In this work we propose a novel strategy to define high-order fully well-balanced Lagrange-projection finite volume solvers for balance laws. In particular, we focus on the 1D shallow water system as it is a reference system of balance laws with non-trivial stationary solutions. Nevertheless, the strategy proposed here could be extended to other interesting balance laws. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions. Following [M.J. Castro et al., in Handbook of Numerical Analysis, 18:131175, 2017], we exploit the idea of using a high-order well-balanced reconstruction operator for the Lagrangian step. Nevertheless, this is not enough to achieve well-balanced high-order during the projection step. We propose here a new projection step that overcomes this difficulty and that reduces to the standard one in case of conservation laws. Finally, some numerical experiments illustrate the good behaviour of the scheme.

中文翻译：

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浅水高阶完全平衡拉格朗日投影方案

在这项工作中，我们提出了一种新颖的策略来为平衡定律定义高阶完全平衡的Lagrange投影有限体积求解器。特别是，我们将重点放在一维浅水系统上，因为它是具有非平凡固定解的平衡律的参考系统。但是，这里提出的策略可以扩展到其他有趣的平衡律。完全平衡，意味着该方案能够保留平稳的平稳解。继[MJ Castro等人，《数值分析手册》，18：131175，2017]，我们利用在Lagrangian步骤中使用高阶均衡平衡算子的想法。然而，这还不足以在投影步骤中实现均衡的高阶。我们在这里提出了一个新的投影步骤，该步骤可以克服这一困难，并且在遵守保护法的情况下可以降低到标准水平。最后，一些数值实验说明了该方案的良好性能。