A bound-preserving multinumerics scheme for steady-state convection-diffusion equations

2025-09-29

AI Summary

For convection-dominated steady-state convection–diffusion equations, standard numerical methods often suffer from spurious oscillations, solution unboundedness, and high computational cost. To address these issues, this paper proposes a multi-method adaptive domain-decomposition strategy. It innovatively couples the cell-centered finite volume method (FVM) with the discontinuous Galerkin (DG) method, employing a *p*-like adaptive partitioning criterion: when the cell-averaged solution violates physical bounds, the local region automatically reverts to FVM discretization. Conservative interface fluxes and slope limiters ensure strong inter-region conservation and global solution boundedness. The resulting hybrid scheme achieves high-resolution accuracy while effectively suppressing non-physical oscillations, robustly resolving boundary and interior layers, and reducing computational overhead compared to pure high-order DG approaches. Comprehensive benchmark tests confirm the method’s unified excellence in accuracy, robustness, and efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Developing a bound-preserving multinumerics scheme for convection-diffusion equations
• Coupling finite volume and discontinuous Galerkin methods adaptively
• Automatically selecting subdomains to eliminate spurious oscillations efficiently

Innovation

• Coupling cell-centered finite volume and discontinuous Galerkin methods
• Novel adaptive partitioning strategy selecting FV/DG subdomains automatically
• Repeated switching to FV until cell averages preserve bounds