Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely remains the status quo for scientists and engineers focusing on applying simulation tools to specific problems in practice. We attribute this growing technical gap to the increasing complexity and knowledge required to pick and assemble state-of-the-art methods. Thus, with this work, we initiate an effort to build a common taxonomy for the most popular grid-based approximation schemes to draw comparisons regarding accuracy and computational efficiency. We then build upon this foundation and introduce a method to systematically guide an application expert through classifying a given PDE problem setting and identifying a suitable numerical scheme. Great care is taken to ensure that making a choice this way is unambiguous, i.e. the goal is to obtain a clear and reproducible recommendation. Our method not only helps to identify and assemble suitable schemes but enables the unique combination of multiple methods on a per-field basis. We demonstrate this process and its effectiveness using different model problems, each comparing the resulting numerical scheme from our method with the next best choice. For both the Allen Cahn and advection equations, we show that substantial computational gains can be attained for the recommended numerical methods regarding accuracy and efficiency. Lastly, we outline how one can systematically analyze and classify a coupled multiphysics problem of considerable complexity with 6 different unknown quantities, yielding an efficient, mixed discretization that in configuration compares well to high-performance implementations from the literature.

翻译：近年来，针对偏微分方程的高性能求解器取得了显著进展，相较工业标准工具可带来潜在效率提升。然而，对于聚焦于具体问题应用仿真工具的科学家和工程师而言，后者仍基本保持现状。我们将这一日益扩大的技术差距归因于选取与组装先进方法所需的知识与复杂度持续攀升。为此，本研究致力于构建主流网格近似格式的通用分类体系，以便对比其精度与计算效率。在此基础之上，我们提出一种系统化方法，引导应用专家对特定偏微分方程问题设定进行分类，进而识别出合适的数值格式。我们特别注重确保该选择过程无歧义性——即目标在于获得清晰且可复现的推荐方案。该方法不仅能帮助识别与组装恰当格式，更能实现基于物理场维度的多方法独特组合。通过不同模型问题的验证，我们比较了本方法推荐的数值格式与次优选择的效果。针对Allen-Cahn方程与对流方程，研究显示推荐数值方法在精度与效率方面均能取得显著计算收益。最后，我们阐述了如何系统分析与分类包含6个不同未知量的高复杂度耦合多物理场问题，所得高效混合离散方案在配置上可与文献中的高性能实现相媲美。