A unified spatiotemporal formulation with physics-preserving structure for time-dependent convection-diffusion problems
James H. Adler, Xiaozhe Hu, Seulip Lee
TL;DR
The paper develops a unified 4D spatiotemporal formulation for time-dependent convection–diffusion problems using differential forms and exterior calculus, extending the space–time approach to $H(\text{grad})$, $H(\text{curl})$, and $H(\text{div})$. It introduces a spatiotemporal diffusion tensor and convection 1-form with a small perturbation $\varepsilon$ to ensure nondegeneracy, along with an exponentially-fitted flux operator that symmetrizes the convection–diffusion operator and preserves key physical constraints. A physics-preserving variational formulation is constructed and shown to be well-posed via an inf–sup condition and continuity, with a convergence analysis proving that the perturbed model converges to the classical time-dependent problem as $\varepsilon\to0$. The framework inherently enforces divergence-free and curl-free properties and provides a path toward robust 4D discretizations for convection-dominated transport, with potential extensions to Lie convection and 4D monotone solvers.
Abstract
We propose a unified four-dimensional (4D) spatiotemporal formulation for time-dependent convection-diffusion problems that preserves underlying physical structures. By treating time as an additional space-like coordinate, the evolution problem is reformulated as a stationary convection-diffusion equation on a 4D space-time domain. Using exterior calculus, we extend this framework to the full family of convection-diffusion problems posed on $H(\textbf{grad})$, $H(\textbf{curl})$, and $H(\text{div})$. The resulting formulation is based on a 4D Hodge-Laplacian operator with a spatiotemporal diffusion tensor and convection field, augmented by a small temporal perturbation to ensure nondegeneracy. This formulation naturally incorporates fundamental physical constraints, including divergence-free and curl-free conditions. We further introduce an exponentially-fitted 4D spatiotemporal flux operator that symmetrizes the convection-diffusion operator and enables a well-posed variational formulation. Finally, we prove that the temporally-perturbed formulation converges to the original time-dependent convection-diffusion model as the perturbation parameter tends to zero.