A unified spatiotemporal formulation with physics-preserving structure for time-dependent convection-diffusion problems

TL;DR

This paper introduces a novel 4D spatiotemporal formulation for time-dependent convection-diffusion problems that preserves physical structures and ensures well-posedness, unifying various problem types through exterior calculus.

Contribution

It develops a unified 4D framework using exterior calculus and a spatiotemporal diffusion tensor, incorporating physical constraints and a flux operator for stable, structure-preserving solutions.

Findings

Formulation preserves divergence and curl constraints.

Converges to classical models as perturbation vanishes.

Provides a symmetrized, well-posed variational formulation.

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.