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

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.