Abstract We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped ℙ 1 {\mathbb{P}_{1}} conforming and non-conforming finite element and on the hybrid finite volume method.

中文翻译：

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线性平流问题的梯度离散化方法

摘要 我们将最初为椭圆和抛物线偏微分方程设计的梯度离散化方法 (GDM) 应用于线性标量双曲方程的情况。这使得各种数值方案的同步设计和收敛分析成为可能，对应于已知的 GDM 方法，例如有限元（符合或不符合、标准或质量集总）、矩形或单纯网格上的有限体积，以及其他最近为一般多面网格开发的方法。该方案是中心型的，增加了线性或非线性数值扩散。我们通过基于质量集总 ℙ 1 {\mathbb{P}_{1}} 符合和非符合有限元以及混合有限体积方法的数值测试来补充收敛分析。