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

TL;DR

This paper introduces a systematic, application-driven method for selecting and assembling numerical schemes for complex multiphysics PDE problems, improving accuracy and efficiency over traditional industry tools.

Contribution

It provides a taxonomy for grid-based schemes and a reproducible method to classify problems and recommend suitable numerical approaches, including combining multiple methods.

Findings

Substantial computational gains for Allen Cahn and advection equations.

Effective classification and scheme selection for complex multiphysics problems.

Comparable performance to high-end literature implementations.

Abstract

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.