The choice of the mesh refinement threshold is a long-standing and problem-dependent issue in adaptive mesh refinement (AMR) methods, often constituting a critical weakness due to the need for ad hoc user calibration. In this work, we propose an automatic mesh refinement thresholding strategy designed to detect all relevant regions requiring refinement without any hand-tuned parameter. The approach is implemented within a block-based adaptive mesh refinement (BB-AMR) finite-volume framework for the multidimensional Saint-Venant system, enabling efficient mesh handling and straightforward parallelisation, while remaining compatible with other numerical discretisations. The automatic mesh refinement threshold (AMRT) is constructed from the decreasing rearrangement (distribution) function of the refinement indicator, enabling a robust identification of dynamically significant scales. This strategy provides a reliable balance between numerical accuracy and computational cost, while removing the dependence on user-defined thresholds. The resulting AMR method is robust, efficient, and nearly parameter-free. Its performance is assessed through a series of demanding numerical benchmarks related to tsunami propagation, including wave propagation over complex bathymetries and three-dimensional run-up configurations, where the method demonstrates both accuracy and significant computational savings.
Ersoy et al. (Mon,) studied this question.