A Systematic Approach to Adaptive Mesh Refinement for Computational Electrodynamics

• Published in: IEEE journal on multiscale and multiphysics computational techniques Vol. 8; pp. 82 - 96
• Main Authors: Balsara, Dinshaw S., Sarris, Costas D.
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive mesh refinement; Algorithms; computational electromagnetics; Control stability; Divergence; Electric fields; Electrodynamics; Finite difference methods; Grid refinement (mathematics); Magnetic flux; Magnetic flux density; Mathematical models; Maxwell equations; Maxwell's equations; Prolongation; Synchronization; Time synchronization; Time-domain analysis
• ISSN: 2379-8815; 2379-8815
• DOI: 10.1109/JMMCT.2022.3233944

There is a great need to solve CED problems on adaptive meshes; referred to here as AMR-CED. The problem was deemed to be susceptible to "long-term instability" and parameterized methods have been used to control the instability. In this paper, we present a new class of AMR-CED methods that are free of this instability because they are based on a more careful understanding of the constraints in Maxwell's equations and their preservation on a single control volume. The important building blocks of these new methods are: 1) Timestep sub-cycling of finer child meshes relative to parent meshes. 2) Restriction of fine mesh facial data to coarser meshes when the two meshes are synchronized in time. 3) Divergence constraint-preserving prolongation of the coarse mesh solution to newly built fine meshes or to the ghost zones of pre-existing fine meshes. 4) Electric and magnetic field intensity-correction strategy at fine-coarse interfaces. Using examples, we show that the resulting AMR-CED algorithm is free of "long-term instability". Unlike previous methods, there are no adjustable parameters. The method is inherently stable because a strict algorithmic consistency is applied at all levels in the AMR mesh hierarchy. We also show that the method preserves order of accuracy, so that high order methods for AMR-CED are indeed possible.