Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks

Implicit and semi-implicit time discretizations are developed for the Cai–Hu model describing the formation of biological transport networks. The model couples a nonlinear elliptic equation for the pressure with a nonlinear reaction-diffusion equation for the network conductance vector. Numerical challenges include the non-linearity and the stiffness, thus an explicit discretization puts severe constraints on the time step. We propose an implicit and a semi-implicit discretizations, which decays the energy unconditionally or under a condition independent of the mesh size respectively, as will be proven in 1D and verified numerically in 2D.