Modal-Rectification-Based Directional Edge Diffusion for Cartesian Convection--Diffusion Problems

Abstract

Centered finite-difference discretizations of convection--diffusion equations may oscillate when convection dominates at the mesh scale. For homogeneous Dirichlet problems with constant coefficients on uniform Cartesian grids, we derive ADSC (Adaptive Directional Sparse Correction), a local directional edge-diffusion correction guided by modal rectification of the centered-stencil Fourier symbol. The ideal modal reference damps modes independently, but its exact nodal action is nonlocal; ADSC replaces it by a nearest-neighbor positive semidefinite correction. For a regularized operator with activation fixed by an auxiliary sequence, we prove consistency, fixed-epsilon energy stability, and conditional discrete H^1-seminorm convergence. The implemented iteration instead uses activation generated by the computed solution. For that fully coupled nonlinear problem we prove existence and qualitative L^2 compactness/convergence only; uniqueness, convergence of activation updates, and energy-norm rates remain open. Numerical tests show selective extrema control, reduced modal-dominance indicators, and a low-cost few-shot variant. Comparisons with upwinding, SUPG, and AFC-inspired strategies are diagnostic rather than claims of uniform superiority.