2025/09/29 by Maurice S. Fabien, Fabien, Maurice S.
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Engineering #Differential Equations and Numerical Methods #FOS: Computer and information sciences #FOS: Mathematics #Finance #Numerical Analysis (math.NA) #Numerical methods for differential equations #and Science (cs.CE)
paper · pdf · doi:10.48550/arxiv.2509.25181
openalex publication_date 2025/09/29 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
We solve the convection-diffusion equation using a coupling of cell-centered finite volume (FV) and discontinuous Galerkin (DG) methods. The domain is divided into disjoint regions assigned to FV or DG, and the two methods are coupled through an interface term. DG is stable and resolves sharp layers in convection-dominated regimes, but it can produce sizable spurious oscillations and is computationally expensive; FV (two-point flux) is low-order and monotone, but inexpensive. We propose a novel adaptive partitioning strategy that automatically selects FV and DG subdomains: whenever the solution's cell average violates the bounds, we switch to FV on a small neighborhood of that element. Viewed as a natural analog of p-adaptivity, this process is repeated until all cell averages are bound-preserving (up to some specified tolerance). Thereafter, standard conservative limiters may be applied to ensure the full solution is bound-preserving. Standard benchmarks confirm the effectiveness of the adaptive technique.