A Convergent Geometry-Aware Reduction for Diffusion in Branched Tubular Networks
Diffusion through tubular networks with variable radius arises in a wide range of biological, engineering, and physical applications. The Fick-Jacobs equation is the standard one-dimensional reduction of this problem, briefly derived nearly a century ago in a classical textbook, but was shown to be unstable and inaccurate when the radial gradient is large by Zwanzig in 1992. Three decades of subsequent modifications have failed to resolve this instability because they all inherit a common structural inconsistency introduced by truncation in the original derivation - one that becomes immediately apparent from novel elementary analysis. In this work, we return to the foundations of the Fick-Jacobs derivation and treat it as a locally defined Taylor expansion, recovering a model with geometry-independent error that contrasts directly with the geometry-dependent instability of past corrections. The result is a new geometry-aware expansion of the Fick-Jacobs model, with a numerical discretization that is provably stable and convergent, and the first method known to the authors to converge spatially to the correct geometry-aware solution. Analysis shows that standard corrections from the literature cannot converge to this solution regardless of spatial refinement. We derive efficient numerical schemes for branched networks at equivalent computational cost, and demonstrate that a geometry-aware one-dimensional reduction can faithfully reproduce full three-dimensional results of a neurobiologically relevant problem that the standard reduction cannot achieve.