Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes

Abstract

We propose a positivity-preserving finite volume scheme on non-conforming quadrilateral distorted meshes with hanging nodes for subdiffusion equations, where the differential equations have a sum of time-fractional derivatives of different orders, and the typical solutions of the problem have a weak singularity at the initial time \(t=0\) for given smooth data. In this paper, a positivity-preserving nonlinear method with centered unknowns is obtained by the two-point flux technique, where a new method to handling vertex unknown including hanging nodes is the highlight of our paper. For each time derivative, we apply the L1 scheme on a temporal graded mesh. Especially, the existence of a solution is strictly proved for the nonlinear system by applying the Brouwer’s fixed point theorem. Numerical results show that the proposed positivity-preserving method is effective for strongly anisotropic and heterogeneous full tensor subdiffusion coefficient problems.