On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations

Abstract

The aim of this work is to develop a conservative, positivity-preserving (PP), nonlinear finite volume (FV) scheme for the multi-term nonlocal Nagumo-type equations on distorted meshes. These equations involve anisotropic diffusion tensor coefficients, and the solution exhibits a weak singularity at t=0. To achieve this, we combine the L1 scheme on a graded mesh with a PP-FV scheme on randomly distorted meshes, resulting in a conservative, positivity-preserving nonlinear L1-PP-FV scheme. By employing a nonlinear-splitting technique, we derive a series of simple nonlinear discrete equations that can be solved implicitly. We also provide proof that both the proposed FV scheme and the nonlinear iteration (NI) scheme are capable of maintaining positivity. Finally, we present numerical results to validate our theoretical analysis.