional dissipation is added to the method. The method is constructed starting from a global flux approach in which an additional flux term is constructed as the primitive of the source. We show that, in the context of nodal spectral finite elements, this can be translated into a simple modification of the integral of the source term. We prove that, when using Gauss-Lobatto nodal finite elements this modified integration is equivalent at steady state to a high order Gauss collocation method applied to an ODE for the flux. This method is superconvergent at the collocation points, thus providing a discrete well-balanced property very similar in spirit to the one proposed in [20], albeit not needing the explicit computation of a local approximation of the steady state. To control the entropy production, we introduce artificial viscosity corrections at the cell level and incorporate them into the scheme. We provide theoretical and numerical characterizations of the accuracy and equilibrium preservation of these corrections. Through extensive numerical benchmarking, we validate our theoretical predictions, with considerable improvements in accuracy for steady states, as well as enhanced robustness for more complex scenarios

翻译：本文提出了一种采用间断伽辽金谱元法求解浅水方程的新方法。该方法具有三个主要特征。首先，它具备离散意义上的完全平衡性质，其思想类似于文献[20]的方法。与参考文献类似，本方案无需先验已知稳态平衡态，也无需通过显式求解任何局部辅助问题来逼近该平衡态。该方案同时具有任意高阶精度，并满足时间连续单元熵等式。当向该方法引入额外耗散时，该等式退化为不等式。该方法基于全局通量方法构建，其中通过源项的原函数构造一个附加通量项。我们证明，在节点谱有限元框架下，这可以转化为对源项积分的简单修正。当使用高斯-洛巴托节点有限元时，该修正积分在稳态时等价于对通量常微分方程应用的高阶高斯配置法。该方法在配置点具有超收敛特性，从而提供与文献[20]方法精神高度相似的离散完全平衡性质，但无需显式计算稳态的局部近似。为控制熵产生，我们在单元层面引入人工黏性修正并将其纳入方案中。我们从理论和数值两方面表征了这些修正的精度与平衡保持特性。通过大量数值基准测试，我们验证了理论预测，在稳态计算中实现了显著的精度提升，同时对更复杂工况展现出更强的鲁棒性。