An Application Driven Method for Assembling Numerical Schemes for the Solution of Complex Multiphysics Problems

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.