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

TL;DR

The work tackles the challenge of choosing stable, efficient grid-based discretizations for complex multiphysics PDEs by introducing an application-driven taxonomy and a unified assembly framework. It shows how the Discontinuous Galerkin baseline can reproduce and interoperate with Continuous Galerkin, Finite Difference, and Finite Volume methods, enabling per-field mixed discretizations. Through model problems including Allen-Cahn, two-phase advection, and Laser Powder Bed Fusion, the authors demonstrate substantial computational gains and reproducible per-field discretization decisions guided by problem class, domain geometry, and hardware. The framework promises practical benefits for practitioners by enabling stable, high-performance simulations without extensive method-specific tuning, while outlining future extensions to finer PDE classes and additional finite element variants.

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.

Figures (13)

• Figure S1: Coupling of global DoFs in the Continuous Galerkin (a) versus Discontinuous Galerkin FEM (b), both of first order. In the latter case, DoFs are entirely local to the cell and thus receive no contribution from neighboring cells. Weak coupling is only introduced by the additional numerical flux. Coupled DoFs are drawn in identical colors.
• Figure S2: Comparison of the nodal nature of the FDM (a) versus the cell-wise assembly used in the CG FEM (b) for an identical, cartesian triangulation with 9 nodes. Both methods are formulated as first-order approximations. Red-colored nodes signify the points where the PDE is evaluated. Contributions to this node are taken from blue nodes, whereas white nodes from no contribution.
• Figure S3: Comparison of FVM (a) versus DGM (b) on an identical quadrilateral triangulation. Coupled DoFs are marked in identical colors. For the FVM, one must first reconstruct the values of the DoFs at the mesh facets to then compute the numerical flux.
• Figure S4: Classification of PDEs up to second order by qualitative nature and types following bitsadze1988some.
• Figure S5: Graphic summary of the proposed process for choosing appropriate numerical schemes. Inputs (I) are given by purple trapezoids, decision points (D) by white diamonds and processes (P) by orange rectangles. Processes with additional vertical bars denote more complex processes and have references to their respective sections. Results are shown in green trapezoids.