This project aims at students with a strong interest in and a basic background of numerical methods for partial differential equations (PDEs). They should like to develop, implement, and explore novel numerical schemes for the simulation of a recent model of cell migration in fibrous environments, e.g. tissues.

We will consider the model as derived in the recent publication by Zhigun & Surulescu [1]. It consists of two coupled equations. Firstly, there is a parabolic equation, Eq. (3.47) in [1], depending on time and space. Its solution provides the input for the second equation, a transport equation for the cell density approximation depending on time, space, and velocity, see Eq. (3.52) in [1].

We will study the spatially one-dimensional case of the model and solve the parabolic equation using Finite Elements. For the transport equation, we use a spectral expansion of the cell density in the velocity variable and thus transform it to a space-time system of linear hyperbolic PDEs. The latter we will solve numerically using Finite Volume techniques. We will also pay attention to structure preserving aspects of the numerical schemes.

Having the numerical scheme at hand, we aim to conduct a series of computational tests that can shed light on properties of the numerical scheme itself but also on characteristics of the model as well as on questions regarding cell migration in fibrous environments. All these are of current research interest and thus publishable.