In this paper, we develop a novel well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) numerical method for solving the two-dimensional shallow water equations with temperature gradients on unstructured triangular meshes. The proposed PAMPA method use a globally continuous representation of the variables, with degree of freedoms (DoFs) consisting of point values on the edges and average values within each triangular element. The update of cell averages is carried out using a conservative form of the partial differential equations (PDEs), while the update of point values -- unconstrained by local conservation -- follows a non-conservative formulation. The powerful PAMPA framework offers great flexibility in the choice of variables for the non-conservative form, including conservative variables, primitive variables, and other possible sets of variables. In order to preserve a wider class of steady-state solutions, we introduce pressure-momentum-temperature variables instead of using the standard conservative or primitive ones. By utilizing these new variables and the associated non-conservative form, along with adopting suitable Gaussian quadrature rules in the discretization of conservative form, we prove that this new class of schemes is well-balanced for both ``lake at rest'' and isobaric steady states. We validate the performance of the proposed well-balanced PAMPA method through a series of numerical experiments, demonstrating their high-order accuracy, well-balancedness, and robustness.

翻译：本文针对非结构三角形网格上具有温度梯度的二维浅水方程，提出了一种新颖的均衡点-平均-矩多项式插值（PAMPA）数值方法。所提出的PAMPA方法采用变量的全局连续表示，其自由度由边上的点值和每个三角形单元内的平均值构成。单元平均值的更新采用偏微分方程的守恒形式进行，而点值的更新——不受局部守恒约束——则遵循非守恒形式。强大的PAMPA框架为选择非守恒形式的变量提供了极大的灵活性，包括守恒变量、原始变量以及其他可能的变量集。为了保持更广泛类别的稳态解，我们引入了压力-动量-温度变量，而非使用标准的守恒变量或原始变量。通过利用这些新变量及其相关的非守恒形式，并在守恒形式的离散化中采用合适的高斯求积法则，我们证明了这类新格式对于"静止湖泊"和等压稳态均具有均衡性。我们通过一系列数值实验验证了所提出的均衡PAMPA方法的性能，展示了其高阶精度、均衡性和鲁棒性。