Spatiotemporal Structure-Preserving Schemes for Transport

Transport and magnetohydrodynamics (MHD) models often involve convection-dominated phenomena where standard discretizations fail, leading to spurious oscillations or excessive numerical dissipation. 

My research develops monotone spatiotemporal schemes that reinterpret time-dependent problems as higher-dimensional stationary ones. This approach preserves the discrete maximum principle, accurately resolves boundary layers, and produces stable, physically consistent simulations.

Four-Dimensional Spatiotemporal Formulations 

Motivated by earlier results, my current work develops four-dimensional spatiotemporal formulations for time-dependent transport and magnetohydrodynamics (MHD). By treating time as an additional spatial-like coordinate, inherently unsteady problems are recast as stationary problems in higher dimensions. This approach naturally requires the language of exterior calculus and unifies transport and MHD models within a single spatiotemporal equation. Importantly, the formulation inherently preserves fundamental physical laws, such as Gauss's law for magnetism (divergence-free condition) and the curl-free condition.  

Edge-Averaged Virtual Element (EAVE) Methods

Standard finite element methods often fail to preserve the maximum principle for convection-diffusion problems, resulting in spurious oscillations in convection-dominated regimes. To address this, my collaborators and I developed monotone polygonal numerical methods, introducing the edge-averaged virtual element (EAVE) scheme. This approach extends classical edge-averaging techniques to polygonal meshes, resulting in an M-matrix that ensures the discrete maximum principle and prevents non-physical oscillations. Beyond stability, the EAVE scheme achieves optimal convergence on complex geometries where irregular or polygonal meshes are essential. It also creates a natural connection to space-time discretizations, where higher-dimensional cylindrical or prismatic meshes play a central role.

Related Publications

• A. Adler, X. Hu, and S. Lee. Unified spatiotemporal formulations with physics-preserving structures for time-dependent convection-diffusion problems. (manuscript nearly completed, to be submitted).

• S. Cao, L. Chen, and S. Lee. Edge-averaged virtual element methods for convection-diffusion and convection-dominated problems. Journal of Scientific Computing, 104, 70 (2025).

• S. Lee, Edge-averaged virtual element methods for convection-diffusion problems, PhD thesis, Department of Mathematics, University of California, Irvine, 2021.