The accurate representation and prediction of physical phenomena through numerical computer codes remains to be a vast and intricate interdisciplinary topic of research. Especially within the last decades, there has been a considerable push toward high performance numerical schemes to solve partial differential equations (PDEs) from the applied mathematics and numerics community. The resulting landscape of choices regarding numerical schemes for a given system of PDEs can thus easily appear daunting for an application expert that is familiar with the relevant physics, but not necessarily with the numerics. Bespoke high performance schemes in particular pose a substantial hurdle for domain scientists regarding their theory and implementation. Here, we propose a unifying scheme for grid based approximation methods to address this issue. We introduce some well defined restrictions to systematically guide an application expert through the process of classifying a given multiphysics problem, identifying suitable numerical schemes and implementing them. We introduce a fixed set of input parameters, amongst them for example the governing equations and the hardware configuration. This method not only helps to identify and assemble suitable schemes, but enables the unique combination of multiple methods on a per field basis. We exemplarily demonstrate this process and its effectiveness using different approaches and systematically show how one should exploit some given properties of a PDE problem to arrive at an efficient compound discretisation.

翻译：通过数值计算机代码对物理现象进行精确表征与预测，始终是一个广泛而复杂的跨学科研究课题。尤其是在近几十年来，应用数学与数值计算领域在求解偏微分方程的高性能数值方案方面取得了显著进展。对于熟悉相关物理机理但未必精通数值计算的领域专家而言，面对给定偏微分方程系统的众多数值方案选择时，往往容易感到无所适从。特别是定制化高性能方案的理论基础与实现方法，对领域科学家构成了重大障碍。本文提出了一种基于网格的近似方法统一框架以解决该问题。我们引入若干明确定义的约束条件，系统引导领域专家完成多物理场问题分类、适用数值方案识别及实现的全过程。通过设定一组固定输入参数（例如控制方程与硬件配置），该方法不仅有助于识别和组装合适的数值方案，还能实现基于不同物理场的多方法独特组合。我们采用不同方法示例性展示了该过程的实施与有效性，并系统论证了如何利用偏微分方程问题的固有特性来构建高效的复合离散化方案。