On positivity preservation in nonlinear finite volume method for multi-term fractional subdiffusion equation on polygonal meshes

Abstract

The positivity preserving is one of the key requirements to discrete schemes for subdiffusion equation. The main goal of our research is to explore a spatial second-order nonlinear finite volume (FV) method solving the multi-term time-fractional subdiffusion equation with this property maintained. Compared to the already published results, our findings have two important special features. First, we prove positivity preservation property of the equation. Second, we construct a nonlinear FV method for fractional subdiffusion equation on star-shaped polygonal meshes and prove that it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients. Numerical experiments are presented to verify our theoretical findings for both smooth and non-smooth highly anisotropic solutions. Moreover, numerical results show that our scheme has approximate second-order accuracy for the solution and first-order accuracy for the flux on various distorted meshes.