Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes (Q6536822)

The paper addresses a significant problem in the numerical analysis of partial differential equations. The authors focus on developing a novel nonlinear finite-volume (NFV) scheme that preserves the discrete maximum principle (DMP) for the two-dimensional sub-diffusion equation on distorted meshes, a problem that is particularly relevant in computational physics and engineering.\N\NSub-diffusion equations are a type of fractional diffusion equation that model various anomalous diffusion processes observed in physical phenomena. These processes are characterised by slower diffusion compared to the classical diffusion governed by the Gaussian distribution. The authors address the specific challenge of ensuring the preservation of the DMP, which is crucial for maintaining the physical bounds of quantities like concentration, temperature, and density in numerical simulations. Previous studies have largely overlooked this aspect, focusing instead on achieving stability and convergence in numerical schemes. The DMP ensures that the numerical solutions do not exhibit spurious oscillations and that they respect the inherent non-negativity of the physical quantities involved.\N\NThe manuscript introduces a new NFV scheme designed to handle distorted meshes, which are common in practical applications where the mesh geometry may not be regular or uniform. The authors' approach involves constructing a conservative flux through a weighted combination of nonconservative fluxes, which is achieved using nonlinear weighted methods. This method is advantageous because it does not impose stringent conditions on the mesh quality, making it applicable to a wide range of mesh configurations, including randomly distorted meshes. The scheme is particularly effective for problems with anisotropic diffusion, where the diffusion properties vary directionally and can be described by a full tensor.\N\NTo establish the theoretical foundation of their method, the authors prove that their scheme satisfies the DMP. This is achieved by deriving the maximum principle for the sub-diffusion problem and demonstrating that the discrete solution of their NFV scheme adheres to this principle. The method is shown to have local conservation properties, relying solely on cell-centered unknowns, which simplifies the computational implementation while ensuring accuracy.\N\NThe significance of this work lies in its ability to preserve the physical integrity of the solutions for sub-diffusion equations on distorted meshes, a feature that is crucial in many scientific and engineering applications. The authors demonstrate the effectiveness of their scheme through numerical experiments that confirm the absence of non-physical oscillations and the preservation of the maximum principle across various mesh types, including both quadrilateral and triangular meshes. The scheme achieves second-order accuracy in space and first-order accuracy in flux computations, further validating its robustness and reliability.\N\NIn conclusion, this manuscript makes a valuable contribution to the field of numerical analysis by addressing a critical gap in the treatment of sub-diffusion equations. The authors' NFV scheme offers a robust and versatile tool for simulating sub-diffusion processes on complex mesh geometries, ensuring that the solutions remain physically meaningful and free from spurious numerical artifacts. This research is expected to have significant implications for the accurate modelling of diffusion processes in various scientific disciplines.