Numerical solution of mixed gradient-diffusion equations modelling axon growth

In the current paper a numerical approach is presented for solving a system of coupled gradient-diffusion equations which acts as a first model for the growth of axons in brain tissue. The presented approach can be applied to a much wider range of problems, but we focus on the axon growth problem. In our approach time stepping is performed with a Rosenbrock solver with approximate matrix factorization. For the Jacobian an approximation is used that simplifies the solution of the coupled parabolic and gradient equations. A possible complication in the implementation of source terms is noted and a criterion that helps to avoid it is presented.